We investigate a family of discrete-time stationary processes defined by multiple stable integrals and renewal processes with infinite means. The model may exhibit behaviors of short-range or long-range dependence, respectively, depending on the parameters. The main contribution is to establish a phase transition in terms of the tail processes that characterize local clustering of extremes. Moreover, in the short-range dependence regime, the model provides an example where the extremal index is different from the candidate extremal index.
{"title":"Tail processes for stable-regenerative multiple-stable model","authors":"Shuyang Bai, Yizao Wang","doi":"10.3150/22-bej1582","DOIUrl":"https://doi.org/10.3150/22-bej1582","url":null,"abstract":"We investigate a family of discrete-time stationary processes defined by multiple stable integrals and renewal processes with infinite means. The model may exhibit behaviors of short-range or long-range dependence, respectively, depending on the parameters. The main contribution is to establish a phase transition in terms of the tail processes that characterize local clustering of extremes. Moreover, in the short-range dependence regime, the model provides an example where the extremal index is different from the candidate extremal index.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47389282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the i.i.d. Bernoulli field $mu_p$ on $mathbb{Z}^d$ with occupation density $pin [0,1]$. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large $p$, as it changes only a small fraction $p(1-p)^{2d}$ of sites, there is $p(d)<1$ such that for all $pin(p(d),1)$ the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small $p$, the Gibbs property is preserved.
{"title":"Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites","authors":"B. Jahnel, C. Kuelske","doi":"10.3150/22-bej1572","DOIUrl":"https://doi.org/10.3150/22-bej1572","url":null,"abstract":"We consider the i.i.d. Bernoulli field $mu_p$ on $mathbb{Z}^d$ with occupation density $pin [0,1]$. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large $p$, as it changes only a small fraction $p(1-p)^{2d}$ of sites, there is $p(d)<1$ such that for all $pin(p(d),1)$ the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small $p$, the Gibbs property is preserved.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45757610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Gaussian noise, including the stochastic heat equation.
{"title":"Chung-type law of the iterated logarithm and exact moduli of continuity for a class of anisotropic Gaussian random fields","authors":"C. Lee, Yimin Xiao","doi":"10.3150/22-bej1467","DOIUrl":"https://doi.org/10.3150/22-bej1467","url":null,"abstract":"We establish a Chung-type law of the iterated logarithm and the exact local and uniform moduli of continuity for a large class of anisotropic Gaussian random fields with a harmonizable-type integral representation and the property of strong local nondeterminism. Compared with the existing results in the literature, our results do not require the assumption of stationary increments and provide more precise upper and lower bounds for the limiting constants. The results are applicable to the solutions of a class of linear stochastic partial differential equations driven by a fractional-colored Gaussian noise, including the stochastic heat equation.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42394550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In [LPW21], the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE$_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE$_2$. The conclusion is a generalization of [HLW20,Theorem 1.6] where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE$_2(-1,-1; -1, -1)$. However, the limiting law is nolonger in the family of SLE$_2(rho)$ process as long as $Nge 3$.
{"title":"Loop-erased random walk branch of uniform spanning tree in topological polygons","authors":"Mingchang Liu, Hao Wu","doi":"10.3150/22-bej1510","DOIUrl":"https://doi.org/10.3150/22-bej1510","url":null,"abstract":"We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In [LPW21], the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE$_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE$_2$. The conclusion is a generalization of [HLW20,Theorem 1.6] where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE$_2(-1,-1; -1, -1)$. However, the limiting law is nolonger in the family of SLE$_2(rho)$ process as long as $Nge 3$.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45142595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper discusses multivariate self- and cross-exciting processes. We define a class of multivariate point processes via their corresponding stochastic intensity processes that are driven by stochastic jumps. Essentially, there is a jump in an intensity process whenever the corresponding point process records an event. An attribute of our modelling class is that not only a jump is recorded at each instance, but also its magnitude. This allows large jumps to influence the intensity to a larger degree than smaller jumps. We give conditions which guarantee that the process is stable, in the sense that it does not explode, and provide a detailed discussion on when the subclass of linear models is stable. Finally, we fit our model to financial time series data from the S&P 500 and Nikkei 225 indices respectively. We conclude that a nonlinear variant from our modelling class fits the data best. This supports the observation that in times of crises (high intensity) jumps tend to arrive in clusters, whereas there are typically longer times between jumps when the markets are calmer. We moreover observe more variability in jump sizes when the intensity is high, than when it is low.
