Theory of Probability and Mathematical Statistics最新文献

{"title":"On the local time of a recurrent random walk on ℤ²","authors":"V. Bohun, A. Marynych","doi":"10.1090/tpms/1156","DOIUrl":"https://doi.org/10.1090/tpms/1156","url":null,"abstract":"We prove a functional limit theorem for the number of visits by a planar random walk on \u0000\u0000 \u0000 \u0000 \u0000 Z\u0000 \u0000 2\u0000 \u0000 mathbb {Z}^2\u0000 \u0000\u0000 with zero mean and finite second moment to the points of a fixed finite set \u0000\u0000 \u0000 \u0000 P\u0000 ⊂\u0000 \u0000 \u0000 Z\u0000 \u0000 2\u0000 \u0000 \u0000 Psubset mathbb {Z}^2\u0000 \u0000\u0000. The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44724360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Aggregation of network traffic and anisotropic scaling of random fields","authors":"R. Leipus, Vytaute Pilipauskaite, D. Surgailis","doi":"10.1090/tpms/1188","DOIUrl":"https://doi.org/10.1090/tpms/1188","url":null,"abstract":"<p>We discuss joint spatial-temporal scaling limits of sums <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript lamda comma gamma\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>A</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>γ<!-- γ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">A_{lambda ,gamma }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> (indexed by <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis x comma y right-parenthesis element-of double-struck upper R Subscript plus Superscript 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>x</mml:mi>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>y</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:msubsup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>+</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:msubsup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(x,y) in mathbb {R}^2_+</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>) of large number <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis lamda Superscript gamma Baseline right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>O</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:msup>\u0000 <mml:mi>λ<!-- λ --></mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>γ<!-- γ --></mml:mi>\u0000 </mml:mrow>\u0000 </mml:msup>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">O(lambda ^{gamma })</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of independent copies of integrated input process <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X equals StartSet upper X left-parenthesis t right-parenthesis comma t element-of double-struck upper R EndSet\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mi>t</mml:mi>\u0000 <mml:mo>∈<!-- ∈ --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">R</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">X = {X(t), t in mathbb {R}}</mml:annotation>\u0000 </mml","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45666244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On quadratic variations for the fractional-white wave equation","authors":"Radomyra Shevchenko","doi":"10.1090/tpms/1192","DOIUrl":"https://doi.org/10.1090/tpms/1192","url":null,"abstract":"This paper studies the behaviour of quadratic variations of a stochastic wave equation driven by a noise that is white in space and fractional in time. Complementing the analysis of quadratic variations in the space component carried out in [Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise, Electron. J. Stat. 12 (2018), no. 2, 3639–3672] and [Generalized \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus, J. Statist. Plann. Inference 207 (2020), 155–180], it focuses on the time component of the solution process. For different values of the Hurst parameter a central and a noncentral limit theorems are proved, allowing to construct parameter estimators and compare them to the findings in the space-dependent case. Finally, rectangular quadratic variations in the white noise case are studied and a central limit theorem is demonstrated.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47831804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gaussian random fields: with and without covariances","authors":"N. Bingham, T. Symons","doi":"10.1090/tpms/1163","DOIUrl":"https://doi.org/10.1090/tpms/1163","url":null,"abstract":"We begin with isotropic Gaussian random fields, and show how the Bochner–Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs). We then turn from this continuous setting to approximating discrete settings, Gaussian Markov random fields (GMRFs), and the computational advantages they bring in handling large data sets, by exploiting the sparseness properties of the relevant precision (concentration) matrices.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45547546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Boundedness of the nodal domains of additive Gaussian fields","authors":"S. Muirhead","doi":"10.1090/tpms/1169","DOIUrl":"https://doi.org/10.1090/tpms/1169","url":null,"abstract":"We study the connectivity of the excursion sets of additive Gaussian fields, i.e. stationary centred Gaussian fields whose covariance function decomposes into a sum of terms that depend separately on the coordinates. Our main result is that, under mild smoothness and correlation decay assumptions, the excursion sets \u0000\u0000 \u0000 \u0000 {\u0000 f\u0000 ≤\u0000 ℓ\u0000 }\u0000 \u0000 {f le ell }\u0000 \u0000\u0000 of additive planar Gaussian fields are bounded almost surely at the critical level \u0000\u0000 \u0000 \u0000 \u0000 ℓ\u0000 c\u0000 \u0000 =\u0000 0\u0000 \u0000 ell _c = 0\u0000 \u0000\u0000. Since we do not assume positive correlations, this provides the first examples of continuous non-positively-correlated stationary planar Gaussian fields for which the boundedness of the nodal domains has been confirmed. By contrast, in dimension \u0000\u0000 \u0000 \u0000 d\u0000 ≥\u0000 3\u0000 \u0000 d ge 3\u0000 \u0000\u0000 the excursion sets have unbounded components at all levels.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47197545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stochastic analysis for vector-valued generalized grey Brownian motion","authors":"W. Bock, M. Grothaus, Karlo S. Orge","doi":"10.1090/tpms/1184","DOIUrl":"https://doi.org/10.1090/tpms/1184","url":null,"abstract":"In this article, we show that the standard vector-valued generalization of a generalized grey Brownian motion (ggBm) has independent components if and only if it is a fractional Brownian motion. In order to extend ggBm with independent components, we introduce a vector-valued generalized grey Brownian motion (vggBm). The characteristic function of the corresponding measure is introduced as the product of the characteristic functions of the one-dimensional case. We show that for this measure, the Appell system and a calculus of generalized functions or distributions are accessible. We characterize these distributions with suitable transformations and give a \u0000\u0000 \u0000 d\u0000 d\u0000 \u0000\u0000-dimensional Donsker’s delta function as an example for such distributions. From there, we show the existence of local times and self-intersection local times of vggBm as distributions under some constraints, and compute their corresponding generalized expectations. At the end, we solve a system of linear SDEs driven by a vggBm noise in \u0000\u0000 \u0000 d\u0000 d\u0000 \u0000\u0000 dimensions.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49499380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Infinitesimal invariance of completely Random Measures for 2D Euler Equations","authors":"Francesco Grotto, G. Peccati","doi":"10.1090/tpms/1178","DOIUrl":"https://doi.org/10.1090/tpms/1178","url":null,"abstract":"We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortices dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler’s equations preserving independently scattered random measures.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46256299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stochastic heat equation with piecewise constant coefficients and generalized fractional type noise","authors":"M. Zili, Eya Zougar","doi":"10.1090/tpms/1150","DOIUrl":"https://doi.org/10.1090/tpms/1150","url":null,"abstract":"","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41923995","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A bound in the stable ($alpha $), $1 < alpha le 2$, limit theorem for associated random variables with infinite variance","authors":"M. Sreehari","doi":"10.1090/tpms/1151","DOIUrl":"https://doi.org/10.1090/tpms/1151","url":null,"abstract":"","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48403756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Estimation of the Hurst and diffusion parameters in fractional stochastic heat equation","authors":"D. A. Avetisian, K. Ralchenko","doi":"10.1090/tpms/1145","DOIUrl":"https://doi.org/10.1090/tpms/1145","url":null,"abstract":"","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45976927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}