Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

By Mariusz Mirek. 2009

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We consider the Markov chain $${\{X_n^x\}_{n=0}^\infty}$$ on $${\mathbb{R}^d}$$ defined by the stochastic recursion $${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$$, starting at $${x\in\mathbb{R}^d}$$, where θ1, θ2, . . . are i.i.d. random variables taking their values in a metric space $${(\Theta, \mathfrak{r})}$$, and $${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ν. Under appropriate assumptions on $${\psi_{\theta_n}}$$, we will show that the measure ν has a heavy tail with the exponent α > 0 i.e. $${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$$. Using this result we show that properly normalized Birkhoff sums $${S_n^x=\sum_{k=1}^n X_k^x}$$, converge in law to an α-stable law for $${\alpha\in(0, 2]}$$.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/65ad2da168c389339c260e026f0a3153e4a5eeca

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