Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval

Let $ \mathcal{T} $ be a positive random variable independent of a real-valued stochastic process $ \left\{ {X(t),t\geqslant 0} \right\} $ . In this paper, we investigate the asymptotic behavior of $ \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right) $ as u→∞ assuming that X is a strongly dependent stationary Gaussian process and $ \mathcal{T} $ has a regularly varying survival function at infinity with index λ ∈ [0, 1). Under asymptotic restrictions on the correlation function of the process, we show that $ \mathrm{P}\left( {{\sup_{{t\in \left[ {0,\mathcal{T}} \right]}}}X(t)>u} \right)={c^{\lambda }}\mathrm{P}\left( {\mathcal{T}>m(u)} \right)\left( {1+o(1)} \right) $ with some positive finite constant c and function m(·) defined in terms of the local behavior of the correlation function and the standard Gaussian distribution.     

Exact asymptotics of supremum of a stationary Gaussian process over a random interval

Let {X(t):t∈[0,∞)} be a centered stationary Gaussian process. We study the exact asymptotics of P(sup_s∈[0,T]X(s)>u), as u→∞, where T is an independent of {X(t)} nonnegative random variable. It appears that the heaviness of T impacts the form of the asymptotics, leading to three scenarios: the case of integrable T, the case of T having regularly varying tail distribution with parameter λ∈(0,1) and the case of T having slowly varying tail distribution.     

Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals

Let {X(t),t ≥ 0} be a centered Gaussian process and let γ be a non-negative constant. In this paper we study the asymptotics of \(\mathbb {P} \left \{\underset {t\in [0,\mathcal {T}/u^{\gamma }]}\sup X(t)>u\right \}\) as \(u\rightarrow \infty \) , with \(\mathcal {T}\) an independent of X non-negative random variable. As an application, we derive the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes.     

Limit theorems with refinements for random processes

One considers the problem of the derivation of limit theorems with refinements in functional spaces. One proves theorems on the expansions of the mathematical expectations of bounded continuous linear functionals of the trajectories of a Gaussian random process. From these theorems one derives a limit theorem with correction terms for the mathematical expectation of a functional of the trajectories of the time-discretized Wiener process, when the step of the discretization tends to zero. One discusses questions regarding generalizations, methods of proof, and the relation of these kind of limit theorems with other problems of the theory of probability, as well as possible applications of these theorems.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 94–114, 1986.     

Limit theorems for random processes with random time substitution

In this paper we prove a theorem on sufficient conditions for the convergence in the Skorokhod space D[0, 1] of a sequence of random processes with random time substitution. We obtain almost sure versions of this theorem.     

Limit theorems for functionals of two independent Gaussian processes

Under certain mild conditions, some limit theorems for functionals of two independent Gaussian processes are obtained. The results apply to general Gaussian processes including fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. A new and interesting phenomenon is that, in comparison with the results for fractional Brownian motion, extra randomness appears in the limiting distributions for Gaussian processes with nonstationary increments, say sub-fractional Brownian motion and bi-fractional Brownian. The results are obtained based on the method of moments, in which Fourier analysis, the chaining argument introduced in [11] and a pairing technique are employed.     

Limit theorems for random transformations and processes in random environments

I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.

     

Limit theorems for random walks

We consider a random walk $S_{\tau }$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau }$ converges in the Skorohod space to the symmetric $\alpha$-stable process $B^{\alpha }$. We also prove asymptotic formula for the transition function of $S_{\tau }$ similar to the Pólya’s asymptotic formula for $B^{\alpha }$.     

Limit theorems for random processes constructed from sums of independent identically distributed random variables

There are proved limit theorems for random processes constructed from sums of independent identically distributed random variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 1, pp. 141–145, January, 1991.     

The supremum of Gaussian processes with a constant variance

Summary We obtain an estimate of the distribution of the large values of the supremum of a sample bounded Gaussian process having a constant variance. This estimate uses the entropy function of the parameter space endowed, as usual, with the pseudo-metric induced by the L ²-norm of the increments of the process.     

Limit theorems for empirical processes

Summary The empirical measure P _n for iid sampling on a distribution P is formed by placing mass n ^–1 at each of the first n observations. Generalizations of the classical Glivenko-Cantelli theorem for empirical measures have been proved by Vapnik and ervonenkis using combinatorial methods. They found simple conditions on a class C to ensure that sup {|P _n (C) – P(C)|: C C} converges in probability to zero. They used a randomization device that reduced the problem to finding exponential bounds on the tails of a hypergeometric distribution. In this paper an alternative randomization is proposed. The role of the hypergeometric distribution is thereby taken over by the binomial distribution, for which the elementary Bernstein inequalities provide exponential boundson the tails. This leads to easier proofs of both the basic results of Vapnik-ervonenkis and the extensions due to Steele. A similar simplification is made in the proof of Dudley's central limit theorem forn ^1/2(P P _n –P)— a result that generalizes Donsker's functional central limit theorem for empirical distribution functions.This research was supported in part by the Air Force Office of Scientific Research, Contract No. F49620-79-C-0164     

The limit theorems for maxima of stationary Gaussian processes with random Index

Let {X(t), t ≥ 0} be a standard(zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M(T) = max{X(t), t∈ [0, T ]} with random index TT, where TT /T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M(TT) exists under some additional conditions related to the correlation function r(·).     

Limit theorems for functionals of Gaussian vectors

Operator self-similar processes, as an extension of self-similar processes, have been studied extensively. In this work, we study limit theorems for functionals of Gaussian vectors. Under some conditions, we determine that the limit of partial sums of functionals of a stationary Gaussian sequence of random vectors is an operator self-similar process.     

Limit theorems for a class of diffusion processes

We consider a diffusion process {x(t)} on a compact Riemannian manifold with generator δ/2 + b. A current‐valued continuous stochastic process {X _t} in the sense of Itô [8] corresponds to {x(t)} by considering the stochastic line integral X _t(a) along {x(t)} for every smooth 1-form a. Furthermore {X _t} is decomposed into the martingale part and the bounded variation part as a current-valued continuous process. We show the central limit theorems for {X _t} and the martingale part of {X _t}. Occupation time laws for recurrent diffusions and homogenization problems of periodic diffusions are closely related to these theorems     

Limit theorems for stationary random evolutions

On a separable Banach space, let A(ξ₁),A(ξ₂),... be a strictly stationary sequence of infinitesimal operators, centered so that EA(ξ_i) = 0, i = 1,2,.... This paper characterizes the limit of the random evolutions

as the solution to a martingale problem. This work is a direct extension of previous work on i.i.d. random evolutions.     

Limit theorems for random symmetric functions

In this paper, based on theorems for limit distributions of empirical power processes for the i.i.d. case and for the case with independent triangular arrays of random variables, we prove limit theorems for U- and V-statistics determined by generalized polynomial kernel functions. We also show that under some natural conditions the limit distributions can be represented as functionals on the limit process of the normed empirical power process. We consider the one-sample case, as well as multi-sample cases. Dedicated to Professor V. M. Zolotarev on his sixty-fifth birthday. Supported by the Hungarian National Foundation for Scientific Research (grant No. T1666). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Estimates of the supremum distribution for a certain class of random processes

Exponential estimates of the tails of supremum distributions are obtained for a certain class of pre-Gaussian random processes. The results obtained are applied to the quadratic forms of Gaussian processes and to processes representable as stochastic integrals of processes with independent increments.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 596–608, May, 1993.