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Persistence probabilities for a bridge of an integrated simple random walk

Aurzada F., Dereich S., Lifshits M.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 1

1--22

We prove that an integrated simple random walk, where random walk and integrated random walk are conditioned to return to zero, has asymptotic probability n−1/2 to stay positive. This question is motivated by random polymer models and proves a conjecture by Caravenna and Deuschel.

Path regularity of Gaussian processes via small deviations

Aurzada F.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

61--78

We study the a.s. sample path regularity of Gaussian processes. To this end we relate the path regularity directly to the theory of small deviations. In particular, we show that if the process is n-times differentiable, then the exponential rate of decay of its small deviations is at most ε-1/n. We also show a similar result if n is not an integer. Further generalizations are given, which parallel the entropy method to determine the small deviations. In particular, the present approach seems to be a probabilistic interpretation of the multiplicativity property of the entropy numbers.

Metric entropy and the small deviation problem for stable processes

Aurzada F.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 2

261--274

The famous connection between metric entropy and small deviation probabilities of Gaussian processes was discovered by Kuelbs and Li in [6] and completed by Li and Linde in [9]. The question whether similar connections exist for other types of processes has remained open ever since. In [10], Li and Linde propose a first approach to this problem for stable processes. The present article clarifies the question completely for symmetric stable processes.