Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices. (English) Zbl 1266.60041
Summary: Given a sequence \((M _{n }, Q _{n })_{n \geq 1}\) of i.i.d. random variables with generic copy \((M, Q) \in \operatorname{GL}(d, \mathbb R) \times \mathbb R^{d }\), we consider the random difference equation (RDE)
\[
R_{n}=M_{n}R_{n-1}+Q_{n},
\]
\(n \geq 1\), and assume the existence of \(\kappa > 0\) such that
\[
\lim_{n \to \infty} (\operatorname{E}\| {M_1 \dotsm M_n}\|^\kappa)^{\frac{1}{n}} = 1.
\]
We prove, under suitable assumptions, that the sequence \(S _{n } = R _{1} + \dotsb + R _{n }\), appropriately normalized, converges in law to a multidimensional stable distribution with index \(\kappa \). As a by-product, we show that the unique stationary solution \(R\) of the RDE is regularly varying with index \(\kappa \) and give a precise description of its tail measure.
MSC:
| 60F05 | Central limit and other weak theorems |
| 60J05 | Discrete-time Markov processes on general state spaces |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
Keywords:
weak limit theorems; random difference equations; stable laws; stochastic recursions; multivariate regular variationReferences:
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