Journal of Theoretical Probability
Journal Prestige (SJR): 0.981 Citation Impact (citeScore): 1 Number of Followers: 3 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15729230  ISSN (Online) 08949840 Published by SpringerVerlag [2468 journals] 
 Probability and Moment Inequalities for Additive Functionals of
Geometrically Ergodic Markov Chains
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Abstract: Abstract In this paper, we establish moment and Bernsteintype inequalities for additive functionals of geometrically ergodic Markov chains. These inequalities extend the corresponding inequalities for independent random variables. Our conditions cover Markov chains converging geometrically to the stationary distribution either in weighted total variation norm or in weighted Wasserstein distances. Our inequalities apply to unbounded functions and depend explicitly on constants appearing in the conditions that we consider.
PubDate: 20240218

 Invariant Measures for the Nonlinear Stochastic Heat Equation with No
Drift Term
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Abstract: Abstract This paper deals with the longterm behavior of the solution to the nonlinear stochastic heat equation \(\frac{\partial u}{\partial t}  \frac{1}{2}\Delta u = b(u){\dot{W}}\) , where b is assumed to be a globally Lipschitz continuous function and the noise \({\dot{W}}\) is a centered and spatially homogeneous Gaussian noise that is white in time. We identify a set of nearly optimal conditions on the initial data, the correlation measure of the noise, and the weight function \(\rho \) , which together guarantee the existence of an invariant measure in the weighted space \(L^2_\rho ({\mathbb {R}}^d)\) . In particular, our result covers the parabolic Anderson model (i.e., the case when \(b(u) = \lambda u\) ) starting from the Dirac delta measure.
PubDate: 20240215

 Nonuniqueness Phase of Percolation on Reflection Groups in
$${\mathbb {H}^3}$$
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Abstract: Abstract We consider Bernoulli bond and site percolation on Cayley graphs of reflection groups in the threedimensional hyperbolic space \({\mathbb {H}^3}\) corresponding to a very large class of Coxeter polyhedra. In such setting, we prove the existence of a nonempty nonuniqueness percolation phase, i.e. that \(p_c < p_u\) . This means that for some values of the Bernoulli percolation parameter there are a.s. infinitely many infinite components in the percolation subgraph. The proof relies on upper estimates for the spectral radius of the graph and on a lower estimate for its growth rate. The latter estimate involves only the number of generators of the group and is proved in the article as well.
PubDate: 20240215

 Fractional Skellam Process of Order k

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Abstract: Abstract We introduce and study a fractional version of the Skellam process of order k by timechanging it with an independent inverse stable subordinator. We call it the fractional Skellam process of order k (FSPoK). An integral representation for its onedimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of FSPoK which are utilized to establish its longrange dependence property. Later, we consider two timechanged versions of the FSPoK. These are obtained by timechanging the FSPoK by an independent Lévy subordinator and its inverse. Some distributional properties and particular cases are discussed for these timechanged processes.
PubDate: 20240212

 Waiting Time for a Small Subcollection in the Coupon Collector Problem
with Universal Coupon
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Abstract: Abstract We consider a generalization of the classical coupon collector problem, where the set of available coupons consists of standard coupons (which can be part of the collection), and two coupons with special purposes: one that speeds up the collection process and one that slows it down. We obtain several asymptotic results related to the expectation and the variance of the waiting time until a portion of the collection is sampled, as the number of standard coupons tends to infinity.
PubDate: 20240121

 A Note on Transience of Generalized MultiDimensional Excited Random Walks

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Abstract: Abstract We consider a variant of the generalized excited random walk (GERW) in dimension \(d\ge 2\) where the lower bound on the drift for excited jumps is timedependent and decays to zero. We show that if the lower bound decays more slowly than \(n^{\beta _0}\) (n is time), where \(\beta _0\) depends on the transitions of the process, the GERW is transient in the direction of the drift.
PubDate: 20240111

