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A strengthened asymptotic uniform distribution property

Weber Michel J. G.

Probability and Mathematical Statistics

2023

Vol. 43, Fasc. 2

263--285

We are concerned with estimating the rate of convergence in (0.1), a question recently discussed by Dolgopyat and Hafouta (2022). We obtain a quantitative estimate under weaker moment assumptions, by using a different approach.

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.

A sharp correlation inequality with application to almost sure local limit theorem

Weber M.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

79--98

We prove a new sharp correlation inequality for sums of i.i.d. square integrable lattice distributed random variables. We also apply it to establish an almost sure version of the local limit theorem for i.i.d. square integrable random variables taking values in an arbitrary lattice. This extends a recent similar result jointly obtained with Giuliano-Antonini under a slightly stronger absolute moment assumption (of order 2+u with u > 0). The approach used to treat the case u > 0 breaks down when u = 0. Mac- Donald’s concept of the Bernoulli part of a random variable is used in a crucial way to remedy this.

Multivariate large deviations for sums of i.i.d. random vectors with compactly supported distribution

Zaigraev A.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

315--335

The sums of i.i.d. random vectors with compactly supported and absolutely continuous distribution are considered. Under some conditions the strong form of the local limit theorem for large deviations is proved. In passing the asymptotic behaviour of the moment generating function as well as possible non-degenerate limit laws for the natural exponential family of distributions are established.

Multivariate large deviations with stable limit laws

Zaigraev A.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

323--335

The large deviation problem for sums of i.i.d. random vectors is considered. It is assumed that the underlying distribution is absolutely continuous and its density is of regular variation. An asymptotic expression for the probability of large deviations is established in the case of a non-normal stable limit law. The role of the maximal summand is also emphasized.