Electronic Journal of Probability

On the rate of convergence in the Kesten renewal theorem

Dariusz Buraczewski, Ewa Damek, and Tomasz Przebinda

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We consider the stochastic recursion $X_{n+1} = M_{n+1}X_n + Q_{n+1}$ on $\mathbb{R}^d$,

where ($M_n, Q_n$) are i.i.d. random variables such that $Q_n$ are translations, $M_n$ are similarities of the Euclidean space $\mathbb{R}^d$. Under some standard assumptions the sequence $X_n$ converges to a random variable $R$ and the law $\nu$ of $R$ is the unique stationary measure of the process. Moreover,

the weak limit of properly dilated measure $\nu$ exists, defining thus a homogeneous tail measure $\Lambda$. In this paper we study the rate of convergence of dilations of $\nu$ to $\Lambda$


In particular in the one dimensional setting, when $(M_n,Q_n) \in \mathbb{R}^+\times \mathbb{R}$, $\mathbb{E} M_n^{\alpha }=1$ and $X_n\in \mathbb{R}$, the Kesten renewal theorem says that $t^\alpha\mathbb{P}[|R|>t]$ converges to some strictly positive constant $C_+$. Our main result says that $$\big|t^\alpha\mathbb{P}[|R|>t]-C_+\big|\le C (\log t)^{-\sigma},$$ for some $\sigma>0$ and large $t$. It generalizes the previous one by Goldie.

Article information

Electron. J. Probab. Volume 20 (2015), paper no. 22, 35 pp.

Accepted: 3 March 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K05: Renewal theory
Secondary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35}

stochastic recursions random difference equation stationary measure rate of convergence renewal theorem spectral gap

This work is licensed under a Creative Commons Attribution 3.0 License.


Buraczewski, Dariusz; Damek, Ewa; Przebinda, Tomasz. On the rate of convergence in the Kesten renewal theorem. Electron. J. Probab. 20 (2015), paper no. 22, 35 pp. doi:10.1214/EJP.v20-3708. https://projecteuclid.org/euclid.ejp/1465067128

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