Probability

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<a href=/list/math.PR/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> <dl id='articles'> <h3>New submissions (showing 6 of 6 entries)</h3> <dt> <a name='item1'>[1]</a> <a href ="/abs/2502.14246" title="Abstract" id="2502.14246"> arXiv:2502.14246 </a> [<a href="/pdf/2502.14246" title="Download PDF" id="pdf-2502.14246" aria-labelledby="pdf-2502.14246">pdf</a>, <a href="https://arxiv.org/html/2502.14246v1" title="View HTML" id="html-2502.14246" aria-labelledby="html-2502.14246" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.14246" title="Other formats" id="oth-2502.14246" aria-labelledby="oth-2502.14246">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Asymptotics of the occupation measure defined on a nonnegative matrix of two-dimensional quasi-birth-and-death type </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Ozawa,+T">Toshihisa Ozawa</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 23 pages, 4 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span> </div> <p class='mathjax'> We consider a nonnegative matrix having the same block structure as that of the transition probability matrix of a two-dimensional quasi-birth-and-death process (2d-QBD process for short) and define two kinds of measure for the nonnegative matrix. One corresponds to the mean number of visits to each state before the 2d-QBD process starting from the level zero returns to the level zero for the first time. The other corresponds to the probabilities that the 2d-QBD process starting from each state visits the level zero. We call the former the occupation measure and the latter the hitting measure. We obtain asymptotic properties of the occupation measure such as the asymptotic decay rate in an arbitrary direction. Those of the hitting measure can be obtained from the results for the occupation measure by using a kind of duality between the two measures. </p> </div> </dd> <dt> <a name='item2'>[2]</a> <a href ="/abs/2502.14395" title="Abstract" id="2502.14395"> arXiv:2502.14395 </a> [<a href="/pdf/2502.14395" title="Download PDF" id="pdf-2502.14395" aria-labelledby="pdf-2502.14395">pdf</a>, <a href="https://arxiv.org/html/2502.14395v1" title="View HTML" id="html-2502.14395" aria-labelledby="html-2502.14395" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.14395" title="Other formats" id="oth-2502.14395" aria-labelledby="oth-2502.14395">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Central Limit Theorem for Irregular Discretization Scheme of Multilevel Monte Carlo Method </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Guo,+Y">Yi Guo</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Guo,+Y">Yuxi Guo</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wang,+H">Hanchao Wang</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span> </div> <p class='mathjax'> In this paper, we study the asymptotic error distribution for a two-level irregular discretization scheme of the solution to the stochastic differential equations (SDE for short) driven by a continuous semimartingale and obtain a central limit theorem for the error processes with the rate $\sqrt{n}$. As an application, in the spirit of the result of Ben Alaya and Kebaier, we get a central limit theorem of the Linderberg-Feller type for the irregular discretization scheme of the multilevel Monte Carlo method. </p> </div> </dd> <dt> <a name='item3'>[3]</a> <a href ="/abs/2502.14771" title="Abstract" id="2502.14771"> arXiv:2502.14771 </a> [<a href="/pdf/2502.14771" title="Download PDF" id="pdf-2502.14771" aria-labelledby="pdf-2502.14771">pdf</a>, <a href="https://arxiv.org/html/2502.14771v1" title="View HTML" id="html-2502.14771" aria-labelledby="html-2502.14771" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.14771" title="Other formats" id="oth-2502.14771" aria-labelledby="oth-2502.14771">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Flows driven by multi-indices Rough Paths </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Bellingeri,+C">Carlo Bellingeri</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Bruned,+Y">Yvain Bruned</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Hou,+Y">Yingtong Hou</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 29 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Classical Analysis and ODEs (math.CA) </div> <p class='mathjax'> In this work, we introduce a solution theory for scalar-valued rough differential equations driven by multi-indices rough paths. To achieve this task, we will show how the flow approach using the log-ODE method introduced by Bailleul fits perfectly in this setting. In addition, we also describe the action of multi-indices rough paths at the level of rough differential equations. </p> </div> </dd> <dt> <a name='item4'>[4]</a> <a href ="/abs/2502.14774" title="Abstract" id="2502.14774"> arXiv:2502.14774 </a> [<a href="/pdf/2502.14774" title="Download PDF" id="pdf-2502.14774" aria-labelledby="pdf-2502.14774">pdf</a>, <a href="https://arxiv.org/html/2502.14774v1" title="View HTML" id="html-2502.14774" aria-labelledby="html-2502.14774" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.14774" title="Other formats" id="oth-2502.14774" aria-labelledby="oth-2502.14774">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Branching with selection and mutation II: Mutant fitness of Gumbel type </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Park,+S">Su-Chan Park</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Krug,+J">Joachim Krug</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=M%C3%B6rters,+P">Peter M枚rters</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 44 pages, 2 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Statistical Mechanics (cond-mat.stat-mech); Biological Physics (physics.bio-ph) </div> <p class='mathjax'> We study a model of a branching process subject to selection, modeled by giving each family an individual fitness acting as a branching rate, and mutation, modeled by resampling the fitness of a proportion of offspring in each generation. For two large classes of fitness distributions of Gumbel type we determine the growth of the population, almost surely on survival. We then study the empirical fitness distribution in a simplified model, which is numerically indistinguishable from the original model, and show the emergence of a Gaussian travelling wave. </p> </div> </dd> <dt> <a name='item5'>[5]</a> <a href ="/abs/2502.14839" title="Abstract" id="2502.14839"> arXiv:2502.14839 </a> [<a href="/pdf/2502.14839" title="Download PDF" id="pdf-2502.14839" aria-labelledby="pdf-2502.14839">pdf</a>, <a href="https://arxiv.org/html/2502.14839v1" title="View HTML" id="html-2502.14839" aria-labelledby="html-2502.14839" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.14839" title="Other formats" id="oth-2502.14839" aria-labelledby="oth-2502.14839">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The law of thin processes: a law of large numbers for point processes </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Aldridge,+M">Matthew Aldridge</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 6 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span> </div> <p class='mathjax'> If you take a superposition of n IID copies of a point process and thin that by a factor of 1/n, then the resulting process tends to a Poisson point process as n tends to infinity. We give a simple proof of this result that highlights its similarity to the law of large numbers and to the law of thin numbers of Harremo毛s et al. </p> </div> </dd> <dt> <a name='item6'>[6]</a> <a href ="/abs/2502.14863" title="Abstract" id="2502.14863"> arXiv:2502.14863 </a> [<a href="/pdf/2502.14863" title="Download PDF" id="pdf-2502.14863" aria-labelledby="pdf-2502.14863">pdf</a>, <a href="https://arxiv.org/html/2502.14863v1" title="View HTML" id="html-2502.14863" aria-labelledby="html-2502.14863" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.14863" title="Other