Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems - Mathematics > ProbabilityReport as inadecuate




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Abstract: Let $\Phi n$ be an i.i.d. sequence of Lipschitz mappings of $\R^d$. We studythe Markov chain $\{X n^x\} {n=0}^\infty$ on $\R^d$ defined by the recursion$X n^x = \Phi nX^x {n-1}$, $n\in\N$, $X 0^x=x\in\R^d$. We assume that$\Phi nx=\PhiA n x,B nx$ for a fixed continuous function $\Phi:\R^d\times\R^d\to\R^d$, commuting with dilations and i.i.d random pairs $A n,B n$,where $A n\in {End}\R^d$ and $B n$ is a continuous mapping of $\R^d$.Moreover, $B n$ is $\alpha$-regularly varying and $A n$ has a faster decay atinfinity than $B n$. We prove that the stationary measure $ u$ of the Markovchain $\{X n^x\}$ is $\alpha$-regularly varying. Using this result we showthat, if $\alpha<2$, the partial sums $S n^x=\sum {k=1}^n X k^x$, appropriatelynormalized, converge to an $\alpha$-stable random variable. In particular, weobtain new results concerning the random coefficient autoregressive process$X n = A n X {n-1}+B n$.



Author: Dariusz Buraczewski, Ewa Damek, Mariusz Mirek

Source: https://arxiv.org/







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