Document Zbl 0751.60037

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Multivariate Anal.</a> <a href="/?q=in%3A156786" title="Articles in this Issue">37, No. 1, 115-134 (1991)</a>. </div> <div class="abstract">Let $T$ be an abstract set, and $V$ be a vector space of real functions on $T$. Let $M(t)$ be a symmetric $\alpha$-stable $(S\alpha S)$ Lévy process on $T$ with a control measure $m(dt)$. Consider a multiple stochastic integral process \[ X(t)=\int^ \infty_{-\infty}\dots\int^ \infty_{-\infty} f_ t(x_ 1,\dots,x_ n)M(dx_ 1)\dots M(dx_ n),\quad t\in T,\quad(t\mapsto f_ t(x_ 1,\dots,x_ n))\in V. \] The paper studies the problem of the existence of a version of the process $X(t)$ with sample paths in the space $V$. Necessary conditions, and sufficient conditions as well, are presented for a wide class of vector spaces $V$. Many classes of interest from the view point of applications [cf. <span class="zbmathjax-textit">M. Taqqu</span>, Random processes with long-range dependence and high variability, J. Geophys. Res. 92, 9683-9686 (1987)] satisfy the suitable conditions (including spaces of bounded, or continuous, or differentiable functions, and numerous alterations of the aforementioned spaces). Besides use of a large variety of techniques, pertinent to multiple integration, proofs are based on subtle measure theoretical arguments. A widely understood separability is essential although the nonseparable case (e.g., related to the space of bounded functions) is treated as well. Assuming the fulfillment of a certain separability condition, the authors prove a zero-one law for the process $X(t)$, $P(\{X(t), t\in T\}\in V)=0$ or 1, in the two-dimensional case (recently, the authors have proved the 0-1 law for an arbitrary dimension).<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A2663">J.Szulga (Auburn)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A9427">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A60G17" title="MSC2020">60G17</a> </td> <td class="space"> Sample path properties </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60E07" title="MSC2020">60E07</a> </td> <td class="space"> Infinitely divisible distributions; stable distributions </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H05" title="MSC2020">60H05</a> </td> <td class="space"> Stochastic integrals </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Asample+path+properties">sample path properties</a>; <a href="/?q=ut%3Arandom+measure">random measure</a>; <a href="/?q=ut%3ABanach+space">Banach space</a>; <a href="/?q=ut%3Amultiple+stochastic+integral+process">multiple stochastic integral process</a>; <a href="/?q=ut%3ARandom+processes+with+long-range+dependence">Random processes with long-range dependence</a>; <a href="/?q=ut%3Azero-one+law">zero-one law</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0751.60037" data-ciurl="/ci/00009427" data-biburl="/bibtex/00009427.bib" data-amsurl="/amsrefs/00009427.bib" data-xmlurl="/xml/00009427.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00009427.pdf" title="Zbl 0751.60037 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/0047-259X(91)90115-I" aria-label="DOI for “Sample path properties of stochastic processes represented as multiple stable integrals”" title="10.1016/0047-259X(91)90115-I">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://hdl.handle.net/1813/8754" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Cambanis, S.; Rajput, B. S., Some zero-one laws for Gaussian processes, Ann. Probab., 1, 304-312 (1973) · <a href="/0262.60017" class="nowrap">Zbl 0262.60017</a></td> </tr><tr> <td>[2]</td> <td class="space">DeAcosta, A., Quadratic zero-one laws for Gaussian measures and the distribution of quadratic forms, (Proc. Amer. Math. Soc., 54 (1976)), 319-325 · <a href="/0295.60018" class="nowrap">Zbl 0295.60018</a></td> </tr><tr> <td>[3]</td> <td class="space">Dudley, R. M.; Kanter, M., Zero-one laws for stable measures, (Proc. Amer. Math. Soc., 45 (1974)), 245-252 · <a href="/0297.60007" class="nowrap">Zbl 0297.60007</a></td> </tr><tr> <td>[4]</td> <td class="space">Doob, J. L., (Stochastic Processes (1953), Wiley: Wiley New York) · <a href="/0053.26802" class="nowrap">Zbl 0053.26802</a></td> </tr><tr> <td>[5]</td> <td class="space">Dunford, N.; Schwartz, J. 