 On Convergence of the Uniform Norm and Approximation for Stochastic
Processes from the Space $${\textbf{F}}_\psi (\Omega )$$
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Abstract: Abstract In this paper, we consider random variables and stochastic processes from the space \({\textbf{F}}_\psi (\Omega )\) and study approximation problems for such processes. The method of series decomposition of a stochastic process from \({\textbf{F}}_\psi (\Omega )\) is used to find an approximating process called a model. The rate of convergence of the model to the process in the uniform norm is investigated. We develop an approach for estimating the cutoff level of the model under given accuracy and reliability of the simulation.
PubDate: 20231220

 The TimeDependent Symbol of a Nonhomogeneous Itô Process and
Corresponding Maximal Inequalities
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Abstract: Abstract The probabilistic symbol is defined as the righthand side derivative at time zero of the characteristic functions corresponding to the onedimensional marginals of a timehomogeneous stochastic process. As described in various contributions to this topic, the symbol contains crucial information concerning the process. When leaving timehomogeneity behind, a modification of the symbol by inserting a time component is needed. In the present article, we show the existence of such a timedependent symbol for nonhomogeneous Itô processes. Moreover, for this class of processes, we derive maximal inequalities which we apply to generalize the Blumenthal–Getoor indices to the nonhomogeneous case. These are utilized to derive several properties regarding the paths of the process, including the asymptotic behavior of the sample paths, the existence of exponential moments and the finiteness of pvariationa. In contrast to many situations where nonhomogeneous Markov processes are involved, the spacetime process cannot be utilized when considering maximal inequalities.
PubDate: 20231219

 The Moduli of Continuity for Operator Fractional Brownian Motion

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Abstract: Abstract The almostsure sample path behavior of the operator fractional Brownian motion with exponent D, including multivariate fractional Brownian motion, is investigated. In particular, the global and the local moduli of continuity of the sample paths are established. These results show that the global and the local moduli of continuity of the sample paths are completely determined by the real parts of the eigenvalues of the exponent D, as well as the covariance matrix at some unit vector. These results are applicable to multivariate fractional Brownian motion.
PubDate: 20231215

 Strong Approximations for a Class of Dependent Random Variables with
SemiExponential Tails
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Abstract: Abstract We give rates of convergence in the almost sure invariance principle for sums of dependent random variables with semiexponential tails, whose coupling coefficients decrease at a subexponential rate. We show that the rates in the strong invariance principle are in powers of \(\log n\) . We apply our results to iid products of random matrices.
PubDate: 20231206

 Quantitative Russo–Seymour–Welsh for Random Walk on Random Graphs and
Decorrelation of Uniform Spanning Trees
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Abstract: Abstract We prove a quantitative Russo–Seymour–Welsh (RSW)type result for random walks on two natural examples of random planar graphs: the supercritical percolation cluster in \(\mathbb {Z}^2\) and the Poisson Voronoi triangulation in \(\mathbb {R}^2\) . More precisely, we prove that the probability that a simple random walk crosses a rectangle in the hard direction with uniformly positive probability is stretched exponentially likely in the size of the rectangle. As an application, we prove a near optimal decorrelation result for uniform spanning trees for such graphs. This is the key missing step in the application of the proof strategy of Berestycki et al. (Ann Probab 48(1):1–52, 2020) for such graphs [in Berestycki et al. (2020), random walk RSW was assumed to hold with probability 1]. Applications to almost sure Gaussianfree field scaling limit for dimers on Temperleyantype modification on such graphs are also discussed.
PubDate: 20231201
DOI: 10.1007/s10959023012487

 On Conditioning Brownian Particles to Coalesce

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Abstract: Abstract We introduce the notion of a conditional distribution to a zeroprobability event in a given direction of approximation and prove that the conditional distribution of a family of independent Brownian particles to the event that their paths coalesce after the meeting coincides with the law of a modified massive Arratia flow, defined in Konarovskyi (Ann Probab 45(5):3293–3335, 2017. https://doi.org/10.1214/16AOP1137).
PubDate: 20231201
DOI: 10.1007/s10959023012674

 Stochastic Dynamics of Generalized Planar Random Motions with Orthogonal
Directions
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Abstract: 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) onedimensional symmetric telegraph processes with rate function proportional to \(\lambda \) and velocity c/2. Finally, we extend the results to a wider class of orthogonaltype evolutions.
PubDate: 20231201
DOI: 10.1007/s10959022012292