{"title":"Multivariate self-exciting jump processes with applications to financial data","authors":"Heidar Eyjolfsson, D. Tjøstheim","doi":"10.3150/22-bej1537","DOIUrl":"https://doi.org/10.3150/22-bej1537","url":null,"abstract":"The paper discusses multivariate self- and cross-exciting processes. We define a class of multivariate point processes via their corresponding stochastic intensity processes that are driven by stochastic jumps. Essentially, there is a jump in an intensity process whenever the corresponding point process records an event. An attribute of our modelling class is that not only a jump is recorded at each instance, but also its magnitude. This allows large jumps to influence the intensity to a larger degree than smaller jumps. We give conditions which guarantee that the process is stable, in the sense that it does not explode, and provide a detailed discussion on when the subclass of linear models is stable. Finally, we fit our model to financial time series data from the S&P 500 and Nikkei 225 indices respectively. We conclude that a nonlinear variant from our modelling class fits the data best. This supports the observation that in times of crises (high intensity) jumps tend to arrive in clusters, whereas there are typically longer times between jumps when the markets are calmer. We moreover observe more variability in jump sizes when the intensity is high, than when it is low.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46888458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive non-asymptotic minimax bounds for the Hausdorff estimation of $d$-dimensional submanifolds $M subset mathbb{R}^D$ with (possibly) non-empty boundary $partial M$. The model reunites and extends the most prevalent $mathcal{C}^2$-type set estimation models: manifolds without boundary, and full-dimensional domains. We consider both the estimation of the manifold $M$ itself and that of its boundary $partial M$ if non-empty. Given $n$ samples, the minimax rates are of order $Obigl((log n/n)^{2/d}bigr)$ if $partial M = emptyset$ and $Obigl((log n/n)^{2/(d+1)}bigr)$ if $partial M neq emptyset$, up to logarithmic factors. In the process, we develop a Voronoi-based procedure that allows to identify enough points $Obigl((log n/n)^{2/(d+1)}bigr)$-close to $partial M$ for reconstructing it.
{"title":"Minimax boundary estimation and estimation with boundary","authors":"Eddie Aamari, C. Aaron, Clément Levrard","doi":"10.3150/23-bej1585","DOIUrl":"https://doi.org/10.3150/23-bej1585","url":null,"abstract":"We derive non-asymptotic minimax bounds for the Hausdorff estimation of $d$-dimensional submanifolds $M subset mathbb{R}^D$ with (possibly) non-empty boundary $partial M$. The model reunites and extends the most prevalent $mathcal{C}^2$-type set estimation models: manifolds without boundary, and full-dimensional domains. We consider both the estimation of the manifold $M$ itself and that of its boundary $partial M$ if non-empty. Given $n$ samples, the minimax rates are of order $Obigl((log n/n)^{2/d}bigr)$ if $partial M = emptyset$ and $Obigl((log n/n)^{2/(d+1)}bigr)$ if $partial M neq emptyset$, up to logarithmic factors. In the process, we develop a Voronoi-based procedure that allows to identify enough points $Obigl((log n/n)^{2/(d+1)}bigr)$-close to $partial M$ for reconstructing it.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43115545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Mat_{mathbb{C}}(K,N)$ be the space of $Ktimes N$ complex matrices. Let $mathbf{B}_t$ be Brownian motion on $Mat_{mathbb{C}}(K,N)$ starting from the zero matrix and $mathbf{M}in Mat_{mathbb{C}}(K,N)$. We prove that, with $Kge N$, the $N$ eigenvalues of $left(mathbf{B}_t+tmathbf{M}right)^*left(mathbf{B}_t+tmathbf{M}right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.
