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)
The surface morphology of Ge0.96Sn0.04/Si(100) heterostructures grown at temperatures from 250 to 450°C by atomic force microscopy (AFM) and scanning tunnel microscopy (STM) ex situ has been studied. The statistical data for the density of Ge0.96Sn0.04 nanodots (ND) depending on their lateral size have been obtained. Maximum density of ND (6 × 1011 cm-2)… (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.