 The Oscillating Random Walk on $$ {\mathbf {\mathbb {Z}}} $$

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Abstract: Abstract The paper is concerned with a new approach for the recurrence properties of the oscillating random walk on \(\mathbb {Z}\) in the sense of Kemperman. In the case when the random walk is ascending on \(\mathbb {Z}^\) and descending on \(\mathbb {Z}^+\) , the invariant measure of the embedded process of successive crossing times is explicitly determined, which yields a sufficient condition for recurrence. Finally, we make use of this result to show that the general oscillating random walk is recurrent under some moment assumptions.
PubDate: 20231201
DOI: 10.1007/s1095902301250z

 The Smallest Singular Value of a Shifted Random Matrix

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Abstract: Abstract Let \(R_n\) be an \(n \times n\) random matrix with i.i.d. subgaussian entries. Let M be an \(n \times n\) deterministic matrix with norm \(\Vert M \Vert \le n^\gamma \) where \(1/2<\gamma <1\) . The goal of this paper is to give a general estimate of the smallest singular value of the sum \(R_n + M\) , which improves an earlier result of Tao and Vu.
PubDate: 20231201
DOI: 10.1007/s10959023012558

 Structural Properties of Gibbsian Point Processes in Abstract Spaces

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Abstract: In the language of random counting measures, many structural properties of the Poisson process can be studied in arbitrary measurable spaces. We provide a similarly general treatise of Gibbs processes. With the GNZ equations as a definition of these objects, Gibbs processes can be introduced in abstract spaces without any topological structure. In this general setting, partition functions, Janossy densities, and correlation functions are studied. While the definition covers finite and infinite Gibbs processes alike, the finite case allows, even in abstract spaces, for an equivalent and more explicit characterization via a familiar series expansion. Recent generalizations of factorial measures to arbitrary measurable spaces, where counting measures cannot be written as sums of Dirac measures, likewise allow to generalize the concept of Hamiltonians. The DLR equations, which completely characterize a Gibbs process, as well as basic results for the local convergence topology, are also formulated in full generality. We prove a new theorem on the extraction of locally convergent subsequences from a sequence of point processes and use this statement to provide existence results for Gibbs processes in general spaces with potentially infinite range of interaction. These results are used to guarantee the existence of Gibbs processes with clusterdependent interactions and to prove a recent conjecture concerning the existence of Gibbsian particle processes.
PubDate: 20231201
DOI: 10.1007/s10959023012629

 Strong Local Nondeterminism and Exact Modulus of Continuity for Isotropic
Gaussian Random Fields on Compact TwoPoint Homogeneous Spaces
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Abstract: Abstract This paper is concerned with sample path properties of realvalued isotropic Gaussian fields on compact twopoint 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: 20231201
DOI: 10.1007/s10959022012318

 Fluctuation Moments Induced by Conjugation with Asymptotically Liberating
Random Matrix Ensembles
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Abstract: Abstract Independent Haarunitary random matrices and independent Haarorthogonal random matrices are known to be asymptotically liberating ensembles, and they give rise to asymptotic free independence when used for conjugation of constant matrices. G. Anderson and B. Farrel showed that a certain family of discrete random unitary matrices can actually be used to the same end. In this paper, we investigate fluctuation moments and higherorder moments induced on constant matrices by conjugation with asymptotically liberating ensembles. We show for the first time that the fluctuation moments associated with secondorder free independence can be obtained from conjugation with an ensemble consisting of signed permutation matrices and the discrete Fourier transform matrix. We also determine fluctuation moments induced by various related ensembles where we do not get known expressions but others related to traffic free independence.
PubDate: 20231201
DOI: 10.1007/s10959023012469

 Discretization of the Ergodic Functional Central Limit Theorem

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Abstract: 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 qweak order approximation (not necessarily stationary). This paper presents both the discretization method of order q for the time integral and the qorder 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: 20231201
DOI: 10.1007/s1095902301237w

 Asymptotic Properties of Random Contingency Tables with Uniform Margin

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Abstract: Abstract Let \(C\ge 2\) be a positive integer. Consider the set of \(n\times n\) nonnegative 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: 20231201
DOI: 10.1007/s10959022012345