formats" id="oth-2502.14863" aria-labelledby="oth-2502.14863">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Najnudel,+J">Joseph Najnudel</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Paquette,+E">Elliot Paquette</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Simm,+N">Nick Simm</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Vu,+T">Truong Vu</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span> </div> <p class='mathjax'> The holomorphic multiplicative chaos (HMC) is a holomorphic analogue of the Gaussian multiplicative chaos. It arises naturally as the limit in large matrix size of the characteristic polynomial of Haar unitary, and more generally circular-$\beta$-ensemble, random matrices. <br>We consider the Fourier coefficients of the holomorphic multiplicative chaos in the $L^1$-phase, and we show that appropriately normalized, this converges in distribution to a complex normal random variable, scaled by the total mass of the Gaussian multiplicative chaos measure on the unit circle. We further generalize this to a process convergence, showing the joint convergence of consecutive Fourier coefficients. As a corollary, we derive convergence in law of the secular coefficients of sublinear index of the circular-$\beta$-ensemble for all $\beta > 2$. </p> </div> </dd> </dl> <dl id='articles'> <h3>Cross submissions (showing 3 of 3 entries)</h3> <dt> <a name='item7'>[7]</a> <a href ="/abs/2502.13978" title="Abstract" id="2502.13978"> arXiv:2502.13978 </a> (cross-list from physics.data-an) [<a href="/pdf/2502.13978" title="Download PDF" id="pdf-2502.13978" aria-labelledby="pdf-2502.13978">pdf</a>, <a href="https://arxiv.org/html/2502.13978v1" title="View HTML" id="html-2502.13978" aria-labelledby="html-2502.13978" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.13978" title="Other formats" id="oth-2502.13978" aria-labelledby="oth-2502.13978">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the perceptions of empirical randomness of an experiment: Extending the Golden Theorem </div> <div class='list-authors'><a href="https://arxiv.org/search/physics?searchtype=author&query=Lobo,+A">Allen Lobo</a>, <a href="https://arxiv.org/search/physics?searchtype=author&query=Arumugam,+S">Saravanan Arumugam</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Data Analysis, Statistics and Probability (physics.data-an)</span>; Probability (math.PR) </div> <p class='mathjax'> In this work, Bernoulli's Law of Large Numbers, also known as the Golden theorem, has been extended to study the relations between empirical probability and empirical randomness of an otherwise random experiment. Using the example of a coin toss and a dice role, some interesting results are drawn. Analytically and using numerical computations, empirical randomness of each outcome has been shown to increase by \textit{ chance}, which itself depends on the growth rate of empirical probabilities. The analyses presented in this work, apart form depicting the nature of flow of random experiments in repetitions, also present dynamical behaviours of the random experiment, and experimental and simulation-based verifications of the mathematical analyses. It also presents an appreciation of the beauty of Bernoulli's Golden theorem and its applications by extension. </p> </div> </dd> <dt> <a name='item8'>[8]</a> <a href ="/abs/2502.14069" title="Abstract" id="2502.14069"> arXiv:2502.14069 </a> (cross-list from math.ST) [<a href="/pdf/2502.14069" title="Download PDF" id="pdf-2502.14069" aria-labelledby="pdf-2502.14069">pdf</a>, <a href="https://arxiv.org/html/2502.14069v1" title="View HTML" id="html-2502.14069" aria-labelledby="html-2502.14069" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.14069" title="Other formats" id="oth-2502.14069" aria-labelledby="oth-2502.14069">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Finite sample bounds for barycenter estimation in geodesic spaces </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Brunel,+V">Victor-Emmanuel Brunel</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Serres,+J">Jordan Serres</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Statistics Theory (math.ST)</span>; Probability (math.PR); Machine Learning (stat.ML) </div> <p class='mathjax'> We study the problem of estimating the barycenter of a distribution given i.i.d. data in a geodesic space. Assuming an upper curvature bound in Alexandrov's sense and a support condition ensuring the strong geodesic convexity of the barycenter problem, we establish finite-sample error bounds in expectation and with high probability. Our results generalize Hoeffding- and Bernstein-type concentration inequalities from Euclidean to geodesic spaces. </p> </div> </dd> <dt> <a name='item9'>[9]</a> <a href ="/abs/2502.14407" title="Abstract" id="2502.14407"> arXiv:2502.14407 </a> (cross-list from math.ST) [<a href="/pdf/2502.14407" title="Download PDF" id="pdf-2502.14407" aria-labelledby="pdf-2502.14407">pdf</a>, <a href="https://arxiv.org/html/2502.14407v1" title="View HTML" id="html-2502.14407" aria-labelledby="html-2502.14407" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.14407" title="Other formats" id="oth-2502.14407" aria-labelledby="oth-2502.14407">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Sharp Phase Transitions in Estimation with Low-Degree Polynomials </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Sohn,+Y">Youngtak Sohn</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Wein,+A+S">Alexander S. Wein</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 64 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Statistics Theory (math.ST)</span>; Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS); Probability (math.PR) </div> <p class='mathjax'> High-dimensional planted problems, such as finding a hidden dense subgraph within a random graph, often exhibit a gap between statistical and computational feasibility. While recovering the hidden structure may be statistically possible, it is conjectured to be computationally intractable in certain parameter regimes. A powerful approach to understanding this hardness involves proving lower bounds on the efficacy of low-degree polynomial algorithms. We introduce new techniques for establishing such lower bounds, leading to novel results across diverse settings: planted submatrix, planted dense subgraph, the spiked Wigner model, and the stochastic block model. Notably, our results address the estimation task -- whereas most prior work is limited to hypothesis testing -- and capture sharp phase transitions such as the "BBP" transition in the spiked Wigner model (named for Baik, Ben Arous, and P茅ch茅) and the Kesten-Stigum threshold in the stochastic block model. Existing work on estimation either falls short of achieving these sharp thresholds or is limited to polynomials of very low (constant or logarithmic) degree. In contrast, our results rule out estimation with polynomials of degree $n^{\delta}$ where $n$ is the dimension and $\delta > 0$ is a constant, and in some cases we pin down the optimal constant $\delta$. Our work resolves open problems posed by Hopkins & Steurer (2017) and Schramm & Wein (2022), and provides rigorous support within the low-degree framework for conjectures by Abbe & Sandon (2018) and Lelarge & Miolane (2019). </p> </div> </dd> </dl> <dl id='articles'> <h3>Replacement submissions (showing 24 of 24 entries)</h3> <dt> <a name='item10'>[10]</a> <a href ="/abs/1904.00578" title="Abstract" id="1904.00578"> arXiv:1904.00578 </a> (replaced) [<a href="/pdf/1904.00578" title="Download PDF" id="pdf-1904.00578" aria-labelledby="pdf-1904.00578">pdf</a>, <a href="https://arxiv.org/html/1904.00578v3" title="View HTML" id="html-1904.00578" aria-labelledby="html-1904.00578" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/1904.00578" title="Other formats" id="oth-1904.00578" aria-labelledby="oth-1904.00578">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> On