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Abstract: The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly appears quite naturally in ergodic theory in the study of measure-preserving transformations arising from various group actions or, equivalently, the study of stationary sequences when adopting a probabilistic perspective as in this paper. Our purpose is to give a new and relatively short proof of the coboundary theorem due to Schmidt (Cocycles on ergodic transformation groups. Macmillan lectures in mathematics, vol 1, Macmillan Company of India, Ltd., Delhi, 1977) which provides a sharp criterion that determines (and rules out) when two stationary processes belong to the same null-homology equivalence class. We also discuss various aspects of null-homology within the class of Markov random walks and compare null-homology with a formally stronger notion which we call strict-sense null-homology. Finally, we also discuss some concrete cases where the notion of null-homology turns up in a relevant manner. PubDate: 2023-03-01

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Abstract: Let \(\Phi \) be a nuclear space, and let \(\Phi '\) denote its strong dual. In this paper, we introduce sufficient conditions for the almost sure uniform convergence on bounded intervals of time for a sequence of \(\Phi '\) -valued processes having continuous (respectively, càdlàg) paths. The main result is formulated first in the general setting of cylindrical processes but later specialized to other situations of interest. In particular, we establish conditions for the convergence to occur in a Hilbert space continuously embedded in \(\Phi '\) . Furthermore, in the context of the dual of an ultrabornological nuclear space (like spaces of smooth functions and distributions) we also include applications to convergence in \(L^{r}\) uniformly on a bounded interval of time, to the convergence of a series of independent càdlàg processes, and to the convergence of solutions to linear stochastic evolution equations driven by Lévy noise. PubDate: 2023-03-01

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Abstract: In this paper, we study the discretization of the ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see Bhattacharya in Z Wahrscheinlichkeitstheorie Verwandte Geb 60:185–201, 1982) which states the following: Given a stationary and ergodic Markov process \((X_t)_{t \geqslant 0}\) with unique invariant measure \(\nu \) and infinitesimal generator A, then, for every smooth enough function f, \((n^{1/2} \frac{1}{n}\int _0^{nt} Af(X_s){\textrm{d}}s)_{t \geqslant 0}\) converges in distribution towards the distribution of the process \((\sqrt{-2 \langle f, Af \rangle _{\nu }} W_{t})_{t \geqslant 0}\) with \((W_{t})_{t \geqslant 0}\) a Wiener process. In particular, we consider the marginal distribution at fixed \(t=1\) , and we show that when \(\int _0^{n} Af(X_s)ds\) is replaced by a well chosen discretization of the time integral with order q (e.g. Riemann discretization in the case \(q=1\) ), then the CLT still holds but with rate \(n^{q/(2q+1)}\) instead of \(n^{1/2}\) . Moreover, our results remain valid when \((X_t)_{t \geqslant 0}\) is replaced by a q-weak order approximation (not necessarily stationary). This paper presents both the discretization method of order q for the time integral and the q-order ergodic CLT we derive from them. We finally propose applications concerning the first order CLT for the approximation of Markov Brownian diffusion stationary regimes with Euler scheme (where we recover existing results from the literature) and the second order CLT for the approximation of Brownian diffusion stationary regimes using Talay’s scheme (Talay in Stoch Stoch Rep 29:13–36, 1990) of weak order two. PubDate: 2023-03-01

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Abstract: In this paper, under suitable assumptions on the Gaussian process \(G=\lbrace G_t,\,t\ge 0\rbrace \) , we establish results on uniform convergence in probability and in law stably for the realized power variation of the Riemann–Stieljes integral \(Z_t=\int _0^t u_s \text {d}Y_{s,G}^{(1)}\) with respect to \({Y_{t,G}^{(1)}}=\int _0^t \text {e}^{-s} \text {d}G_{a(s)}\) , where u is a process of finite q-variation with \(q<1/(1-\alpha )\) , \(\alpha \in (0,1)\) and \(a(t)=\alpha \text {e}^{\frac{t}{\alpha }}\) . To illustrate the results, we show that the required conditions on G are satisfied for processes including fractional Brownian motion with Hurst parameter \(\alpha \in (0,1)\) , subfractional Brownian motion of index \(\alpha \in (0,1/2)\) and bifractional Brownian motion of parameters \((\alpha , K)\in (0,1/2)\times (0,1]\) . Furthermore, we apply our results to construct an estimator for the integrated volatility parameter in an Ornstein–Uhlenbeck model driven by \(Y_{G}^{(1)}\) . PubDate: 2023-02-15