{"title":"On the singular values of complex matrix Brownian motion with a matrix drift","authors":"T. Assiotis","doi":"10.3150/22-bej1517","DOIUrl":"https://doi.org/10.3150/22-bej1517","url":null,"abstract":"Let $Mat_{mathbb{C}}(K,N)$ be the space of $Ktimes N$ complex matrices. Let $mathbf{B}_t$ be Brownian motion on $Mat_{mathbb{C}}(K,N)$ starting from the zero matrix and $mathbf{M}in Mat_{mathbb{C}}(K,N)$. We prove that, with $Kge N$, the $N$ eigenvalues of $left(mathbf{B}_t+tmathbf{M}right)^*left(mathbf{B}_t+tmathbf{M}right)$ form a Markov process with an explicit transition kernel. This generalizes a classical result of Rogers and Pitman for multidimensional Brownian motion with drift which corresponds to $N=1$. We then give two more descriptions for this Markov process. First, as independent squared Bessel diffusion processes in the wide sense, introduced by Watanabe and studied by Pitman and Yor, conditioned to never intersect. Second, as the distribution of the top row of interacting squared Bessel type diffusions in some interlacting array. The last two descriptions also extend to a general class of one-dimensional diffusions.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49483795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In his work cite{Ti80}, Tikhomirov combined elements of Stein's method with the theory of characteristic functions to derive Kolmogorov bounds for the convergence rate in the central limit theorem for a normalized sum of a stationary sequence of random variables satisfying one of several weak dependency conditions. The combination of elements of Stein's method with the theory of characteristic functions is sometimes called emph{Stein-Tikhomirov method}. citet*{AMPS17} successfully used the Stein-Tikhomirov method to bound the convergence rate in contexts with non-Gaussian targets. citet*{Ro17} used the Stein-Tikhomirov method to bound the convergence rate in the Kolmogorov distance for normal approximation of normalized triangle counts in the Erd"os-R'enyi random graph.
{"title":"Kolmogorov bounds for decomposable random variables and subgraph counting by the Stein–Tikhomirov method","authors":"P. Eichelsbacher, Benedikt Rednoss","doi":"10.3150/22-bej1522","DOIUrl":"https://doi.org/10.3150/22-bej1522","url":null,"abstract":"In his work cite{Ti80}, Tikhomirov combined elements of Stein's method with the theory of characteristic functions to derive Kolmogorov bounds for the convergence rate in the central limit theorem for a normalized sum of a stationary sequence of random variables satisfying one of several weak dependency conditions. The combination of elements of Stein's method with the theory of characteristic functions is sometimes called emph{Stein-Tikhomirov method}. citet*{AMPS17} successfully used the Stein-Tikhomirov method to bound the convergence rate in contexts with non-Gaussian targets. citet*{Ro17} used the Stein-Tikhomirov method to bound the convergence rate in the Kolmogorov distance for normal approximation of normalized triangle counts in the Erd\"os-R'enyi random graph.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48909584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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, $alphain (0,1)$. 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 $(X(t))_{tgeq 0}$ 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 $0$.
{"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":"https://doi.org/10.3150/22-bej1515","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, $alphain (0,1)$. 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 $(X(t))_{tgeq 0}$ 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":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47447993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with α ∈ (0 , 2] , to include both linear multifractional Brownian sheets ( α = 2 ) and linear multifractional stable sheets ( α < 2 ). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local Hölder condition of the local times in the set variable. Among the main results of this paper, Theorem 2.4 provides a sufficient and necessary condition for the existence of local times of LMSS; Theorem 3.1 shows a sufficient condition for the joint continuity of local times; and Theorem 4.1 proves a sharp local Hölder condition for the local times in the set variable. All these theorems improve significantly the existing results for the local times of multifractional Brownian sheets and linear multifractional stable sheets in the literature.
{"title":"Linear multifractional stable sheets in the broad sense: Existence and joint continuity of local times","authors":"Yujia Ding, Qidi Peng, Yimin Xiao","doi":"10.3150/22-bej1479","DOIUrl":"https://doi.org/10.3150/22-bej1479","url":null,"abstract":"We introduce the notion of linear multifractional stable sheets in the broad sense (LMSS) with α ∈ (0 , 2] , to include both linear multifractional Brownian sheets ( α = 2 ) and linear multifractional stable sheets ( α < 2 ). The purpose of the present paper is to study the existence and joint continuity of the local times of LMSS, and also the local Hölder condition of the local times in the set variable. Among the main results of this paper, Theorem 2.4 provides a sufficient and necessary condition for the existence of local times of LMSS; Theorem 3.1 shows a sufficient condition for the joint continuity of local times; and Theorem 4.1 proves a sharp local Hölder condition for the local times in the set variable. All these theorems improve significantly the existing results for the local times of multifractional Brownian sheets and linear multifractional stable sheets in the literature.","PeriodicalId":55387,"journal":{"name":"Bernoulli","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44725503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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