the circle, Gaussian Multiplicative Chaos and Beta Ensembles match exactly </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Chhaibi,+R">Reda Chhaibi</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Najnudel,+J">Joseph Najnudel</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 65 pages, no figures. v2: Added comments on the supercritical phase, and more bibliographic references. v3: Published version at JEMS </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Mathematical Physics (math-ph); Spectral Theory (math.SP) </div> <p class='mathjax'> We identify an equality between two objects arising from different contexts of mathematical physics: Kahane's Gaussian Multiplicative Chaos ($GMC^\gamma$) on the circle, and the Circular Beta Ensemble $(C\beta E)$ from Random Matrix Theory. This is obtained via an analysis of related random orthogonal polynomials, making the approach spectral in nature. In order for the equality to hold, the simple relationship between coupling constants is $\gamma = \sqrt{\frac{2}{\beta}}$, which we establish only when $\gamma \leq 1$ or equivalently $\beta \geq 2$. This corresponds to the sub-critical and critical phases of the $GMC$. <br>As a side product, we answer positively a question raised by Virag. We also give an alternative proof of the Fyodorov-Bouchaud formula concerning the total mass of the $GMC^\gamma$ on the circle. This conjecture was recently settled by R茅my using Liouville conformal field theory. We can go even further and describe the law of all moments. <br>Furthermore, we notice that the ``spectral construction'' has a few advantages. For example, the Hausdorff dimension of the support is efficiently described for all $\beta>0$, thanks to existing spectral theory. Remarkably, the critical parameter for $GMC^\gamma$ corresponds to $\beta=2$, where the geometry and representation theory of unitary groups lie. </p> </div> </dd> <dt> <a name='item11'>[11]</a> <a href ="/abs/2211.15609" title="Abstract" id="2211.15609"> arXiv:2211.15609 </a> (replaced) [<a href="/pdf/2211.15609" title="Download PDF" id="pdf-2211.15609" aria-labelledby="pdf-2211.15609">pdf</a>, <a href="https://arxiv.org/html/2211.15609v4" title="View HTML" id="html-2211.15609" aria-labelledby="html-2211.15609" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2211.15609" title="Other formats" id="oth-2211.15609" aria-labelledby="oth-2211.15609">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Holden,+N">Nina Holden</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Yuan,+Y">Yizheng Yuan</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> paper is extended in v3 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Complex Variables (math.CV) </div> <p class='mathjax'> We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the SLE with its natural parametrisation. More precisely, denoting by $d\in(0,2]$ the dimension of the curve, we show the following. <br>1. The optimal $\psi$-variation is $\psi(x)=x^d(\log\log x^{-1})^{-(d-1)}$ in the sense that $\eta$ is a.s. of finite $\psi$-variation for this $\psi$ and not for any function decaying more slowly as $x \downarrow 0$. <br>2. The optimal modulus of continuity is $\omega(s) = c\,s^{1/d}(\log s^{-1})^{1-1/d}$, i.e. for some random $c>0$ we have $|\eta(t)-\eta(s)| \le \omega(t-s)$ a.s., while this does not hold for any function $\omega$ decaying faster as $s \downarrow 0$. <br>3. $\limsup_{t\downarrow 0} |\eta(t)|\,\big(t^{1/d}(\log\log t^{-1})^{1-1/d}\big)^{-1}$ is a.s. equal to a deterministic constant in $(0,\infty)$. <br>We also show that the natural parametrisation of SLE is given by the fine mesh limit of the $\psi$-variation. As part of our proof, we show that every stochastic process whose increments satisfy a particular moment condition attains a certain variation regularity. </p> </div> </dd> <dt> <a name='item12'>[12]</a> <a href ="/abs/2307.14931" title="Abstract" id="2307.14931"> arXiv:2307.14931 </a> (replaced) [<a href="/pdf/2307.14931" title="Download PDF" id="pdf-2307.14931" aria-labelledby="pdf-2307.14931">pdf</a>, <a href="https://arxiv.org/html/2307.14931v2" title="View HTML" id="html-2307.14931" aria-labelledby="html-2307.14931" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2307.14931" title="Other formats" id="oth-2307.14931" aria-labelledby="oth-2307.14931">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> How long are the arms in DBM? </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Losev,+I">Ilya Losev</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Smirnov,+S">Stanislav Smirnov</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 19 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Mathematical Physics (math-ph) </div> <p class='mathjax'> Diffusion Limited Aggregation and its generalization, Dielectric Breakdown model play an important role in physics, approximating a range of natural phenomena. Yet little is known about them, with the famous Kesten's estimate on the DLAs growth being perhaps the most important result. Using a different approach we prove a generalisation of this result for the DBM in $\mathbb{Z}^2$ and $\mathbb{Z}^3$. The obtained estimate depends on the DBM parameter, and matches with the best known results for DLA. In particular, since our methods are different from Kesten's, our argument provides a new proof for Kesten's result both in $\mathbb{Z}^2$ and $\mathbb{Z}^3$. </p> </div> </dd> <dt> <a name='item13'>[13]</a> <a href ="/abs/2311.04041" title="Abstract" id="2311.04041"> arXiv:2311.04041 </a> (replaced) [<a href="/pdf/2311.04041" title="Download PDF" id="pdf-2311.04041" aria-labelledby="pdf-2311.04041">pdf</a>, <a href="https://arxiv.org/html/2311.04041v3" title="View HTML" id="html-2311.04041" aria-labelledby="html-2311.04041" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2311.04041" title="Other formats" id="oth-2311.04041" aria-labelledby="oth-2311.04041">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Eckstein,+S">Stephan Eckstein</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> This version: Added discussion of main results in introduction, simplified notation and fixed many typos. Accepted for publication in PTRF </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Optimization and Control (math.OC); Machine Learning (stat.ML) </div> <p class='mathjax'> Motivated by the entropic optimal transport problem in unbounded settings, we study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These versions of Hilbert's metric originate from cones which are relaxations of the cone of all non-negative functions, in the sense that they include all functions having non-negative integral values when multiplied with certain test functions. We show that kernel integral operators are contractions with respect to suitable specifications of such metrics even for kernels which are not bounded away from zero, provided that the decay to zero of the kernel is controlled. As an application to entropic optimal transport, we show exponential convergence of Sinkhorn's algorithm in settings where the marginal distributions have sufficiently light tails compared to the growth of the cost function. </p> </div> </dd> <dt> <a name='item14'>[14]</a> <a href ="/abs/2312.16008" title="Abstract" id="2312.16008"> arXiv:2312.16008 </a> (replaced) [<a href="/pdf/2312.16008" title="Download PDF" id="pdf-2312.16008" aria-labelledby="pdf-2312.16008">pdf</a>, <a href="/format/2312.16008" title="Other formats" id="oth-2312.16008" aria-labelledby="oth-2312.16008">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Potts and random cluster measures on locally regular-tree-like graphs </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Basak,+A">Anirban Basak</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Dembo,+A">Amir Dembo</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Sly,+A">Allan Sly</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 48 pages, minor