Large deviations for some Poisson random integrals. (English) Zbl 0959.60022

Jurek, Zbigniew J.; Wu, Liming

Azéma, J. (ed.) et al., Séminaire de Probabilités XXXIV. Berlin: Springer. Lect. Notes Math. 1729, 185-197 (2000).

The well-known Schilder’s theorem gives the large deviation estimates for the convergence \(\sqrt\varepsilon W\Rightarrow \delta_0\) on \(C([0, \infty),\mathbb{R}^d)\) as \(\varepsilon\to 0\), for the Brownian motion \(W\). The authors investigate analogous problems for some integrals \(\varepsilon N(f):= \varepsilon \int f dN\), where \(N\) is a Poisson point process and \(f\) is a deterministic function. They find that this large deviation estimation depends strongly on the tail behaviour of \(f\). This differs from the Brownian motion case where only the norm of \(f\) in \(L^2\) is involved. In particular, they get the large deviation principle for the Levi class \(L\) distributions (called also self-decomposable measures). The question about large deviations for the multiple Poisson integrals is not discussed here.
For the entire collection see [Zbl 0940.00007].

Reviewer: Serguey Victorovich Zhulenev (Moskva)

MSC:

60F10	Large deviations
60H05	Stochastic integrals

Keywords:

large deviation; Poisson point process; Levi class; multiple Poisson integrals