arXiv subjects

Ewa Damek

Publications and source records attributed to Ewa Damek.

At least 19 recordsLinked to original sources

arXiv2025

Hidden regular variation for stochastic recursions with diagonal matrices

We consider random vectors $X$ that satisfy the equation in law $X=AX+B$, where $A$ is a given random diagonal matrix and $B$ a given random vector, both independent of $X$. It is well known by the works of Kesten and Goldie that the marginals of $X$ may exhibit heavy tails, with possibly different tail indices. In recent works (Damek 2025, Mentemeier and Wintenberger 2022) it was observed that asymptotic independence may occur despite strong dependencies in the entries of $A$: The probability that both marginals are simultaneously large decays faster than the marginal probability of an extreme event; the tail measure is concentrated on the axis. In this work, we analyse the hidden regular variation properties of $X$, that is, we find the proper scaling for which one observes simultaneous extremes.

math.PR↗

arXiv2024

Analysing heavy-tail properties of Stochastic Gradient Descent by means of Stochastic Recurrence Equations

In recent works on the theory of machine learning, it has been observed that heavy tail properties of Stochastic Gradient Descent (SGD) can be studied in the probabilistic framework of stochastic recursions. In particular, Gürbüzbalaban et al. (arXiv:2006.04740) considered a setup corresponding to linear regression for which iterations of SGD can be modelled by a multivariate affine stochastic recursion $X_k=A_k X_{k-1}+B_k$, for independent and identically distributed pairs $(A_k, B_k)$, where $A_k$ is a random symmetric matrix and $B_k$ is a random vector. In this work, we will answer several open questions of the quoted paper and extend their results by applying the theory of irreducible-proximal (i-p) matrices.

stat.ML↗

arXiv2022

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown-Resnick random field and regularly varying max-moving averages.

math.ST↗

arXiv2022

Stochastic recurrence equation with diagonal matrices

Multivariate process satisfying affine stochastic recurrence equation with generic diagonal matrices is considered. We prove that the stationary solution is regularly varying. The results are applicable to diagonal autoregressive models.

math.PR↗

arXiv2022

Tails of bivariate stochastic recurrence equation with triangular matrices

We study bivariate stochastic recurrence equations with triangular matrix coefficients and we characterize the tail behavior of their stationary solutions ${\bf W} =(W_1,W_2)$. Recently it has been observed that $W_1,W_2$ may exhibit regularly varying tails with different indices, which is in contrast to well-known Kesten-type results. However, only partial results have been derived. Under typical "Kesten-Goldie" and "Grey" conditions, we completely characterize tail behavior of $W_1,W_2$. The tail asymptotics we obtain has not been observed in previous settings of stochastic recurrence equations.

math.PR↗

arXiv2021

Limit theorems for supercritical branching processes in random environment

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We focus on the supercritical case, when the process survives with a positive probability and grows exponentially fast on the nonextinction set. Our main is goal is establish Fourier techniques for this model, which allow to obtain a number of precise estimates related to limit theorems. As a consequence we provide new results concerning central limit theorem, Edgeworth expansions and renewal theorem for $\log Z_n$.

math.PR↗

arXiv2019

Stochastic recursions: between Kesten's and Grincevičius-Grey's assumptions

We study the stochastic recursion $X_n=Ψ_n(X_{n-1})$, where $(Ψ_n)_{n\geq 1}$ is a sequence of i.i.d. random Lipschitz mappings close to the random affine transformation $x\mapsto Ax+B$. We describe the tail behaviour of the stationary solution $X$ under the assumption that there exists $α>0$ such that $\mathbb{E} |A|^α=1$ and the tail of $B$ is regularly varying with index $-α<0$. We also find the second order asymptotics of the tail of $X$ when $Ψ(x)=Ax+B$.

math.PR↗

arXiv2018

Large versus bounded solutions to sublinear elliptic problems

Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $Ω\subset \mathbb{R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - φ(x, u(x))=0$ on $Ω$, where $φ$ is in the Kato class with respect to the first variable and it grows sublinearly with respect to the second variable. Under fairly general assumptions we prove that if there is a bounded non zero solution then there is no large solution.

math.AP↗

arXiv2018

Sublinear elliptic problems under radiality. Harmonic $NA$ groups and Euclidean spaces

Let $Ł$ be the Laplace operator on $\R ^d$, $d\geq 3$ or the Laplace Beltrami operator on the harmonic $NA$ group (in particular on a rank one noncompact symmetric space). For the equation $ Łu - φ(\cdot,u)=0$ we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of $φ$ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.

math.DG↗

arXiv2018

A renewal theorem and supremum of a perturbed random walk

We study tails of the supremum of a perturbed random walk under regime which was not yet considered in the literature. Our approach is based on a new renewal theorem, which is of independent interest. We obtain first and second order asymptotics of the solution to renewal equation under weak assumptions and we apply these results to obtain first and second order asymptotics of the tail of the supremum of a perturbed random walk.

