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2015 Functional limit theorems for divergent perpetuities in the contractive case
Dariusz Buraczewski, Alexander Iksanov
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Electron. Commun. Probab. 20: 1-14 (2015). DOI: 10.1214/ECP.v20-3915


Let $\big(M_k, Q_k\big)_{k\in\mathbb{N}}$ be independent copies of an $\mathbb{R}^2$-valued random vector. It is known that if $Y_n:=Q_1+M_1Q_2+\ldots+M_1\cdot\ldots\cdot M_{n-1}Q_n$ converges a.s. to a random variable $Y$, then the law of $Y$ satisfies the stochastic fixed-point equation $Y \overset{d}{=} Q_1+M_1Y$, where $(Q_1, M_1)$ is independent of $Y$. In the present paper we consider the situation when $|Y_n|$ diverges to $\infty$ in probability because $|Q_1|$ takes large values with high probability, whereas the multiplicative random walk with steps $M_k$'s tends to zero a.s. Under a regular variation assumption we show that $\log |Y_n|$, properly scaled and normalized, converge weakly in the Skorokhod space equipped with the $J_1$-topology to an extremal process. A similar result also holds for the corresponding Markov chains. Proofs rely upon a deterministic result which establishes the $J_1$-convergence of certain sums to a maximal function and subsequent use of the Skorokhod representation theorem.


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Dariusz Buraczewski. Alexander Iksanov. "Functional limit theorems for divergent perpetuities in the contractive case." Electron. Commun. Probab. 20 1 - 14, 2015.


Accepted: 31 January 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1307.60026
MathSciNet: MR3314645
Digital Object Identifier: 10.1214/ECP.v20-3915

Primary: 60F17
Secondary: 60G50

Keywords: Extremal process , Functional limit theorem , perpetuity , random difference equation

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