changes in v2 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph) </div> <p class='mathjax'> Fixing $\beta \ge 0$ and an integer $q \ge 2$, consider the ferromagnetic $q$-Potts measures $\mu_n^{\beta,B}$ on finite graphs ${\sf G}_n$ on $n$ vertices, with external field strength $B \ge 0$ and the corresponding random cluster measures $\varphi^{q,\beta,B}_{n}$. Suppose that as $n \to \infty$ the uniformly sparse graphs ${\sf G}_n$ converge locally to an infinite $d$-regular tree ${\sf T}_{d}$, $d \ge 3$. We show that the convergence of the Potts free energy density to its Bethe replica symmetric prediction (which has been proved in case $d$ is even, or when $B=0$), yields the local weak convergence of $\varphi^{q,\beta,B}_n$ and $\mu_n^{\beta,B}$ to the corresponding free or wired random cluster measure, Potts measure, respectively, on ${\sf T}_{d}$. The choice of free versus wired limit is according to which has the larger Potts Bethe functional value, with mixtures of these two appearing {as limit points on} the critical line $\beta_c(q,B)$ where these two values of the Bethe functional coincide. For $B=0$ and $\beta>\beta_c$, we further establish a pure-state decomposition by showing that conditionally on the same dominant color $1 \le k \le q$, the $q$-Potts measures on such edge-expander graphs ${\sf G}_n$ converge locally to the $q$-Potts measure on ${\sf T}_{d}$ with a boundary wired at color $k$. </p> </div> </dd> <dt> <a name='item15'>[15]</a> <a href ="/abs/2403.12941" title="Abstract" id="2403.12941"> arXiv:2403.12941 </a> (replaced) [<a href="/pdf/2403.12941" title="Download PDF" id="pdf-2403.12941" aria-labelledby="pdf-2403.12941">pdf</a>, <a href="https://arxiv.org/html/2403.12941v4" title="View HTML" id="html-2403.12941" aria-labelledby="html-2403.12941" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2403.12941" title="Other formats" id="oth-2403.12941" aria-labelledby="oth-2403.12941">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Sina沫 excursions </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Donderwinkel,+S">Serte Donderwinkel</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Kolesnik,+B">Brett Kolesnik</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> v4: many small improvements; added Corollary 5.1 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Combinatorics (math.CO) </div> <p class='mathjax'> Sina沫 initiated the study of random walks with persistently positive area processes, motivated by shock waves in solutions to the inviscid Burgers' equation. We find the precise asymptotic probability that the area process of a random walk bridge is an excursion. A key ingredient is an analogue of Sparre Andersen's classical formula. The asymptotics are related to von Sterneck's subset counting formulas from additive number theory. Our results sharpen bounds by Aurzada, Dereich and Lifshits and respond to a question of Caravenna and Deuschel, which arose in their study of the wetting model. In this context, Sina\u谋 excursions are a class of random polymer chains exhibiting entropic repulsion. </p> </div> </dd> <dt> <a name='item16'>[16]</a> <a href ="/abs/2407.01841" title="Abstract" id="2407.01841"> arXiv:2407.01841 </a> (replaced) [<a href="/pdf/2407.01841" title="Download PDF" id="pdf-2407.01841" aria-labelledby="pdf-2407.01841">pdf</a>, <a href="https://arxiv.org/html/2407.01841v2" title="View HTML" id="html-2407.01841" aria-labelledby="html-2407.01841" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2407.01841" title="Other formats" id="oth-2407.01841" aria-labelledby="oth-2407.01841">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> An infinite server system with packing constraints and ranked servers </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Stolyar,+A">Alexander Stolyar</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> Revision. 18 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Networking and Internet Architecture (cs.NI) </div> <p class='mathjax'> A service system with multiple types of customers, arriving as Poisson processes, is considered. The system has infinite number of servers, ranked by $1,2,3, \ldots$; a server rank is its ``location." Each customer has an independent exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to ``packing'' constraints. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered, such that the mean number of customers $r$ goes to infinity. <br>We seek algorithms with the underlying objective of minimizing the location (rank) $U$ of the right-most (highest ranked) occupied (non-empty) server. Therefore, this objective seeks to minimize the total number $Q$ of occupied servers {\em and} keep the set of occupied servers as far at the ``left'' as possible, i.e., keep $U$ close to $Q$. In previous work, versions of {\em Greedy Random} (GRAND) algorithm have been shown to asymptotically minimize $Q/r$ as $r\to\infty$. In this paper we show that when these algorithms are combined with the First-Fit rule for ``taking'' empty servers, they asymptotically minimize $U/r$ as well. </p> </div> </dd> <dt> <a name='item17'>[17]</a> <a href ="/abs/2407.02289" title="Abstract" id="2407.02289"> arXiv:2407.02289 </a> (replaced) [<a href="/pdf/2407.02289" title="Download PDF" id="pdf-2407.02289" aria-labelledby="pdf-2407.02289">pdf</a>, <a href="https://arxiv.org/html/2407.02289v4" title="View HTML" id="html-2407.02289" aria-labelledby="html-2407.02289" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2407.02289" title="Other formats" id="oth-2407.02289" aria-labelledby="oth-2407.02289">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Some properties of a non-hydrostatic stochastic oceanic primitive equations model </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Debussche,+A">Arnaud Debussche</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=M%C3%A9min,+%C3%89">脡tienne M茅min</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Moneyron,+A">Antoine Moneyron</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Mathematical Physics (math-ph) </div> <p class='mathjax'> In this paper, we study how relaxing the classical hydrostatic balance hypothesis affects theoretical aspects of the LU primitive equations well-posedness. We focus on models that sit between incompressible 3D LU Navier-Stokes equations and standard LU primitive equations, aiming for numerical manageability while capturing non-hydrostatic phenomena. Our main result concerns the well-posedness of a specific stochastic interpretation of the LU primitive equations. This holds with rigid-lid type boundary conditions, and when the horizontal component of noise is independent of z. In fact these conditions can be related to the dynamical regime in which the primitive equations remain valid. Moreover, under these conditions, we show that the LU primitive equations solution tends toward the one of the deterministic primitive equations for a vanishing noise, thus providing a physical coherence to the LU stochastic model. </p> </div> </dd> <dt> <a name='item18'>[18]</a> <a href ="/abs/2409.15787" title="Abstract" id="2409.15787"> arXiv:2409.15787 </a> (replaced) [<a href="/pdf/2409.15787" title="Download PDF" id="pdf-2409.15787" aria-labelledby="pdf-2409.15787">pdf</a>, <a href="/format/2409.15787" title="Other formats" id="oth-2409.15787" aria-labelledby="oth-2409.15787">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Rates of convergence in the central limit theorem for Banach valued dependent variables </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Bigot,+A">Aur茅lie Bigot</a> (LAMA)</div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span> </div> <p class='mathjax'> We provide rates of convergence in the central limit theorem in terms of projective criteria for adapted stationary sequences of centered random variables taking values in Banach spaces, with finite moment of order $p \in ]2,3]$ as soon as the central limit theorem holds for the partial sum normalized