360,464 research outputs found
Limit properties of exceedances point processes of scaled stationary Gaussian sequences
We derive the limiting distributions of exceedances point processes of
randomly scaled weakly dependent stationary Gaussian sequences under some mild
asymptotic conditions. In the literature analogous results are available only
for contracted stationary Gaussian sequences. In this paper, we include
additionally the case of randomly inflated stationary Gaussian sequences with a
Weibullian type random scaling. It turns out that the maxima and minima of both
contracted and inflated weakly dependent stationary Gaussian sequences are
asymptotically independent.Comment: 1
Multivariate limit theorems in the context of long-range dependence
We study the limit law of a vector made up of normalized sums of functions of
long-range dependent stationary Gaussian series. Depending on the memory
parameter of the Gaussian series and on the Hermite ranks of the functions, the
resulting limit law may be (a) a multivariate Gaussian process involving
dependent Brownian motion marginals, or (b) a multivariate process involving
dependent Hermite processes as marginals, or (c) a combination. We treat cases
(a), (b) in general and case (c) when the Hermite components involve ranks 1
and 2. We include a conjecture about case (c) when the Hermite ranks are
arbitrary
Couplings and Strong Approximations to Time Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions
We define a time dependent empirical process based on i.i.d.~fractional
Brownian motions and establish Gaussian couplings and strong approximations to
it by Gaussian processes. They lead to functional laws of the iterated
logarithm for this process.Comment: To appear in the Journal of Theoretical Probability. 37 pages.
Corrected version. The results on quantile processes are taken out and it
will appear elsewher
Functional central limit theorem for negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows
We prove a functional central limit theorem for partial sums of symmetric
stationary long range dependent heavy tailed infinitely divisible processes
with a certain type of negative dependence. Previously only positive dependence
could be treated. The negative dependence involves cancellations of the
Gaussian second order. This leads to new types of limiting processes involving
stable random measures, due to heavy tails, Mittag-Leffler processes, due to
long memory, and Brownian motions, due to the Gaussian second order
cancellations.Comment: 35 page
Short-range dependent processes subordinated to the Gaussian may not be strong mixing
There are all kinds of weak dependence. For example, strong mixing.
Short-range dependence (SRD) is also a form of weak dependence. It occurs in
the context of processes that are subordinated to the Gaussian. Is a SRD
process strong mixing if the underlying Gaussian process is long-range
dependent? We show that this is not necessarily the case.Comment: 3 page
Central Limit Theorems for Supercritical Superprocesses
In this paper, we establish a central limit theorem for a large class of
general supercritical superprocesses with spatially dependent branching
mechanisms satisfying a second moment condition. This central limit theorem
generalizes and unifies all the central limit theorems obtained recently in
Mi{\l}o\'{s} (2012, arXiv:1203:6661) and Ren, Song and Zhang (2013, to appear
in Acta Appl. Math., DOI 10.1007/s10440-013-9837-0) for supercritical super
Ornstein-Uhlenbeck processes. The advantage of this central limit theorem is
that it allows us to characterize the limit Gaussian field. In the case of
supercritical super Ornstein-Uhlenbeck processes with non-spatially dependent
branching mechanisms, our central limit theorem reveals more independent
structures of the limit Gaussian field
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