Convergence to stable laws and a local limit theorem for stochastic recursions

Mariusz Mirek ¹

¹ Institute of Mathematics University of Wrocław Plac Grunwaldzki 2/4 50-384 Wrocław, Poland

Colloquium Mathematicum, Tome 118 (2010) no. 2, pp. 705-720

Résumé

We consider the random recursion $X_{n}^{x}=M_nX_{n-1}^{x}+Q_n+N_n(X_{n-1}^{x})$, where $x\in\mathbb R$ and $(M_n, Q_n, N_n)$ are i.i.d., $Q_n$ has a heavy tail with exponent $\alpha>0$, the tail of $M_n$ is lighter and $N_n(X_{n-1}^{x})$ is smaller at infinity, than $M_nX_{n-1}^{x}$. Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $S_n^x=\sum_{k=0}^n X_k^x$ converge weakly to an $\alpha$-stable law for $\alpha\in(0, 2]$. The related local limit theorem is also proved.

Détail

DOI : 10.4064/cm118-2-21

Keywords: consider random recursion n n n where mathbb has heavy tail exponent alpha tail lighter n smaller infinity n using asymptotics stationary solutions properly normalized birkhoff sums sum x converge weakly alpha stable law alpha related local limit theorem proved