Log-level comparison principle for small ball probabilities

Log-level Comparison Principle for Small Ball Probabilities

(if μ is the Lebesgue measure, the index μ will be omitted). The problem is to define the behavior of P{||X||μ ≤ ε} as ε → 0. The study of small deviation problem was initiated by Sytaya [S] and continued by many authors. The history of the problem in 20th century is described in reviews by Lifshits [Lf2] and by Li and Shao [LS]. Latest results can be found in [Lf3]. According to the well-known...

Log-level Comparison for Small Deviation Probabilities

Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L2 norm, multiple sums motivated by tensor product of Gaussian processes, and various integrated fractional Brownian motions under the sup-norm. ∗Department of Mathematics, University of Idaho, Moscow, ID 83844-1103, [email protected]. Research partially ...

Logarithmic Level Comparison for Small Deviation Probabilities

Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L 2 norm, multiple sums motivated by tensor product of Gaussian processes, and various integrated fractional Brownian motions under the sup-norm.

Small Ball Probabilities for Gaussian Markov

Small Ball Probabilities for Linear Images of High Dimensional Distributions

We study concentration properties of random vectors of the form AX, where X = (X1, . . . , Xn) has independent coordinates and A is a given matrix. We show that the distribution of AX is well spread in space whenever the distributions of Xi are well spread on the line. Specifically, assume that the probability that Xi falls in any given interval of length t is at most p. Then the probability th...