arXiv subjects
Publications and source records attributed to Bartosz Kołodziejek.
In the paper we present a characterization theorem of the Riesz measure and a Wishart exponential family on homogeneous cones through the invariance property of a natural exponential family under the action of the triangular group.
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 $Ψ(x)=Ax+B$.
We study solutions to the stochastic fixed point equation $X\stackrel{d}{=}AX+B$ when the coefficients are nonnegative and $B$ is an "inverse exponential decay" (IED) random variable. We provide theorems on the left tail of $X$ which complement well-known tail results of Kesten and Goldie. We generalize our results to ARMA processes with nonnegative coefficients whose noise terms are from the IED class. We describe the lower envelope for these ARMA processes.
In the paper we consider the asymptotics of logarithmic tails of a perpetuity $$R \stackrel{d}{=}\sum_{j=1}^\infty Q_j \prod_{k=1}^{j-1}M_k,\qquad(M_n,Q_n)_{n=1}^\infty \mbox{ are i.i.d. copies of }(M,Q),$$ in the case when $\mathbb{P}(M\in[0,1))=1$ and $Q$ has all exponential moments. If $M$ and $Q$ are independent, under regular variation assumptions, we find the precise asymptotics of $-\log\mathbb{P}(R>x)$ as $x\to\infty$. Moreover, we deal with the case of dependent $M$ and $Q$ and give asymptotic bounds for $-\log\mathbb{P}(R>x)$. It turns out that dependence structure between $M$ and $Q$ has a significant impact on the asymptotic rate of logarithmic tails of $R$. Such phenomenon is not observed in the case of heavy-tailed perpetuities.
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.
Let $P_0$ be a probability on the real line generating a natural exponential family $(P_t)_{t\in \mathbb {R}}$. Fix $α$ in $ (0,1).$ We show that the property that $P_t((-\infty,t)) \leq α\leq P_t((-\infty,t])$ for all $t$ implies that there exists a number $μ_α$ such that $P_0$ is the Gaussian distribution $N(μ_α,1).$ In other terms, if for all $t$, $t$ is a quantile of $P_t$ associated to some threshold $α\in (0,1)$, then the exponential family must be Gaussian. The case $α=1/2$, \textit{i.e.} $t$ is always a median of $P_t,$ has been considered in Letac \textit{et al.} (2018). Analogously let $Q$ be a measure on $[0,\infty)$ generating a natural exponential family $(Q_{-t})_{t>0}$. We show that $Q_{-t}([0,t^{-1}))\leq α\leq Q_{-t}([0,t^{-1}])$ for all $t>0$ implies that there exists a number $p=p_α>0$ such that $Q(dx)\propto x^{p-1}dx,$ and thus $Q_{-t}$ has to be a gamma distribution with parameters $p$ and $t.$
We generalize the following univariate characterization of the Kummer and Gamma distributions to the cone of symmetric positive definite matrices: let $X$ and $Y$ be independent, non-degenerate random variables valued in $(0, \infty)$, then $U= Y/(1+X)$ and $V = X(1+U)$ are independent if and only if $X$ follows the Kummer distribution and $Y$ follows the the Gamma distribution with appropriate parameters. We solve a related functional equation in the cone of symmetric positive definite matrices, which is our first main result and apply its solution to prove the characterization of Wishart and matrix-Kummer distributions, which is our second main result.
We present a systematic study of Riesz measures and their natural exponential families of Wishart laws on a homogeneous cone. We compute explicitly the inverse of the mean map and the variance function of a Wishart exponential family.
In the paper we resolve positively the conjecture on a characterization of matrix Kummer and Wishart laws through independence property, which was posed in [Koudou, Statist. Probab. Lett. 82 (2012), 1903--1907] . Apart from the probabilistic result, we determine the general solution of the functional equation associated to the characterization problem under weak assumptions.
In the paper, we find exact asymptotics of the left tail of renewal measure for a broad class of two-sided random walks. We only require that an exponential moment of the left tail is finite. Through a simple change of measure approach, our result turns out to be almost equivalent to Blackwell's Theorem.
In the paper we generalize the following characterization of beta distribution to the symmetric cone setting: let $X$ and $Y$ be independent, non-degenerate random variables with values in $(0,1)$, then $U=1-XY$ and $V=\frac{1-X}{U}$ are independent if and only if there exist positive numbers $p_i$, $i=1,2,3$, such that $X$ and $Y$ follow beta distributions with parameters $(p_1+p_3,p_2)$ and $(p_3,p_1)$, respectively.
The Matsumoto--Yor (MY) property of the generalized inverse Gaussian and gamma distributions has many generalizations. As it was observed in (Letac and Wesołowski in Ann Probab 28:1371--1383, 2000) the natural framework for the multivariate MY property is symmetric cones; however they prove their results for the cone of symmetric positive definite real matrices only. In this paper, we prove the converse to the symmetric cone-variate MY property, which extends some earlier results. The smoothness assumption for the densities of respective variables is reduced to the continuity only. This enhancement was possible due to the new solution of a related functional equation for real functions defined on symmetric cones.
In this paper we generalize the fundamental equation of information to the symmetric cone domain and find general solution under the assumption of continuity of respective functions.
We prove the Lukacs-Olkin-Rubin theorem without invariance of the distribution of the "quotient", which was the key assumption in the original proof of [Olkin--Rubin, Ann. Math. Stat. 33 (1962), 1272--1280]. Instead we assume existence of strictly positive continuous densities of respective random variables. We consider the (cone variate) "quotient" for any division algorithm satisfying some natural conditions. For that purpose, the new proof of the Olkin--Baker functional equation on symmetric cones is given.
We solve the multiplicative Cauchy functional equation on symmetric cones with respect to two different multiplication algorithms. We impose no regularity assumptions on respective functions.
We prove the Lukacs characterization of the Wishart distribution on non-octonion symmetric cones of rank greater than 2. We weaken the smoothness assumptions in the version of the Lukacs theorem of [Bobecka-Wesołowski, Studia Math. 152 (2002), 147-160]. The main tool is a new solution of the Olkin-Baker functional equation on symmetric cones, under the assumption of continuity of respective functions. It was possible thanks to the use of Gleason's theorem.