by $n^{-1/2}$. This result applies to the empirical distribution function in $L^p(\mu)$, where $p\geq 2$ and $\mu$ is a real $\sigma$-finite measure: under some $\tau$-mixing conditions we obtain a rate of order $O(n^{-(p-2)/2})$. In the real case, our result leads to new conditions to reach the optimal rates of convergence in terms of Wasserstein distances of order $p\in ]2,3]$. </p> </div> </dd> <dt> <a name='item19'>[19]</a> <a href ="/abs/2410.11836" title="Abstract" id="2410.11836"> arXiv:2410.11836 </a> (replaced) [<a href="/pdf/2410.11836" title="Download PDF" id="pdf-2410.11836" aria-labelledby="pdf-2410.11836">pdf</a>, <a href="https://arxiv.org/html/2410.11836v2" title="View HTML" id="html-2410.11836" aria-labelledby="html-2410.11836" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2410.11836" title="Other formats" id="oth-2410.11836" aria-labelledby="oth-2410.11836">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The Paquette-Zeitouni law of fractional logarithms for the GUE minor process and the Plancherel growth process </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Baslingker,+J">Jnaneshwar Baslingker</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Basu,+R">Riddhipratim Basu</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Bhattacharjee,+S">Sudeshna Bhattacharjee</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Krishnapur,+M">Manjunath Krishnapur</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 24 pages, 4 figures. Law of fractional logarithm for Plancherel growth process is added </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span> </div> <p class='mathjax'> It is well-known that the largest eigenvalue of an $n\times n$ GUE matrix and the length of a longest increasing subsequence in a uniform random permutation of length $n$, both converge weakly to the GUE Tracy-Widom distribution as $n\to \infty$. We consider the sequences of the largest eigenvalues of the $n\times n$ principal minor of an infinite GUE matrix, and the the lengths of longest increasing subsequences of a growing sequence of random permutations (which, by the RSK bijection corresponds to the top row of the Young diagrams growing according to the Plancherel growth process), and establish laws of fractional logarithms for these. That is, we show that, under a further scaling of $(\log n)^{2/3}$ and $(\log n)^{1/3}$, the $\limsup$ and $\liminf$ respectively of these scaled quantities converge almost surely to explicit non-zero and finite constants. Our results provide complete solutions to two questions raised by Kalai in 2013. We affirm a conjecture of Paquette and Zeitouni (Ann. Probab., 2017), and give a new proof of $\limsup$, due to Paquette and Zeitouni (Ann. Probab., 2017), who provided a partial solution in the case of GUE minor process. </p> </div> </dd> <dt> <a name='item20'>[20]</a> <a href ="/abs/2411.07878" title="Abstract" id="2411.07878"> arXiv:2411.07878 </a> (replaced) [<a href="/pdf/2411.07878" title="Download PDF" id="pdf-2411.07878" aria-labelledby="pdf-2411.07878">pdf</a>, <a href="https://arxiv.org/html/2411.07878v2" title="View HTML" id="html-2411.07878" aria-labelledby="html-2411.07878" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2411.07878" title="Other formats" id="oth-2411.07878" aria-labelledby="oth-2411.07878">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Bernstein-type and Bennett-type inequalities for unbounded matrix martingales </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Kroshnin,+A">Alexey Kroshnin</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Suvorikova,+A">Alexandra Suvorikova</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 32 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Statistics Theory (math.ST) </div> <p class='mathjax'> We derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued martingale processes with unbounded observations from the Hermitian space $\mathbb{H}(d)$. Specifically, we assume that the $\psi_{\alpha}$-Orlicz (quasi-)norms of their difference process are bounded for some $\alpha > 0$. Further, we generalize the obtained result by replacing the ambient dimension $d$ with the effective rank of the covariance of the observations. To illustrate the applicability of the results, we prove several corollaries, including an empirical version of Bernstein's inequality and an extension of the bounded difference inequality, also known as McDiarmid's inequality. </p> </div> </dd> <dt> <a name='item21'>[21]</a> <a href ="/abs/2412.16952" title="Abstract" id="2412.16952"> arXiv:2412.16952 </a> (replaced) [<a href="/pdf/2412.16952" title="Download PDF" id="pdf-2412.16952" aria-labelledby="pdf-2412.16952">pdf</a>, <a href="https://arxiv.org/html/2412.16952v2" title="View HTML" id="html-2412.16952" aria-labelledby="html-2412.16952" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2412.16952" title="Other formats" id="oth-2412.16952" aria-labelledby="oth-2412.16952">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Mixing Phases and Metastability for the Glauber Dynamics on the p-Spin Curie-Weiss Model </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Samanta,+R+J">Ramkrishna Jyoti Samanta</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Mukherjee,+S">Somabha Mukherjee</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Zhang,+J">Jiang Zhang</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Statistics Theory (math.ST) </div> <p class='mathjax'> The Glauber dynamics for the classical $2$-spin Curie-Weiss model on $N$ nodes with inverse temperature $\beta$ and zero external field is known to mix in time $\Theta(N\log N)$ for $\beta < \frac{1}{2}$, in time $\Theta(N^{3/2})$ at $\beta = \frac{1}{2}$, and in time $\exp(\Omega(N))$ for $\beta >\frac{1}{2}$. In this paper, we consider the $p$-spin generalization of the Curie-Weiss model with an external field $h$, and identify three disjoint regions almost exhausting the parameter space, with the corresponding Glauber dynamics exhibiting three different orders of mixing times in these regions. The construction of these disjoint regions depends on the number of local maximizers of a certain function $H_{\beta,h,p}$, and the behavior of the second derivative of $H_{\beta,h,p}$ at such a local maximizer. Specifically, we show that if $H_{\beta,h,p}$ has a unique local maximizer $m_*$ with $H_{\beta,h,p}''(m_*) < 0$ and no other stationary point, then the Glauber dynamics mixes in time $\Theta(N\log N)$, and if $H_{\beta,h,p}$ has multiple local maximizers, then the mixing time is $\exp(\Omega(N))$. Finally, if $H_{\beta,h,p}$ has a unique local maximizer $m_*$ with $H_{\beta,h,p}''(m_*) = 0$, then the mixing time is $\Theta(N^{3/2})$. We provide an explicit description of the geometry of these three different phases in the parameter space, and observe that the only portion of the parameter plane that is left out by the union of these three regions, is a one-dimensional curve, on which the function $H_{\beta,h,p}$ has a stationary inflection point. Finding out the exact order of the mixing time on this curve remains an open question. Finally, we show that if $H_{\beta,h,p}$ has multiple local maximizers (metastable states), then one can create a restricted version of the original Glauber dynamics, which still mixes in time $\Theta(N\log N)$. </p> </div> </dd> <dt> <a name='item22'>[22]</a> <a href ="/abs/2502.05890" title="Abstract" id="2502.05890"> arXiv:2502.05890 </a> (replaced) [<a href="/pdf/2502.05890" title="Download PDF" id="pdf-2502.05890" aria-labelledby="pdf-2502.05890">pdf</a>, <a href="https://arxiv.org/html/2502.05890v4" title="View HTML" id="html-2502.05890" aria-labelledby="html-2502.05890" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.05890" title="Other formats" id="oth-2502.05890" aria-labelledby="oth-2502.05890">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Uniqueness of generalized conformal restriction measures and Malliavin-Kontsevich-Suhov