Pub Date : 2023-11-18DOI: 10.1007/s10959-023-01303-3
Guangying Lv, Wei Wang, Jinlong Wei
In this paper, the reflected McKean–Vlasov diffusion ov a convex domain is studied. We first establish the well-posedness of a coupled system of nonlinear stochastic differential equations via a fixed point theorem which is similar to that for partial differential equations. Moreover, the reason why we make different assumptions on drift and cross terms is given. Then, the propagation of chaos for the particle system is also obtained.
{"title":"Coupled McKean–Vlasov Equations Over Convex Domains","authors":"Guangying Lv, Wei Wang, Jinlong Wei","doi":"10.1007/s10959-023-01303-3","DOIUrl":"https://doi.org/10.1007/s10959-023-01303-3","url":null,"abstract":"<p>In this paper, the reflected McKean–Vlasov diffusion ov a convex domain is studied. We first establish the well-posedness of a coupled system of nonlinear stochastic differential equations via a fixed point theorem which is similar to that for partial differential equations. Moreover, the reason why we make different assumptions on drift and cross terms is given. Then, the propagation of chaos for the particle system is also obtained.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"58 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138506807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-18DOI: 10.1007/s10959-023-01299-w
Xiaohan Wu, Anyue Chen, Junping Li
Based on the resolvent decomposition theorems presented very recently by Chen (J Theor Probab 33:2089–2118, 2020), in this paper we focus on investigating the fundamental problems of existence and uniqueness criteria for Denumerable Markov Processes with finitely many instantaneous states. Some elegant sufficient and necessary conditions are obtained for this less-investigated topic. A few important examples including the generalized Kolmogorov models are presented to illustrate our general results.
{"title":"Existence and Uniqueness of Denumerable Markov Processes with Instantaneous States","authors":"Xiaohan Wu, Anyue Chen, Junping Li","doi":"10.1007/s10959-023-01299-w","DOIUrl":"https://doi.org/10.1007/s10959-023-01299-w","url":null,"abstract":"<p>Based on the resolvent decomposition theorems presented very recently by Chen (J Theor Probab 33:2089–2118, 2020), in this paper we focus on investigating the fundamental problems of existence and uniqueness criteria for Denumerable Markov Processes with finitely many instantaneous states. Some elegant sufficient and necessary conditions are obtained for this less-investigated topic. A few important examples including the generalized Kolmogorov models are presented to illustrate our general results.</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-16DOI: 10.1007/s10959-023-01301-5
Marc Arnaudon, Koléhè Coulibaly-Pasquier, Laurent Miclo
Given an intertwining relation between two finite Markov chains, we investigate how it can be transformed by conditioning the primal Markov chain to stay in a proper subset. A natural assumption on the underlying link kernel is put forward. The three classical examples of discrete Pitman, top-to-random shuffle and absorbed birth-and-death chain intertwinings serve as illustrations.
