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- W4301232658 abstract "Let $Phi_n$ be an i.i.d. sequence of Lipschitz mappings of $R^d$. We study the Markov chain ${X_n^x}_{n=0}^infty$ on $R^d$ defined by the recursion $X_n^x = Phi_n(X^x_{n-1})$, $ninN$, $X_0^x=xinR^d$. We assume that $Phi_n(x)=Phi(A_n x,B_n(x))$ for a fixed continuous function $Phi:R^dtimes R^dtoR^d$, commuting with dilations and i.i.d random pairs $(A_n,B_n)$, where $A_nin {End}(R^d)$ and $B_n$ is a continuous mapping of $R^d$. Moreover, $B_n$ is $alpha$-regularly varying and $A_n$ has a faster decay at infinity than $B_n$. We prove that the stationary measure $nu$ of the Markov chain ${X_n^x}$ is $alpha$-regularly varying. Using this result we show that, if $alpha<2$, the partial sums $S_n^x=sum_{k=1}^n X_k^x$, appropriately normalized, converge to an $alpha$-stable random variable. In particular, we obtain new results concerning the random coefficient autoregressive process $X_n = A_n X_{n-1}+B_n$." @default.
- W4301232658 created "2022-10-05" @default.
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- W4301232658 date "2010-11-07" @default.
- W4301232658 modified "2023-09-30" @default.
- W4301232658 title "Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems" @default.
- W4301232658 doi "https://doi.org/10.48550/arxiv.1011.1685" @default.
- W4301232658 hasPublicationYear "2010" @default.
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