math.PR↗

arXiv2018

Affine stochastic equation with triangular matrices

We study solution X of the stochastic equation X = AX +B, where A is a random matrix and B,X are random vectors, the law of (A,B) is given and X is independent of (A,B). The equation is meant in law, the matrix A is 2x2 upper triangular, A_{11}=A_{22}>0, A_{12} is real. A sharp asymptotics of the tail of X =(X _1,X_2) is obtained. We show that under "so called" Kesten-Goldie conditions P (X_2>t)\sim t^{-a} and P (X_1>t )\sim t^{-a}(\log t)^b, where b =a or a\2.

math.PR↗

arXiv2018

Absolute continuity of the martingale limit in branching processes in random environment

We consider a supercritical branching process $Z_n$ in a stationary and ergodic random environment $ξ=(ξ_n)_{n\ge0}$. Due to the martingale convergence theorem, it is known that the normalized population size $W_n=Z_n/ (\mathbb E (Z_n|ξ))$ converges almost surely to a random variable $W$. We prove that if $W$ is not concentrated at $0$ or $1$ then for almost every environment $ξ$ the law of $W$ conditioned on the environment $ξ$ is absolutely continuous with a possible atom at $0$. The result generalizes considerably the main result of \cite{kaplan:1974}, and of course it covers the well-known case of the martingale limit of a Galton-Watson process. Our proof combines analytical arguments with the recursive description of $W$.

math.PR↗

arXiv2018

Absolute Continuity of Complex Martingales and of Solutions to Complex Smoothing Equations

Let $X$ be a $\mathbb{C}$-valued random variable with the property that $$X \ \text{ has the same law as }\ \sum_{j\ge1} T_j X_j$$ where $X_j$ are i.i.d.\ copies of $X$, which are independent of the (given) $\mathbb{C}$-valued random variables $ (T_j)_{j\ge1}$. We provide a simple criterion for the absolute continuity of the law of $X$ that requires, besides the known conditions for the existence of $X$, only finiteness of the first and second moment of $N$ - the number of nonzero weights $T_j$. Our criterion applies in particular to Biggins' martingale with complex parameter.

math.PR↗

arXiv2017

Componentwise different tail solutions for bivariate stochastic recurrence equations -- with application to GARCH(1,1) processes --

We study bivariate stochastic recurrence equations (SREs) motivated by applications to GARCH(1,1) processes. If coefficient matrices of SREs have strictly positive entries, then the Kesten result applies and it gives solutions with regularly varying tails. Moreover, the tail indices are the same for all coordinates. However, for applications, this framework is too restrictive. We study SREs when coefficients are triangular matrices and prove that the coordinates of the solution may exhibit regularly varying tails with different indices. We also specify each tail index together with its constant. The results are used to characterize regular variations of bivariate stationary GARCH(1,1) processes.

math.PR↗

arXiv2017

Iterated random functions and regularly varying tails

We consider solutions to so-called stochastic fixed point equation $R \stackrel{d}{=} Ψ(R)$, 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 $x \mapsto Ax+B$ we show that the tail of $R$ is comparable with the one of $A$, provided that the distribution of $\log (A\vee 1) $ is tail equivalent. In particular we obtain new results for the random difference equation.

math.PR↗

arXiv2017

Pointwise estimates for first passage times of perpetuity sequences

We consider first passage times $τ_u = \inf\{n:\; Y_n>u\}$ for the perpetuity sequence $$ Y_n = B_1 + A_1 B_2 + \cdots + (A_1\ldots A_{n-1})B_n, $$ where $(A_n,B_n)$ are i.i.d. random variables with values in ${\mathbb R} ^+\times {\mathbb R}$. Recently, a number of limit theorems related to $τ_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems. We obtain a precise asymptotics of the sequence ${\mathbb P}[τ_u = \log u/ρ]$, $ρ>0$, $u\to \infty $ which considerably improves the previous results. There, probabilities ${\mathbb P}[τ_u \in I_u]$ were identified, for some large intervals $I_u$ around $k_u$, with lengths growing at least as $\log\log u$. Remarkable analogies and differences to random walks are discussed.

math.PR↗

arXiv2016

A simple proof of heavy tail estimates for affine type Lipschitz recursions

We study the affine recursion $X_n = A_nX_{n-1}+B_n$ where $(A_n,B_n)\in {\mathbb R}^+ \times {\mathbb R} $ is an i.i.d. sequence and recursions $X_n = Φ_n(X_{n-1})$ defined by Lipschitz transformations such that $Φ(x)\geq Ax+B$. It is known that under appropriate hypotheses the stationary solution $X$ has regularly varying tail, i.e. $$\lim_{t\to\infty} t^α {\mathbb P}[X>t] = C. $$ However positivity of $C$ in general is either unknown or requires some additional involved arguments. In this paper we give a simple proof that $C>0$. This applies, in particular, to the case when Kesten-Goldie assumptions are satisfied.

math.PR↗

arXiv2015

Two-sided bounds for $L_p$-norms of combinations of products of independent random variables

We show that for every positive p, the L_p-norm of linear combinations (with scalar or vector coefficients) of products of i.i.d. random variables, whose moduli have a nondegenerate distribution with the p-norm one, is comparable to the l_p-norm of the coefficients and the constants are explicit. As a result the same holds for linear combinations of Riesz products. We also establish the upper and lower bounds of the L_p-moments of partial sums of perpetuities.

math.PR↗