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Vitali Wachtel

Publications and source records attributed to Vitali Wachtel.

67 records · Page 4Linked to original sources
arXiv2011

Ordered random walks with heavy tails

This note continues paper of Denisov and Wachtel (2010), where we have constructed a $k$-dimensional random walk conditioned to stay in the Weyl chamber of type $A$. The construction was done under the assumption that the original random walk has $k-1$ moments. In this note we continue the study of killed random walks in the Weyl chamber, and assume that the tail of increments is regularly varying of index $α<k-1$. It appears that the asymptotic behaviour of random walks is different in this case. We determine the asymptotic behaviour of the exit time, and, using thisinformation, construct a conditioned process which lives on a partial compactification of the Weyl chamber.

math.PR↗
arXiv2010

Optimal local Hölder index for density states of superprocesses with $(1+β)$-branching mechanism

For $0<α\leq2$, a super-$α$-stable motion $X$ in $\mathsf{R}^d$ with branching of index $1+β\in(1,2)$ is considered. Fix arbitrary $t>0$. If $d<α/β$, a dichotomy for the density function of the measure $X_t$ holds: the density function is locally Hölder continuous if $d=1$ and $α>1+β$ but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal local Hölder index.

math.PR↗
arXiv2010

Trap models with vanishing drift: Scaling limits and aging regimes

We discuss the long term behaviour of trap models on the integers with asymptotically vanishing drift, providing scaling limit theorems and ageing results. Depending on the tail behaviour of the traps and the strength of the drift, we identify three different regimes, one of which features a previously unobserved limit process.

math.PR↗
arXiv2009

Transition phenomena for ladder epochs of random walks with small negative drift

For a family of random walks $\{S^{(a)}\}$ satisfying $\mathbf{E}S_1^{(a)}=-a<0$ we consider ladder epochs $τ^{(a)}=\min\{k\geq1: S_k^{(a)}<0\}$. We study the asymptotic, as $a\to0$, behaviour of $\mathbf{P}(τ^{(a)}>n)$ in the case when $n=n(a)\to\infty$. As a consequence we obtain also the growth rates of the moments of $τ^{(a)}$.

math.PR↗
arXiv2009

Heavy-traffic analysis of the maximum of an asymptotically stable random walk

For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka < 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for asymptotically stable random walks. This problem appeared first in the 1960's in the analysis of a single-server queue when the traffic load tends to 1 and since then is referred to as the heavy-traffic approximation problem. Kingman and Prokhorov suggested two different approaches which were later followed by many authors. We give two elementary proofs of our main result, using each of these approaches. It turns out that the main technical difficulties in both proofs are rather similar and may be resolved via a generalisation of the Kolmogorov inequality to the case of an infinite variance. Such a generalisation is also obtained in this note.

math.PR↗
arXiv2007

Local probabilities for random walks conditioned to stay positive

Let S_0=0,{S_n, n>0} be a random walk generated by a sequence of i.i.d. random variables X_1,X_2,... and let τ^{-} be the first descending ladder epoch. Assuming that the distribution of X_1 belongs to the domain of attraction of an α-stable law we study the asymptotic behavior of the local probabilities P(τ^{-}=n) and the conditional local probabilities P(S_n\in [x,x+y)|τ^{-}>n) for fixed y and x=x(n)\in (0,\infty).

math.PR↗
arXiv2007

Local limit theorems for ladder epochs

Let {S_n, n=0,1,2,...} be a random walk generated by a sequence of i.i.d. random variables X_1, X_2,... and let tau be the first descending ladder epoch. Assuming that the distribution of X_1 belongs to the domain of attraction of an alpha-stable law, we study the asymptotic behavior of P(tau=n).

math.PR↗
arXiv2006

Moderate deviations for random walk in random scenery

We investigate random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramer's condition. We prove moderate deviation principles in dimensions two and larger, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. In the case of dimension four and larger we even obtain precise asymptotics for the annealed probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. In dimension three and larger, an important ingredient in the proofs are new concentration inequalities for self-intersection local times of random walks, which are of independent interest, whilst in dimension two we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

math.PR↗
arXiv2006

Large deviations for sums defined on a Galton-Watson process

In this paper we study the large deviation behavior of sums of i.i.d. random variables X_i defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX_1^2 and EZ_1log Z_1. The underlying interplay of the partial sums of the X_i and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on Z we recently published in [FW06].

math.PR↗
arXiv2006

Critical Galton-Watson processes: The maximum of total progenies within a large window

Consider a critical Galton-Watson process Z={Z_n: n=0,1,...} of index 1+alpha, alpha in (0,1]. Let S_k(j) denote the sum of the Z_n with n in the window [k,...,k+j), and M_m(j) the maximum of the S_k with k moving in [0,m-j]. We describe the asymptotic behavior of the expectation EM_m(j) if the window width j=j_m is such that j/m converges in [0,1] as m tends to infinity. This will be achieved via establishing the asymptotic behavior of the tail probabilities of M_{infinity}(j).

math.PR↗
arXiv2005

Hydrodynamic limit fluctuations of super-Brownian motion with a stable catalyst

We consider the behaviour of a continuous super-Brownian motion catalysed by a random medium with infinite overall density under the hydrodynamic scaling of mass, time, and space. We show that, in supercritical dimensions, the scaled process converges to a macroscopic heat flow, and the appropriately rescaled random fluctuations around this macroscopic flow are asymptotically bounded, in the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck processes. The most interesting new effect we observe is the occurrence of an index-jump from a 'Gaussian' situation to stable fluctuations of index 1+gamma, where gamma is an index associated to the medium.

math.PR↗
arXiv2005

Lower deviation probabilities for supercritical Galton-Watson processes

There is a well-known sequence of constants c_n describing the growth of supercritical Galton-Watson processes Z_n. With 'lower deviation probabilities' we refer to P(Z_n=k_n) with k_n=o(c_n) as n increases. We give a detailed picture of the asymptotic behavior of such lower deviation probabilities. This complements and corrects results known from the literature concerning special cases. Knowledge on lower deviation probabilities is needed to describe large deviations of the ratio Z_{n+1}/Z_n. The latter are important in statistical inference to estimate the offspring mean. For our proofs, we adapt the well-known Cramer method for proving large deviations of sums of independent variables to our needs.

math.PR↗