#### 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. arXivAbstract: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 arXivAbstract:We consider the branching process in random environment {Zn}n≥0, 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 x↦Ax+B we show that the tail of R is comparable with the one of A, provided that the distribution of log(A∨1) 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 version, arXiv)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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