ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60F10, 60F11. Rebolledo, Rolando. Facultad de Matemáticas, Universidad Católica de Chile. Stopping times and Large-Deviation Principles Consider a sequence of processes (X^{n})_{n} whose trajectories are in the Skorokhod space D = D([0, ∞[, ℝ). Assume that each process X^{n} is adapted to a filtration 𝔽^{n}. Then the following criterion on Large Deviations holds: Theorem. The sequence (X^{n})_{n} admits a subsequence which satisfies a Large Deviation Principle whenever the two hypotheses below are satisfied: (H1) For any N ≥ 1 and all α > 0 there exist numbers n_{0} ≥ 1 and a > 0 such that ℙ(sup_{t≤N} |X_{t}^{n}| > a) ≤ exp(−nα), for all n ≥ n_{0}; (H2) Given any N ≥ 1, β, ϵ > 0 and, in addition, any arbitrary sequence of stopping times (T^{n})_{n} bounded by N; any sequence (δ^{n})_{n} of strictly positive real numbers decreasing to 0, there exists an n_{0} ≥ 1 such that ℙ(|X_{T^{n}+δ^{n}}^{n} − X_{T^{n}}^{n}| > ϵ) ≤ exp(−nβ), for all n ≥ n_{0}.