{"title":"On Markov Intertwining Relations and Primal Conditioning","authors":"Marc Arnaudon, Koléhè Coulibaly-Pasquier, Laurent Miclo","doi":"10.1007/s10959-023-01301-5","DOIUrl":"https://doi.org/10.1007/s10959-023-01301-5","url":null,"abstract":"<p>Given an intertwining relation between two finite Markov chains, we investigate how it can be transformed by conditioning the primal Markov chain to stay in a proper subset. A natural assumption on the underlying link kernel is put forward. The three classical examples of discrete Pitman, top-to-random shuffle and absorbed birth-and-death chain intertwinings serve as illustrations.\u0000</p>","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"58 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138506806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.1007/s10959-023-01300-6
Qian Yu, Xianye Yu
{"title":"Limit Theorem for Self-intersection Local Time Derivative of Multidimensional Fractional Brownian Motion","authors":"Qian Yu, Xianye Yu","doi":"10.1007/s10959-023-01300-6","DOIUrl":"https://doi.org/10.1007/s10959-023-01300-6","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":" 23","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-03DOI: 10.1007/s10959-023-01298-x
Lianzi Jiang, Gechun Liang
Abstract This article fills a gap in the literature by relaxing the integrability condition for the robust $$alpha $$ α -stable central limit theorem under sublinear expectation. Specifically, for $$alpha in (0,1]$$ α∈(0,1] , we prove that the normalized sums of i.i.d. non-integrable random variables $$big {n^{-frac{1}{alpha }}sum _{i=1}^{n}Z_{i}big }_{n=1}^{infty }$$ {n-1α∑i=1nZi}n=1∞ converge in law to $${tilde{zeta }}_{1}$$ ζ~1 , where $$({tilde{zeta }}_{t})_{tin [0,1]}$$ (ζ~t)t∈[0,1] is a multidimensional nonlinear symmetric $$alpha $$ α -stable process with jump uncertainty set $${mathcal {L}}$$ L . The limiting $$alpha $$ α -stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE): $$begin{aligned} left{ begin{array}{l} displaystyle partial _{t}u(t,x)-sup limits _{F_{mu }in {mathcal {L}}}left{ int _{{mathbb {R}}^{d}}delta _{lambda }^{alpha }u(t,x)F_{mu }(dlambda )right} =0, displaystyle u(0,x)=phi (x),quad forall (t,x)in [0,1]times {mathbb {R}}^{d}, end{array} right. end{aligned}$$
摘要本文通过放宽次线性期望下稳健$$alpha $$ α -稳定中心极限定理的可积性条件，填补了文献的空白。具体而言，对于$$alpha in (0,1]$$ α∈(0,1)，证明了i.i.d不可积随机变量$$big {n^{-frac{1}{alpha }}sum _{i=1}^{n}Z_{i}big }_{n=1}^{infty }$$ n- 1 α∑i = 1 n zi n = 1{∞的归一化和规律收敛于}$${tilde{zeta }}_{1}$$ ζ 1，其中$$({tilde{zeta }}_{t})_{tin [0,1]}$$ (ζ t) t∈[0,1]是一个具有跳跃不确定性集$${mathcal {L}}$$ L的多维非线性对称$$alpha $$ α稳定过程。极限$$alpha $$ α稳定过程进一步表征为一个完全非线性的偏积分微分方程(PIDE): $$begin{aligned} left{ begin{array}{l} displaystyle partial _{t}u(t,x)-sup limits _{F_{mu }in {mathcal {L}}}left{ int _{{mathbb {R}}^{d}}delta _{lambda }^{alpha }u(t,x)F_{mu }(dlambda )right} =0, displaystyle u(0,x)=phi (x),quad forall (t,x)in [0,1]times {mathbb {R}}^{d}, end{array} right. end{aligned}$$∂t u (t, x) - sup F μ∈L∫R d δ λ α u (t, x) F μ (d λ) = 0, u (0, x) = ϕ (x)，∀(t, x)∈[0,1]× R d，其中$$begin{aligned} delta _{lambda }^{alpha }u(t,x):=left{ begin{array}{ll} u(t,x+lambda )-u(t,x)-langle D_{x}u(t,x),lambda mathbbm {1}_{{|lambda |le 1}}rangle , &{}quad alpha =1, u(t,x+lambda )-u(t,x), &{}quad alpha in (0,1). end{array} right. end{aligned}$$ δ λ α u (t, x):= u (t, x + λ) - u (t, x) -⟨dx u (t, x)， λ 1 {| λ |≤1}⟩，α = 1, u (t, x + λ) - u (t, x)， α∈(0,1)。本研究中使用的方法涉及到几种工具的利用，包括弱收敛方法来获得极限过程，非线性$$alpha $$ α稳定过程的l - khintchine表示和截断技术来估计相应的$$alpha $$ α稳定l测量。此外，本文还给出了证明上述全非线性PIDE解的存在性的概率方法。
{"title":"A Robust $$alpha $$-Stable Central Limit Theorem Under Sublinear Expectation without Integrability Condition","authors":"Lianzi Jiang, Gechun Liang","doi":"10.1007/s10959-023-01298-x","DOIUrl":"https://doi.org/10.1007/s10959-023-01298-x","url":null,"abstract":"Abstract This article fills a gap in the literature by relaxing the integrability condition for the robust $$alpha $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>α</mml:mi> </mml:math> -stable central limit theorem under sublinear expectation. Specifically, for $$alpha in (0,1]$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , we prove that the normalized sums of i.i.d. non-integrable random variables $$big {n^{-frac{1}{alpha }}sum _{i=1}^{n}Z_{i}big }_{n=1}^{infty }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>{</mml:mo> </mml:mrow> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>α</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>n</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msubsup> <mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> converge in law to $${tilde{zeta }}_{1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mover> <mml:mi>ζ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , where $$({tilde{zeta }}_{t})_{tin [0,1]}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mover> <mml:mi>ζ</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:msub> </mml:math> is a multidimensional nonlinear symmetric $$alpha $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>α</mml:mi> </mml:math> -stable process with jump uncertainty set $${mathcal {L}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>L</mml:mi> </mml:math> . The limiting $$alpha $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>α</mml:mi> </mml:math> -stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE): $$begin{aligned} left{ begin{array}{l} displaystyle partial _{t}u(t,x)-sup limits _{F_{mu }in {mathcal {L}}}left{ int _{{mathbb {R}}^{d}}delta _{lambda }^{alpha }u(t,x)F_{mu }(dlambda )right} =0, displaystyle u(0,x)=phi (x),quad forall (t,x)in [0,1]times {mathbb {R}}^{d}, end{array} right. end{aligned}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mfenced> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:ms","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"39 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-02DOI: 10.1007/s10959-023-01291-4
Amos Nevo, Felix Pogorzelski
{"title":"Shannon–McMillan–Breiman Theorem Along Almost Geodesics in Negatively Curved Groups","authors":"Amos Nevo, Felix Pogorzelski","doi":"10.1007/s10959-023-01291-4","DOIUrl":"https://doi.org/10.1007/s10959-023-01291-4","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"74 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135933122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1007/s10959-023-01296-z
Luca Avena, Conrado da Costa
Abstract We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples.
{"title":"Laws of Large Numbers for Weighted Sums of Independent Random Variables: A Game of Mass","authors":"Luca Avena, Conrado da Costa","doi":"10.1007/s10959-023-01296-z","DOIUrl":"https://doi.org/10.1007/s10959-023-01296-z","url":null,"abstract":"Abstract We consider weighted sums of independent random variables regulated by an increment sequence and provide operative conditions that ensure a strong law of large numbers for such sums in both the centred and non-centred case. The existing criteria for the strong law are either implicit or based on restrictions on the increment sequence. In our setup we allow for an arbitrary sequence of increments, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centred case, convergence can be translated into the behaviour of a deterministic sequence and it becomes a game of mass when the expectation of the random variables is a function of the increment sizes. We identify various classes of increments and illustrate them with a variety of concrete examples.","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"225 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-31DOI: 10.1007/s10959-023-01297-y
Endre Csáki, Antónia Földes
{"title":"On the Local Time of Anisotropic Random Walk on $$mathbb Z^2$$","authors":"Endre Csáki, Antónia Földes","doi":"10.1007/s10959-023-01297-y","DOIUrl":"https://doi.org/10.1007/s10959-023-01297-y","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135808263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-29DOI: 10.1007/s10959-023-01295-0
Aurelien Gribinski
{"title":"A Theory of Singular Values for Finite Free Probability","authors":"Aurelien Gribinski","doi":"10.1007/s10959-023-01295-0","DOIUrl":"https://doi.org/10.1007/s10959-023-01295-0","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-19DOI: 10.1007/s10959-023-01292-3
Yan-Xia Ren, Renming Song, Rui Zhang
{"title":"Lower Deviation for the Supremum of the Support of Super-Brownian Motion","authors":"Yan-Xia Ren, Renming Song, Rui Zhang","doi":"10.1007/s10959-023-01292-3","DOIUrl":"https://doi.org/10.1007/s10959-023-01292-3","url":null,"abstract":"","PeriodicalId":54760,"journal":{"name":"Journal of Theoretical Probability","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135728558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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