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Probab., 2, 729-737 (1973) · <a href="/0286.60027" class="nowrap">Zbl 0286.60027</a></td> </tr><tr> <td>[9]</td> <td class="space">Krakowiak, W.; Szulga, J., Hypercontraction principle and random multilinear forms, Probab. Theory Related Fileds, 77, 325-342 (1988) · <a href="/0621.60004" class="nowrap">Zbl 0621.60004</a></td> </tr><tr> <td>[10]</td> <td class="space">Krakowiak, W.; Szulga, J., A multiple stochastic integral with respect to a strictly p-stable measure, Ann. Probab., 16, 764-777 (1988) · <a href="/0648.60064" class="nowrap">Zbl 0648.60064</a></td> </tr><tr> <td>[11]</td> <td class="space">Krakowiak, W.; Szulga, J., On a p-stable multiple integral, I, Probab. Theory Related Fields, 78, 437-448 (1988) · <a href="/0628.60066" class="nowrap">Zbl 0628.60066</a></td> </tr><tr> <td>[12]</td> <td class="space">Kwapienń, S.; Woyczyński, W. A., Double stochastic integrals, random quadratic forms and random series in Orlicz spaces, Ann. Probab., 15, 1072-1096 (1987) · <a href="/0622.60054" class="nowrap">Zbl 0622.60054</a></td> </tr><tr> <td>[13]</td> <td class="space">LePage, R., Multidimensional infinitely divisible variables and processes. Part I. Stable case. (1980), Preprint</td> </tr><tr> <td>[14]</td> <td class="space">McConnell, T.; Taqqu, M. S., (Double Integration with Respect to Symmetric Stable Processes (1986), Cornell University), Technical Report 618</td> </tr><tr> <td>[15]</td> <td class="space">Rosiński, J., On stochastic integral representation of stable processes with sample paths in Banach spaces, J. Multivariate Anal., 20, 277-302 (1986) · <a href="/0606.60041" class="nowrap">Zbl 0606.60041</a></td> </tr><tr> <td>[16]</td> <td class="space">Rosiński, J., On path properties of certain infinitely divisible processes, Stochastic Process. Appl., 33, 73-87 (1989) · <a href="/0715.60051" class="nowrap">Zbl 0715.60051</a></td> </tr><tr> <td>[17]</td> <td class="space">Rosiński, J., An application of series representations to zero-one laws for infinitely divisible random vectors, (Progress in Probability, Vol. 21 (1990), Birkhäuser: Birkhäuser Boston) · <a href="/0709.60005" class="nowrap">Zbl 0709.60005</a></td> </tr><tr> <td>[18]</td> <td class="space">Rosiński, J.; Woyczyński, W. A., On Itô stochastic integration with respect to p-stable motion: Inner clock, integrability of sample paths, double and multiple integrals, Ann. Probab., 14, 271-286 (1986) · <a href="/0594.60056" class="nowrap">Zbl 0594.60056</a></td> </tr><tr> <td>[19]</td> <td class="space">Samorodnitsky, G.; Szulga, J., An asymptotic evaluation of the tail of a multiple symmetric α-stable integral, Ann. Probab., 17, 1503-1520 (1989) · <a href="/0689.60051" class="nowrap">Zbl 0689.60051</a></td> </tr><tr> <td>[20]</td> <td class="space">Samorodnitsky, G.; Taqqu, M. S., Multiple stable integrals of Banach-valued function, J. Theoret. Probab., 3, 267-287 (1990) · <a href="/0701.60051" class="nowrap">Zbl 0701.60051</a></td> </tr><tr> <td>[21]</td> <td class="space">Samorodnitsky, G.; Taqqu, M. S., Construction of multiple stable measures and integrals using LePage representation, (Samorodnitsky, G.; Cambanis, S.; Taqqu, M. S., Stable Processes and Related Topics (1991), Birkhäuser: Birkhäuser Boston), To appear · <a href="/0751.60013" class="nowrap">Zbl 0751.60013</a></td> </tr><tr> <td>[22]</td> <td class="space">Taqqu, M. S., Random processes with long-range dependence and high variability, J. Geophys. Res., 92, 9683-9686 (1987)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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