measures for $c \in (0,1]$ </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Cai,+G">Gefei Cai</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Gao,+Y">Yifan Gao</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 21 pages. Minor changes on the last version </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span> </div> <p class='mathjax'> In this paper, we present a unified approach to establish the uniqueness of generalized conformal restriction measures with central charge $c \in (0, 1]$ in both chordal and radial cases, by relating these measures to the Brownian loop soup. Our method also applies to the uniqueness of the Malliavin-Kontsevich-Suhov loop measures for $c \in (0,1]$, which was recently obtained in [Baverez-Jego, <a href="https://arxiv.org/abs/2407.09080" data-arxiv-id="2407.09080" class="link-https">arXiv:2407.09080</a>] for all $c \leq 1$ from a CFT framework of SLE loop measures. In contrast, though only valid for $c \in (0,1]$, our approach provides additional probabilistic insights, as it directly links natural quantities of MKS measures to loop-soup observables. </p> </div> </dd> <dt> <a name='item23'>[23]</a> <a href ="/abs/2502.06754" title="Abstract" id="2502.06754"> arXiv:2502.06754 </a> (replaced) [<a href="/pdf/2502.06754" title="Download PDF" id="pdf-2502.06754" aria-labelledby="pdf-2502.06754">pdf</a>, <a href="https://arxiv.org/html/2502.06754v3" title="View HTML" id="html-2502.06754" aria-labelledby="html-2502.06754" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.06754" title="Other formats" id="oth-2502.06754" aria-labelledby="oth-2502.06754">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> A switching identity for cable-graph loop soups and Gaussian free fields </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Werner,+W">Wendelin Werner</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Mathematical Physics (math-ph) </div> <p class='mathjax'> We derive a "switching identity" that can be stated for critical Brownian loop-soups or for the Gaussian free field on a cable graph: It basically says that at the level of cluster configurations and at the more general level of the occupation time fields, conditioning two points on the cable-graph to belong to the same cluster of Brownian loops (or equivalently to the same sign-cluster of the GFF) amounts to adding a random odd number of independent Brownian excursions between these points to an otherwise unconditioned configuration. This explicit simple description of the conditional law of the clusters when a connection occurs has various direct consequences, in particular about the large scale behaviour of these sign-clusters on infinite graphs. </p> </div> </dd> <dt> <a name='item24'>[24]</a> <a href ="/abs/2502.08799" title="Abstract" id="2502.08799"> arXiv:2502.08799 </a> (replaced) [<a href="/pdf/2502.08799" title="Download PDF" id="pdf-2502.08799" aria-labelledby="pdf-2502.08799">pdf</a>, <a href="/format/2502.08799" title="Other formats" id="oth-2502.08799" aria-labelledby="oth-2502.08799">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Strong completeness of SDEs and non-explosion for RDEs with coefficients having unbounded derivatives </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Li,+X">Xue-Mei Li</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Ying,+K">Kexing Ying</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 31 pages; Added figure for Lemma 4.6 </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Probability (math.PR)</span>; Classical Analysis and ODEs (math.CA) </div> <p class='mathjax'> We establish a non-explosion result for rough differential equations (RDEs) in which both the noise and drift coefficients together with their derivatives are allowed to grow at infinity. Additionally, we prove the existence of a bi-continuous solution flow for stochastic differential equations (SDEs). In the case of RDEs with additive noise, we show that our result is optimal by providing a counterexample. </p> </div> </dd> <dt> <a name='item25'>[25]</a> <a href ="/abs/1212.3817" title="Abstract" id="1212.3817"> arXiv:1212.3817 </a> (replaced) [<a href="/pdf/1212.3817" title="Download PDF" id="pdf-1212.3817" aria-labelledby="pdf-1212.3817">pdf</a>, <a href="/format/1212.3817" title="Other formats" id="oth-1212.3817" aria-labelledby="oth-1212.3817">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Probability Bracket Notation: Markov Sequence Projector of Visible and Hidden Markov Models in Dynamic Bayesian Networks </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Wang,+X+M">Xing M. Wang</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 32 pages. Added Sec. 5.4, Sec. 6, keywords, abbreviations, and several references; edited text </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Artificial Intelligence (cs.AI)</span>; Probability (math.PR) </div> <p class='mathjax'> With the symbolic framework of Probability Bracket Notation (PBN), the Markov Sequence Projector (MSP) is introduced to expand the evolution formula of Homogeneous Markov Chains (HMCs). The well-known weather example, a Visible Markov Model (VMM), illustrates that the full joint probability of a VMM corresponds to a specifically projected Markov state sequence in the expanded evolution formula. In a Hidden Markov Model (HMM), the probability basis (P-basis) of the hidden Markov state sequence and the P-basis of the observation sequence exist in the sequential event space. The full joint probability of an HMM is the product of the (unknown) projected hidden sequence of Markov states and their transformations into the observation P-bases. The Viterbi algorithm is applied to the famous Weather-Stone HMM example to determine the most likely weather-state sequence given the observed stone-state sequence. Our results are verified using the Elvira software package. Using the PBN, we unify the evolution formulas for Markov models like VMMs, HMMs, and factorial HMMs (with discrete time). We briefly investigated the extended HMM, addressing the feedback issue, and the continuous-time VMM and HMM (with discrete or continuous states). All these models are subclasses of Dynamic Bayesian Networks (DBNs) essential for Machine Learning (ML) and Artificial Intelligence (AI). </p> </div> </dd> <dt> <a name='item26'>[26]</a> <a href ="/abs/2304.06060" title="Abstract" id="2304.06060"> arXiv:2304.06060 </a> (replaced) [<a href="/pdf/2304.06060" title="Download PDF" id="pdf-2304.06060" aria-labelledby="pdf-2304.06060">pdf</a>, <a href="https://arxiv.org/html/2304.06060v3" title="View HTML" id="html-2304.06060" aria-labelledby="html-2304.06060" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2304.06060" title="Other formats" id="oth-2304.06060" aria-labelledby="oth-2304.06060">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis </div> <div class='list-authors'><a href="https://arxiv.org/search/q-fin?searchtype=author&query=Nzokem,+A">A.H. Nzokem</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 15 page </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Pricing of Securities (q-fin.PR)</span>; Probability (math.PR) </div> <p class='mathjax'> The paper investigates the performance of the European option price when the log asset price follows a rich class of Generalized Tempered Stable (GTS) distribution. The GTS distribution is an alternative to Normal distribution and $\alpha$-stable distribution for modeling asset return and many physical and economic systems. The data used in the option pricing computation comes from fitting the GTS distribution to the underlying S\&P 500 Index return distribution. The Esscher transform method shows that the GTS distribution preserves its structure. The extended Black-Scholes formula and the Generalized Black-Scholes Formula are applied in the study. The 12-point rule Composite Newton-Cotes Quadrature and the Fractional Fast Fourier (FRFT) algorithms were implemented, and they yield the same European option