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

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].

### MSC:

 60F10 Large deviations 60H05 Stochastic integrals
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