139 research outputs found
Convergence of the all-time supremum of a L\'evy process in the heavy-traffic regime
In this paper we derive a technique of obtaining limit theorems for suprema
of L\'evy processes from their random walk counterparts. For each , let
be a sequence of independent and identically distributed
random variables and be a L\'evy processes such that
, and as . Let .
Then, under some mild assumptions, , for some random variable and some function
. We utilize this result to present a number of limit theorems
for suprema of L\'evy processes in the heavy-traffic regime
Homomorphisms from functional equations: the Goldie equation
The theory of regular variation, in its Karamata and Bojani´c-Karamata/de Haan forms, is long established and makes essential use of the Cauchy functional equation. Both forms are subsumed within the recent theory of Beurling regular variation, developed elsewhere. Various generalizations of the Cauchy equation, including the Gołab–Schinzel functional equation (GS) and Goldie's equation (GBE) below, are prominent there. Here we unify their treatment by algebraicization: extensive use of group structures introduced by Popa and Javor in the 1960s turn all the various (known) solutions into homomorphisms, in fact identifying them 'en passant', and show that (GS) is present everywhere, even if in a thick disguise
On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data
Consider an inviscid Burgers equation whose initial data is a Levy a-stable
process Z with a > 1. We show that when Z has positive jumps, the Hausdorff
dimension of the set of Lagrangian regular points associated with the equation
is strictly smaller than 1/a, as soon as a is close to 1. This gives a negative
answer to a conjecture of Janicki and Woyczynski. Along the way, we contradict
a recent conjecture of Z. Shi about the lower tails of integrated stable
processes
Stationary Random Fields on the Unitary Dual of a Compact Group
We generalise the notion of wide-sense stationarity from sequences of complex-valued random variables indexed by the integers, to fields of random variables that are labelled by elements of the unitary dual of a compact group. The covariance is positive definite, and so it is the Fourier transform of a finite central measure (the spectral measure of the field) on the group. Analogues of the Cramer and Kolmogorov theorems are extended to this framework. White noise makes sense in this context and so, for some classes of group, we can construct time series and investigate their stationarity. Finally we indicate how these ideas fit into the general theory of stationary random fields on hypergroups
Weak convergence of Vervaat and Vervaat Error processes of long-range dependent sequences
Following Cs\"{o}rg\H{o}, Szyszkowicz and Wang (Ann. Statist. {\bf 34},
(2006), 1013--1044) we consider a long range dependent linear sequence. We
prove weak convergence of the uniform Vervaat and the uniform Vervaat error
processes, extending their results to distributions with unbounded support and
removing normality assumption
Gaussian queues in light and heavy traffic
In this paper we investigate Gaussian queues in the light-traffic and in the
heavy-traffic regime. The setting considered is that of a centered Gaussian
process with stationary increments and variance
function , equipped with a deterministic drift ,
reflected at 0: We
study the resulting stationary workload process
in the limiting regimes (heavy
traffic) and (light traffic). The primary contribution is that we
show for both limiting regimes that, under mild regularity conditions on the
variance function, there exists a normalizing function such that
converges to a non-trivial
limit in
Limit theorems for weakly subcritical branching processes in random environment
For a branching process in random environment it is assumed that the
offspring distribution of the individuals varies in a random fashion,
independently from one generation to the other. Interestingly there is the
possibility that the process may at the same time be subcritical and,
conditioned on nonextinction, 'supercritical'. This so-called weakly
subcritical case is considered in this paper. We study the asymptotic survival
probability and the size of the population conditioned on non-extinction. Also
a functional limit theorem is proven, which makes the conditional
supercriticality manifest. A main tool is a new type of functional limit
theorems for conditional random walks.Comment: 35 page
Fractional moment bounds and disorder relevance for pinning models
We study the critical point of directed pinning/wetting models with quenched
disorder. The distribution K(.) of the location of the first contact of the
(free) polymer with the defect line is assumed to be of the form
K(n)=n^{-\alpha-1}L(n), with L(.) slowly varying. The model undergoes a
(de)-localization phase transition: the free energy (per unit length) is zero
in the delocalized phase and positive in the localized phase. For \alpha<1/2 it
is known that disorder is irrelevant: quenched and annealed critical points
coincide for small disorder, as well as quenched and annealed critical
exponents. The same has been proven also for \alpha=1/2, but under the
assumption that L(.) diverges sufficiently fast at infinity, an hypothesis that
is not satisfied in the (1+1)-dimensional wetting model considered by Forgacs
et al. (1986) and Derrida et al. (1992), where L(.) is asymptotically constant.
Here we prove that, if 1/21, then quenched and annealed
critical points differ whenever disorder is present, and we give the scaling
form of their difference for small disorder. In agreement with the so-called
Harris criterion, disorder is therefore relevant in this case. In the marginal
case \alpha=1/2, under the assumption that L(.) vanishes sufficiently fast at
infinity, we prove that the difference between quenched and annealed critical
points, which is known to be smaller than any power of the disorder strength,
is positive: disorder is marginally relevant. Again, the case considered by
Forgacs et al. (1986) and Derrida et al. (1992) is out of our analysis and
remains open.Comment: 20 pages, 1 figure; v2: few typos corrected, references revised. To
appear on Commun. Math. Phy
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