price at two decimal places. Compared to the option price under the GTS distribution, the Black-Scholes (BS) model is underpriced for the Near-The-Money (NTM) and the in-the-money (ITM) options. However, the BS model and GTS European options yield the same option price for the deep out-of-the-money (OTM) and the deep-in-the-money (ITM) options. </p> </div> </dd> <dt> <a name='item27'>[27]</a> <a href ="/abs/2305.14545" title="Abstract" id="2305.14545"> arXiv:2305.14545 </a> (replaced) [<a href="/pdf/2305.14545" title="Download PDF" id="pdf-2305.14545" aria-labelledby="pdf-2305.14545">pdf</a>, <a href="https://arxiv.org/html/2305.14545v3" title="View HTML" id="html-2305.14545" aria-labelledby="html-2305.14545" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2305.14545" title="Other formats" id="oth-2305.14545" aria-labelledby="oth-2305.14545">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Liouville property for groups and conformal dimension </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Bon,+N+M">Nicol谩s Matte Bon</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Nekrashevych,+V">Volodymyr Nekrashevych</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Zheng,+T">Tianyi Zheng</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 41 pages, 5 figures, v3: revised version after referee reports </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Group Theory (math.GR)</span>; Dynamical Systems (math.DS); Probability (math.PR) </div> <p class='mathjax'> Conformal dimension is a fundamental invariant of metric spaces, particularly suited to the study of self-similar spaces, such as spaces with an expanding self-covering (e.g. Julia sets of complex rational functions). The dynamics of these systems are encoded by the associated iterated monodromy groups, which are examples of contracting self-similar groups. Their amenability is a well-known open question. We show that if $G$ is an iterated monodromy group, and if the (Alfhors-regular) conformal dimension of the underlying space is strictly less than 2, then every symmetric random walk with finite second moment on $G$ has the Liouville property. As a corollary, every such group is amenable. This criterion applies to all examples of contracting groups previously known to be amenable, and to many new ones. In particular, it implies that for every sub-hyperbolic complex rational function $f$ whose Julia set is not the whole sphere, the iterated monodromy group of $f$ is amenable. </p> </div> </dd> <dt> <a name='item28'>[28]</a> <a href ="/abs/2307.09108" title="Abstract" id="2307.09108"> arXiv:2307.09108 </a> (replaced) [<a href="/pdf/2307.09108" title="Download PDF" id="pdf-2307.09108" aria-labelledby="pdf-2307.09108">pdf</a>, <a href="https://arxiv.org/html/2307.09108v2" title="View HTML" id="html-2307.09108" aria-labelledby="html-2307.09108" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2307.09108" title="Other formats" id="oth-2307.09108" aria-labelledby="oth-2307.09108">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Stochastic dynamics of particle systems on unbounded degree graphs </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Chargaziya,+G">Georgy Chargaziya</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Daletskii,+A">Alexei Daletskii</a></div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Functional Analysis (math.FA)</span>; Probability (math.PR) </div> <p class='mathjax'> We consider an infinite system of coupled stochastic differential equations (SDE) describing dynamics of the following infinite particle system. Each partricle is characterised by its position $x\in \mathbb{R}^{d}$ and internal parameter (spin) $\sigma _{x}\in \mathbb{R}$. While the positions of particles form a fixed ("quenched") locally-finite set (configuration) $ \gamma \subset $ $\mathbb{R}^{d}$, the spins $\sigma _{x}$ and $\sigma _{y}$ interact via a pair potential whenever $\left\vert x-y\right\vert <\rho $, where $\rho >0$ is a fixed interaction radius. The number $n_{x}$ of particles interacting with a particle in positionn $x$ is finite but unbounded in $x$. The growth of $n_{x}$ as $x\rightarrow \infty $ creates a major technical problem for solving our SDE system. To overcome this problem, we use a finite volume approximation combined with a version of the Ovsjannikov method, and prove the existence and uniqueness of the solution in a scale of Banach spaces of weighted sequences. As an application example, we construct stochastic dynamics associated with Gibbs states of our particle system. </p> </div> </dd> <dt> <a name='item29'>[29]</a> <a href ="/abs/2311.03013" title="Abstract" id="2311.03013"> arXiv:2311.03013 </a> (replaced) [<a href="/pdf/2311.03013" title="Download PDF" id="pdf-2311.03013" aria-labelledby="pdf-2311.03013">pdf</a>, <a href="https://arxiv.org/html/2311.03013v3" title="View HTML" id="html-2311.03013" aria-labelledby="html-2311.03013" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2311.03013" title="Other formats" id="oth-2311.03013" aria-labelledby="oth-2311.03013">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Generalizations of Koga's version of the Wiener-Ikehara theorem </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Chen,+B">Bin Chen</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Vindas,+J">Jasson Vindas</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 14 pages. Theorem 1.2, Theorem 1.3, Corollary 1.4, and Theorem 1.5 have been substantially improved in this new version </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Complex Variables (math.CV)</span>; Classical Analysis and ODEs (math.CA); Number Theory (math.NT); Probability (math.PR) </div> <p class='mathjax'> We establish new versions of the Wiener-Ikehara theorem where only boundary assumptions on the real part of the Laplace transform are imposed. Our results generalize and improve a recent theorem of T. Koga [J. Fourier Anal. Appl. 27 (2021), Article No. 18]. As an application, we give a quick Tauberian proof of Blackwell's renewal theorem. </p> </div> </dd> <dt> <a name='item30'>[30]</a> <a href ="/abs/2403.01421" title="Abstract" id="2403.01421"> arXiv:2403.01421 </a> (replaced) [<a href="/pdf/2403.01421" title="Download PDF" id="pdf-2403.01421" aria-labelledby="pdf-2403.01421">pdf</a>, <a href="https://arxiv.org/html/2403.01421v2" title="View HTML" id="html-2403.01421" aria-labelledby="html-2403.01421" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2403.01421" title="Other formats" id="oth-2403.01421" aria-labelledby="oth-2403.01421">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> Predicting the Unpredictable under Subjective Expected Utility </div> <div class='list-authors'><a href="https://arxiv.org/search/econ?searchtype=author&query=Schipper,+B+C">Burkhard C. Schipper</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 35 pages, 4 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Theoretical Economics (econ.TH)</span>; Probability (math.PR) </div> <p class='mathjax'> We consider a decision maker who is unaware of objects to be sampled and thus cannot form beliefs about the occurrence of particular objects. Ex ante she can form beliefs about the occurrence of novelty and the frequencies of yet to be known objects. Conditional on any sampled objects, she can also form beliefs about the next object being novel or being one of the previously sampled objects. We characterize behaviorally such beliefs under subjective expected utility. In doing so, we relate "reverse" Bayesianism, a central property in the literature on decision making under growing awareness, with exchangeable random partitions, the central property in the literature on the discovery of species problem and mutations in statistics, combinatorial probability theory, and population genetics. Partition exchangeable beliefs do not necessarily satisfy "reverse" Bayesianism. Yet, the most prominent models of exchangeable random partitions, the model by De Morgan (1838), the