Precise large deviations for random walk in random environment

posted 12 Oct 2017, 00:53 by Piotr Dyszewski   [ updated 12 Oct 2017, 00:53 ]

An article written in collaboration with D. Buraczewski. arXiv

Abstract:
We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime.

Precise large deviation estimates for branching process in random environment

posted 15 Jun 2017, 01:49 by Piotr Dyszewski   [ updated 15 Jun 2017, 01:50 ]

Article written in collabortaion with D. Buraczewski avaliable on arXiv

Abstract:
We consider the branching process in random environment {Zn}n0, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We describe precise asymptotics of upper large deviations, i.e. P[Zn>eρn]. Moreover in the subcritical case, under the Cram\'er condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size.

Iterated random functions and regularly varying tails

posted 15 Jun 2017, 01:44 by Piotr Dyszewski   [ updated 15 Jun 2017, 01:46 ]

Article written in collaboration with E. Damek available on arXiv

Abstract:
We consider solutions to so-called stochastic fixed point equation R=dΨ(R), where Ψ is a random Lipschitz function and R is a random variable independent of Ψ. Under the assumption that Ψ can be approximated by the function xAx+B we show that the tail of R is comparable with the one of A, provided that the distribution of log(A1) is tail equivalent. In particular we obtain new results for the random difference equation.

On perpetuities with gamma-like tails

posted 8 Mar 2017, 00:13 by Piotr Dyszewski   [ updated 8 Mar 2017, 00:13 ]

An article written together with D. Buraczewski, A. Marynych and A. Iksanov available on arXiv.


Abstract: 
An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We give three disjoint groups of sufficient conditions which ensure that the distribution right tail of a perpetuity P{X>x} is asymptotic to axce−bx as x→∞ for some a,b>0 and c∈R. Our results complement those of Denisov and Zwart [J. Appl. Probab. 44 (2007), 1031--1046]. As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential moments of perpetuities. Several examples are given in which the distributions of perpetuities are explicitly identified.

Local fluctuations of critical Mandelbrot cascades

posted 13 Apr 2016, 01:12 by Piotr Dyszewski   [ updated 13 Apr 2016, 01:12 ]

An article written together with Dariusz Buraczewski and Konrad Kolesko available on arXiv.

Abstract:
We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure μ fluctuates. For any given point x, denoting by Bn(x) the ball of radius 2−n centered around x we present optimal lower and upper estimates of μ(Bn(x)) as n→∞.


Thin tails of fixed points of the nonhomogeneous smoothing transform

posted 23 Oct 2015, 07:51 by Piotr Dyszewski   [ updated 25 Dec 2015, 05:13 ]

An article written together with Gerold Alsmeyer available on arXiv.

Abstract:
For a given random sequence (C,T1,T2,…) with nonzero C and a.s. finite number of nonzero Tk, the nonhomogeneous smoothing transform S maps the law of a real random variable X to the law of ∑k≥1TkXk+C, where X1,X2,… are independent copies of X and also independent of (C,T1,T2,…). This law is a fixed point of S if the stochastic fixed-point equation (SFPE) X=d∑k≥1TkXk+C holds true, where =d denotes equality in law. Under suitable conditions including EC=0, S possesses a unique fixed point within the class of centered distributions, called the canonical solution to the above SFPE because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on (C,T1,T2,…) such that the canonical solution exhibits right and/or left Poisson tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found.

Talks given on the subject:
  • Smoothing transform and thin tails, Probability theory and stochastic modeling seminar, Wrocław, Poland, 2015 (slides)
  • Exponential moments of fixed points of the nonhomogeneous smoothing transform, Warsaw Summer School in Probability, Warszawa, Poland, 2015 (slides, www)
  • Exponential moments of fixed points of the nonhomogeneous smoothing transform, Probability and Analisys, Będlewo, Poland 2015 (slides, www)
  • Exponential moments of fixed points of the nonhomogeneous smoothing transform, Functional Analysis seminar, Wrocław, Poland, 2015 (www

Iterated random functions and slowly varying tails

posted 9 Sep 2015, 13:55 by Piotr Dyszewski   [ updated 15 Jun 2017, 01:52 ]

Paper published in Stochastic Processes and their Applications (final versionarXiv)

Abstract:
Consider a sequence of i.i.d. random Lipschitz functions {Ψn}n≥0. Using this sequence we can define a Markov chain via the recursive formula Rn+1=Ψn+1(Rn). It is a well known fact that under some mild moment assumptions this Markov chain has a unique stationary distribution. We are interested in the tail behaviour of this distribution in the case when Ψ(t)≈At+B. We will show that under subexponential assumptions on the random variable log+(A∨B) the tail asymptotic in question can be described using the integrated tail function of log+(A0∨B0). In particular we will obtain new results for the random difference equation Rn+1=An+1Rn+Bn+1.

Talks given on the subject:
  • Iterated Lipschitz maps with slowly varying tails, 44th Probability Summer School, Saint Flour, France, 2014 
  • Perpetuities with slowly varying tails, Probabilistic Aspect of Harmonic Analysis, Będlewo, Poland, 2014, 
  • Perpetuities with slowly varying tails, seminarium ”Analiza Funkcjonalna”, Wrocław, Poland, 2014 

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