Let f be a polynomial in k variables and S n be a normed sum of independent identically distributed random vectors X 1 ; X 2 ; : : : ; X n taking values in R k. Upper bounds for jE expfit f (S n)gj are derived provided that either the distribution of X 1 has a nondegenerate discrete component or the distribution of S n 0 has an absolutely continuous… (More)
We prove a stochastic analogue of the well-known Vinogradov mean value theorem. In the special case of random variables taking values 1; 2; : : : ; P with equal probabilities our result yields an estimate of the same order in P as the original Vinogradov estimate.
Let X be a random variable with probability distribution P X concentrated on ?1; 1] and Q(x) be a polynomial of degree k 2. The characteristic function of a random variable Y = Q(X) is of order O(1=jtj 1=k) as jtj ! 1 if P X is suuciently smooth. In comparison for every " : 1=k > " > 0 there exists a singular distribution P X such that every convolution P… (More)
For a statistic S whose distribution can be approximated by χ 2-distributions, there is a considerable interest in constructing improved χ 2-approximations. A typical approach is to consider a transformation T = T (S) based on the Bartlett correction or a Bartlett type correction. In this paper we consider two cases in which S is expressed as a scale… (More)
Spin coherent states play a crucial role in defining QESM (quasi-exactly solvable models) establishing a strict correspondence between energy spectra of spin systems and low-lying quantum states for a particle moving in a potential field of a certain form. Spin coherent states are also used for finding the Wigner-Kirkwood expansion and quantum corrections… (More)
We prove a partial extension of Vinogradovs estimates of trigonome-tric sums to the case of random variables. These generalizations are useful in limit theorems in probability theory and mathematical statistics.