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Abstract: The Sylvester index of a random Hermitian matrix in the Gaussian ensemble has been considered by Dean and Majumdar. We consider this Sylvester index for a matrix ensemble of random Hermitian matrices defined by a probability density of the form \(\exp \bigl (-\textrm{tr}\, Q(x))\bigr )\) , where Q is a convex polynomial. The main result is the determination of the statistical distribution of the eigenvalues under the condition of a prescribed Sylvester index. We revisit some known results, giving complete proofs, for which we use logarithmic potential theory and complex analysis. PubDate: 2023-02-03

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Abstract: In this paper, we study the numerical method for approximating the random periodic solution of semilinear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating infinite-dimensional objects. We first show the existence and uniqueness of the random periodic solution to the equation as the limit of the pull-back flows of the equation, and observe that its mild form is well defined in the intersection of a family of decreasing Hilbert spaces. Then, we propose a Galerkin-type exponential integrator scheme and establish its convergence rate of the strong error to the mild solution, where the order of convergence directly depends on the space (among the family of Hilbert spaces) for the initial point to live. We finally conclude with a best order of convergence that is arbitrarily close to 0.5. PubDate: 2023-01-25

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Abstract: In this paper, we study the moderate deviations principle (MDP) for slow–fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm) with Hurst parameter \(H\in (1/2,1)\) . We derive conditions on the moderate deviations scaling and on the Hurst parameter H under which the MDP holds. In addition, we show that in typical situations the resulting action functional is discontinuous in H at \(H=1/2\) , suggesting that the tail behavior of stochastic dynamical systems perturbed by fBm can have different characteristics than the tail behavior of such systems that are perturbed by standard Brownian motion. PubDate: 2023-01-16

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Abstract: Let \(C\ge 2\) be a positive integer. Consider the set of \(n\times n\) non-negative integer matrices whose row sums and column sums are all equal to Cn and let \(X=(X_{ij})_{1\le i,j\le n}\) be uniformly distributed on this set. This X is called the random contingency table with uniform margin. In this paper, we study various asymptotic properties of \(X=(X_{ij})_{1\le i,j\le n}\) as \(n\rightarrow \infty \) . PubDate: 2023-01-09

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Abstract: We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and nonnegative integrands, our estimates involve only the third moment of the integrand in addition to a variance term using a squared norm of the integrand. As a consequence, we are able to observe a “third moment phenomenon” in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time or have been obtained only for specific kernels such as the exponential or Erlang kernels. PubDate: 2023-01-07 DOI: 10.1007/s10959-022-01233-6

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Abstract: We study the Freidlin–Wentzell large deviation principle for the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise: $$\begin{aligned} \frac{\partial u^{{\varepsilon }}(t,x)}{\partial t}=\frac{\partial ^2 u^{{\varepsilon }}(t,x)}{\partial x^2}+\sqrt{{\varepsilon }}\sigma (t, x, u^{{\varepsilon }}(t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb {R}, \end{aligned}$$ where \(\dot{W}\) is white in time and fractional in space with Hurst parameter \(H\in \left( \frac{1}{4},\frac{1}{2}\right) \) . Recently, Hu and Wang (Ann Inst Henri Poincaré Probab Stat 58(1):379–423, 2022) have studied the well-posedness of this equation without the technical condition of \(\sigma (0)=0\) which was previously assumed in Hu et al. (Ann Probab 45(6):4561–4616, 2017). We adopt a new sufficient condition proposed by Matoussi et al. (Appl Math Optim 83(2):849–879, 2021) for the weak convergence criterion of the large deviation principle. PubDate: 2023-01-06 DOI: 10.1007/s10959-022-01228-3

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Abstract: This paper studies transition probabilities from a Borel subset of a Polish space to a product of two Borel subsets of Polish spaces. For such transition probabilities it introduces and studies the property of semi-uniform Feller continuity. This paper provides several equivalent definitions of semi-uniform Feller continuity and establishes its preservation under integration. The motivation for this study came from the theory of Markov decision processes with incomplete information, and this paper provides the fundamental results useful for this theory. PubDate: 2023-01-06 DOI: 10.1007/s10959-022-01230-9

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Abstract: We study planar random motions with finite velocities, of norm \(c>0\) , along orthogonal directions and changing at the instants of occurrence of a nonhomogeneous Poisson process with rate function \(\lambda = \lambda (t),\ t\ge 0\) . We focus on the distribution of the current position \(\bigl (X(t), Y(t)\bigr ),\ t\ge 0\) , in the case where the motion has orthogonal deviations and where also reflection is admitted. In all the cases, the process is located within the closed square \(S_{ct}=\{(x,y)\in {\mathbb {R}}^2\,:\, x + y \le ct\}\) and we obtain the probability law inside \(S_{ct}\) , on the edge \(\partial S_{ct}\) and on the other possible singularities, by studying the partial differential equations governing all the distributions examined. A fundamental result is that the vector process (X, Y) is probabilistically equivalent to a linear transformation of two (independent or dependent) one-dimensional symmetric telegraph processes with rate function proportional to \(\lambda \) and velocity c/2. Finally, we extend the results to a wider class of orthogonal-type evolutions. PubDate: 2023-01-04 DOI: 10.1007/s10959-022-01229-2