one parameter model of Ewens (1972), and the two parameter model of Pitman (1995) and Zabell (1997), do satisfy "reverse" Bayesianism. Our characterization allows us to interpret these models as subjective beliefs of a decision maker and to derive the parameters from choice behavior. </p> </div> </dd> <dt> <a name='item31'>[31]</a> <a href ="/abs/2408.08370" title="Abstract" id="2408.08370"> arXiv:2408.08370 </a> (replaced) [<a href="/pdf/2408.08370" title="Download PDF" id="pdf-2408.08370" aria-labelledby="pdf-2408.08370">pdf</a>, <a href="https://arxiv.org/html/2408.08370v2" title="View HTML" id="html-2408.08370" aria-labelledby="html-2408.08370" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2408.08370" title="Other formats" id="oth-2408.08370" aria-labelledby="oth-2408.08370">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> When invariance implies exchangeability (and applications to invariant Keisler measures) </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Braunfeld,+S">Samuel Braunfeld</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Jahel,+C">Colin Jahel</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Marimon,+P">Paolo Marimon</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 58 pages, 5 figures </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Logic (math.LO)</span>; Combinatorics (math.CO); Dynamical Systems (math.DS); Probability (math.PR) </div> <p class='mathjax'> We study the problem of when, given a countable homogeneous structure $M$ and a space $S$ of expansions of $M$, every $\mathrm{Aut}(M)$-invariant probability measure on $S$ is exchangeable (i.e. invariant under all permutations of the domain). We show, for example, that if $M$ is a finitely bounded homogeneous $3$-hypergraph with free amalgamation (including the generic tetrahedron-free $3$-hypergraph), all $\mathrm{Aut}(M)$-invariant random expansions by graphs are exchangeable. Moreover, we extend and recover both the work of Angel, Kechris, and Lyons on invariant random orderings and some of the work of Crane and Towsner, and Ackerman on relative exchangeability. <br>In the second part of the paper, we apply our results to the study of invariant Keisler measures, which we prove to be particular invariant random expansions. Thus, we describe the spaces of invariant Keisler measures of various homogeneous structures, obtaining the first results of this kind since the work of Albert and Ensley. We also show there are $2^{\aleph_0}$ supersimple homogeneous ternary structures for which there are non-forking formulas which are universally measure zero. </p> </div> </dd> <dt> <a name='item32'>[32]</a> <a href ="/abs/2501.16449" title="Abstract" id="2501.16449"> arXiv:2501.16449 </a> (replaced) [<a href="/pdf/2501.16449" title="Download PDF" id="pdf-2501.16449" aria-labelledby="pdf-2501.16449">pdf</a>, <a href="https://arxiv.org/html/2501.16449v2" title="View HTML" id="html-2501.16449" aria-labelledby="html-2501.16449" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2501.16449" title="Other formats" id="oth-2501.16449" aria-labelledby="oth-2501.16449">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> The Gaussian Minkowski-type problems for $C$-pseudo-cones </div> <div class='list-authors'><a href="https://arxiv.org/search/math?searchtype=author&query=Shan,+J">Junjie Shan</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Hu,+W">Wenchuan Hu</a>, <a href="https://arxiv.org/search/math?searchtype=author&query=Xu,+W">Wenxue Xu</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> This version merges the initially submitted article with the authors separate work titled "The Gaussian log-Minkowski problem for C-pseudo-cones'' into a unified paper </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Metric Geometry (math.MG)</span>; Probability (math.PR) </div> <p class='mathjax'> The Gaussian surface area measure and the Gaussian cone measure for $C$-pseudo-cones are introduced as a new family of geometric measures. The corresponding Gaussian Minkowski-type problems for $C$-pseudo-cones are posed, and the existence and uniqueness results are established. </p> </div> </dd> <dt> <a name='item33'>[33]</a> <a href ="/abs/2502.06072" title="Abstract" id="2502.06072"> arXiv:2502.06072 </a> (replaced) [<a href="/pdf/2502.06072" title="Download PDF" id="pdf-2502.06072" aria-labelledby="pdf-2502.06072">pdf</a>, <a href="https://arxiv.org/html/2502.06072v2" title="View HTML" id="html-2502.06072" aria-labelledby="html-2502.06072" rel="noopener noreferrer" target="_blank">html</a>, <a href="/format/2502.06072" title="Other formats" id="oth-2502.06072" aria-labelledby="oth-2502.06072">other</a>] </dt> <dd> <div class='meta'> <div class='list-title mathjax'><span class='descriptor'>Title:</span> ID policy (with reassignment) is asymptotically optimal for heterogeneous weakly-coupled MDPs </div> <div class='list-authors'><a href="https://arxiv.org/search/cs?searchtype=author&query=Zhang,+X">Xiangcheng Zhang</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Hong,+Y">Yige Hong</a>, <a href="https://arxiv.org/search/cs?searchtype=author&query=Wang,+W">Weina Wang</a></div> <div class='list-comments mathjax'><span class='descriptor'>Comments:</span> 37 pages </div> <div class='list-subjects'><span class='descriptor'>Subjects:</span> <span class="primary-subject">Machine Learning (cs.LG)</span>; Optimization and Control (math.OC); Probability (math.PR) </div> <p class='mathjax'> Heterogeneity poses a fundamental challenge for many real-world large-scale decision-making problems but remains largely understudied. In this paper, we study the fully heterogeneous setting of a prominent class of such problems, known as weakly-coupled Markov decision processes (WCMDPs). Each WCMDP consists of $N$ arms (or subproblems), which have distinct model parameters in the fully heterogeneous setting, leading to the curse of dimensionality when $N$ is large. We show that, under mild assumptions, a natural adaptation of the ID policy, although originally proposed for a homogeneous special case of WCMDPs, in fact achieves an $O(1/\sqrt{N})$ optimality gap in long-run average reward per arm for fully heterogeneous WCMDPs as $N$ becomes large. This is the first asymptotic optimality result for fully heterogeneous average-reward WCMDPs. Our techniques highlight the construction of a novel projection-based Lyapunov function, which witnesses the convergence of rewards and costs to an optimal region in the presence of heterogeneity. </p> </div> </dd> </dl> <div class='paging'>Total of 33 entries </div> <div class='morefewer'>Showing up to 2000 entries per page: <a href=/list/math.PR/new?skip=0&show=1000 rel="nofollow"> fewer</a> | <span style="color: #454545">more</span> | <span style="color: #454545">all</span> </div> </div> </div> </div> </main> <footer style="clear: both;"> <div class="columns is-desktop" role="navigation" aria-label="Secondary" style="margin: -0.75em -0.75em 0.75em -0.75em">  <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul style="list-style: none; line-height: 2;"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 9.2-18.9v-19c0-26.5-21.5-48-48-48H48C21.5 64 0 85.5 0 112v19c0 7.4 3.4 14.3 9.2 18.9 30.6 23.9 40.7 32.4 173.4 128.7 16.8 12.2 50.2 41.8 73.4 41.4z"/></svg> <a href="https://info.arxiv.org/help/contact.html"> Contact</a> </li> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>subscribe to arXiv mailings</title><desc>Click here to subscribe</desc><path d="M476 3.2L12.5 270.6c-18.1 10.4-15.8 35.6 2.2 43.2L121 358.4l287.3-253.2c5.5-4.9 13.3 2.6 8.6 8.3L176 407v80.5c0 23.6 28.5 32.9 42.5 15.8L282 426l124.6 52.2c14.2 6 30.4-2.9 33-18.2l72-432C515 7.8 493.3-6.8 476 3.2z"/></svg> <a href="https://info.arxiv.org/help/subscribe"> Subscribe</a> </li> </ul> </div> </div> </div>   <div class="column" style="padding: 0;"> <div class="columns"> <div class="column"> <ul style="list-style: none; 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