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Universality of the asymptotics of the one-sided exit problem for integrated processes

Frank Aurzada;  Steffen Dereich — 2013

Annales de l'I.H.P. Probabilités et statistiques

We consider the one-sided exit problem – also called one-sided barrier problem – for ( α -fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function α ↦ θ ( α ) such that the probability that any α -fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time T behaves as T - θ ( α ) + o ( 1 ) for large T . We also investigate when the fixed level can be replaced by a different barrier...

Constructive quantization: approximation by empirical measures

Steffen Dereich;  Michael Scheutzow;  Reik Schottstedt — 2013

Annales de l'I.H.P. Probabilités et statistiques

In this article, we study the approximation of a probability measure μ on ℝ d by its empirical measure μ ^ N interpreted as a random quantization. As error criterion we consider an averaged p th moment Wasserstein metric. In the case where 2 p l t ; d , we establish fine upper and lower bounds for the error, a. Moreover, we provide a universal estimate based on moments, a . In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

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