{"title":"由正稳定跳变扰动的随机漫步的泛函极限定理","authors":"A. Iksanov, A. Pilipenko, B. Povar","doi":"10.3150/22-bej1515","DOIUrl":null,"url":null,"abstract":"Let $\\xi_1$, $\\xi_2,\\ldots$ be i.i.d. random variables of zero mean and finite variance and $\\eta_1$, $\\eta_2,\\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $\\alpha$-stable distribution, $\\alpha\\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $\\xi_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $\\eta_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process $(X(t))_{t\\geq 0}$ satisfying a stochastic equation ${\\rm d}X(t)={\\rm d}W(t)+ {\\rm d}U_\\alpha(L_X^{(0)}(t))$, where $W$ is a Brownian motion, $U_\\alpha$ is an $\\alpha$-stable subordinator which is independent of $W$, and $L_X^{(0)}$ is a local time of $X$ at $0$. Also, we explain that $X$ is a Feller Brownian motion with a `jump-type' exit from $0$.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":" ","pages":""},"PeriodicalIF":1.5000,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":"{\"title\":\"Functional limit theorems for random walks perturbed by positive alpha-stable jumps\",\"authors\":\"A. Iksanov, A. Pilipenko, B. Povar\",\"doi\":\"10.3150/22-bej1515\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $\\\\xi_1$, $\\\\xi_2,\\\\ldots$ be i.i.d. random variables of zero mean and finite variance and $\\\\eta_1$, $\\\\eta_2,\\\\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $\\\\alpha$-stable distribution, $\\\\alpha\\\\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $\\\\xi_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $\\\\eta_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process $(X(t))_{t\\\\geq 0}$ satisfying a stochastic equation ${\\\\rm d}X(t)={\\\\rm d}W(t)+ {\\\\rm d}U_\\\\alpha(L_X^{(0)}(t))$, where $W$ is a Brownian motion, $U_\\\\alpha$ is an $\\\\alpha$-stable subordinator which is independent of $W$, and $L_X^{(0)}$ is a local time of $X$ at $0$. Also, we explain that $X$ is a Feller Brownian motion with a `jump-type' exit from $0$.\",\"PeriodicalId\":55387,\"journal\":{\"name\":\"Bernoulli\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2021-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bernoulli\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3150/22-bej1515\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bernoulli","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3150/22-bej1515","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Functional limit theorems for random walks perturbed by positive alpha-stable jumps
Let $\xi_1$, $\xi_2,\ldots$ be i.i.d. random variables of zero mean and finite variance and $\eta_1$, $\eta_2,\ldots$ positive i.i.d. random variables whose distribution belongs to the domain of attraction of an $\alpha$-stable distribution, $\alpha\in (0,1)$. The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump $\xi_k$ occurs; if the present position of the Markov chain is nonpositive, then its next position is $\eta_j$. We prove a functional limit theorem for this Markov chain under Donsker's scaling. The weak limit is a nonnegative process $(X(t))_{t\geq 0}$ satisfying a stochastic equation ${\rm d}X(t)={\rm d}W(t)+ {\rm d}U_\alpha(L_X^{(0)}(t))$, where $W$ is a Brownian motion, $U_\alpha$ is an $\alpha$-stable subordinator which is independent of $W$, and $L_X^{(0)}$ is a local time of $X$ at $0$. Also, we explain that $X$ is a Feller Brownian motion with a `jump-type' exit from $0$.
期刊介绍:
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