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Abstract: We completely characterize when the free effective resistance of an infinite graph whose vertices have finite degrees can be expressed in terms of simple hitting probabilities of the random walk on the graph. PubDate: 2023-01-04 DOI: 10.1007/s10959-022-01218-5

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Abstract: This paper is concerned with sample path properties of real-valued isotropic Gaussian fields on compact two-point homogeneous spaces. In particular, we establish the property of strong local nondeterminism of an isotropic Gaussian field and then exploit this result to establish an exact uniform modulus of continuity for its sample paths. PubDate: 2023-01-03 DOI: 10.1007/s10959-022-01231-8

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Abstract: In the paper by Fan (Inf Dim Anal Quant Probab Rel Topics 9:451–469, 2006), he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved that the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan’s in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although the two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan’s result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free Lévy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free Lévy processes in terms of their free cumulant transforms is also given, and several examples are discussed. PubDate: 2022-12-29 DOI: 10.1007/s10959-022-01227-4

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Abstract: The purpose of this paper is to establish the convergence in distribution of the normalized error in the Euler approximation scheme for stochastic Volterra equations driven by a standard Brownian motion, with a kernel of the form \((t-s)^\alpha \) , where \(\alpha \in \left( -\frac{1}{2}, \frac{1}{2}\right) \) . PubDate: 2022-12-23 DOI: 10.1007/s10959-022-01222-9

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Abstract: We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables. (For the strong law of large numbers, we suppose only pairwise independence instead of (total) independence.) The class of deviation means is a special class of M-estimators or more generally extremum estimators, which are well studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for M-estimators in the literature. In particular, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasi-arithmetic means due to de Carvalho (Am Stat 70(3):270–274, 2016) and the ones for Bajraktarević means due to Barczy and Burai (Aequ Math 96(2):279–305, 2022). PubDate: 2022-12-23 DOI: 10.1007/s10959-022-01225-6

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Abstract: Let X(t) be an ergodic stationary random process or an ergodic homogeneous random field on \({\mathbb {R}}^m,m\ge 2\) , and let M(B) be a mixing homogeneous locally finite random Borel measure with mean density \(\gamma \) on \({\mathbb {R}}^m,m\ge 1\) . We assume that X and M are independent and possess finite expectations. If \(\{T_n\}\) is an increasing sequence of bounded convex sets, containing balls of radii \(r_n\rightarrow \infty \) , then $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\lambda (T_n)}\int _{T_n}X(t)M(\textrm{d}t,w)=\gamma E[X(0)]\text { a.s. and in } L^1. \end{aligned}$$ Special cases are ergodic theorems with averages over finite random sets. Example: If S is an independent-of-X Poisson random set in \({\mathbb {R}}^m\) with mean density \(\gamma \) , then $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{ {\lambda (T_n)}}\sum _{t\in S \cap T_n}X(t)=\gamma E[X(0 )] \ \text {a.s. and in } L^1 \ \ (\text {card} (S\cap T_n)<\infty \ \text {a.s.}). \end{aligned}$$ These theorems offer a universal way of constructing consistent estimators using observations on finite sets. PubDate: 2022-12-23 DOI: 10.1007/s10959-022-01226-5

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Abstract: In this paper, we study multi-dimensional reflected backward stochastic differential equations (BSDEs) with diagonally quadratic generators. Using the comparison theorem for diagonally quadratic BSDEs established recently in Luo (Disc Contin Dyn Syst 41(6):2543–2557, 2021), we obtain the existence and uniqueness of a solution by a penalization method. Moreover, we provide a comparison theorem. PubDate: 2022-12-19 DOI: 10.1007/s10959-022-01224-7

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Abstract: Under natural conditions, we prove exponential ergodicity in the \( L_1\) -Wasserstein distance of two-type continuous-state branching processes in Lévy random environments with immigration. Furthermore, we express precisely the parameters of the exponent. The coupling method and the conditioned branching property play an important role in the approach. Using the tool of superprocesses, ergodicity in total variation distance is also proved. PubDate: 2022-12-19 DOI: 10.1007/s10959-022-01211-y