A functional stable limit theorem for Gibbs-Markov maps. (English. French summary) Zbl 1522.37005

Kocheim, David; Pühringer, Fabian; Zweimüller, Roland

Ann. Inst. Henri Poincaré, Probab. Stat. 59, No. 1, 185-207 (2023).

Summary: For a class of locally (but not necessarily uniformly) Lipschitz continuous \(d\)-dimensional observables over a Gibbs-Markov system, we show that convergence of (suitably normalized and centered) ergodic sums to a non-Gaussian stable vector is equivalent to the distribution belonging to the classical domain of attraction, and that it implies a weak invariance principle in the (strong) Skorohod \(\mathcal{J}_1\)-topology on \(\mathcal{D}([0,\infty),\mathbb{R}^d)\). The argument uses the classical approach via finite-dimensional marginals and \(\mathcal{J}_1\)-tightness. As applications, we record a Spitzer-type arcsine law for certain \(\mathbb{Z}\)-extensions of Gibbs-Markov systems, and prove an asymptotic independence property of excursion processes of intermittent interval maps.

MSC:

37A30	Ergodic theorems, spectral theory, Markov operators
37A50	Dynamical systems and their relations with probability theory and stochastic processes
37A25	Ergodicity, mixing, rates of mixing
37H12	Random iteration
37C30	Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
60G51	Processes with independent increments; Lévy processes

Keywords:

stable laws; stable Lévy processes; stationary sequences; weak invariance principle; weakly dependent processes

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