On the Kesten–Goldie constant

Abstract

We consider the stochastic difference equation on $\mathbb{R}$$$X_n=A_n{X}_{n-1}+B_n,n\ge 1,$$

where $(A_n,B_n)\in \mathbb{R}\times \mathbb{R}$ is an i.i.d. sequence of random variables and $X_0$ is an initial distribution. Under mild contractivity hypotheses the sequence $X_n$ converges in law to a random variable S, which is the unique solution of the random difference equation $S{=}_{d}AS+B$. We prove that under the Kesten–Goldie conditions$$\underset{n\to \infty }{lim}\frac{\mathbb{E}|X_n{|}^{\mathit{\alpha }}}{n}=\mathit{\alpha }{m}_{\mathit{\alpha }}{C}_{\infty },$$

where $C_{\infty }$ is the Kesten–Goldie constant$$C_{\infty }=\underset{t\to \infty }{lim}{t}^{\mathit{\alpha }}\mathbb{P}[|S|>t],$$$\mathit{\alpha }$ is the Cramér coefficient of $log|A_1|$ and $m_{\mathit{\alpha }}=\mathbb{E}[|A_1{|}^{\mathit{\alpha }}log|A_1|]$. Thus, on one side we describe the behaviour of the $\mathit{\alpha }$th moments of the process $\{X_n\}$, and on the other we obtain an alternative formula for $C_{\infty }$. The results are further extended to a class of Lipschitz iterated systems and to a multidimensional setting.

Keywords:

• Random difference equations
• affine recursion
• iterated functions system
• Lipschitz recursion
• heavy tails
• tail estimates
• limiting constant

Acknowledgements

We thank the reviewer for their constructive comments, which helped us to improve the manuscript.

Notes

No potential conflict of interest was reported by the authors.

1 That is $\mathbb{P}[M>t]\sim C^{\ast }{t}^{-\mathit{\alpha }}$, where $M={sup}_n{A}_1\dots A_n$

2 We will also use the abbreviation: Lipschitz iterated system

3 The convergence follows from the subadditive ergodic theorem.

Additional information

Funding

The authors were partially supported by NCN [grant number UMO-2011/01/M/ST1/04604].