Limit theorems for numbers satisfying a class of triangular arrays

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays, defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain the partial differential equation and special analytical expressions for the numbers using a semi-exponential generating function. We apply the results to prove the asymptotic normality of special classes of the numbers and specify the convergence rate to the limiting distribution. We demonstrate that the limiting distribution is not always Gaussian.

A central limit theorem for numbers satisfying a class of triangular arrays associated with Hermite polynomials

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays. We obtain analytical expressions for the semiexponential generating function the numbers, associated with Hermite polynomials. We apply the results to prove the asymptotic normality of the numbers and specify the convergence rate to the limiting distribution.

Approximations for sums of three-valued 1-dependent symmetric random variables

The sum of symmetric three-point 1-dependent nonidentically distributed random variables is approximated by a compound Poisson distribution. The accuracy of approximation is estimated in the local and total variation norms. For distributions uniformly bounded from zero,the accuracy of approximation is of the order O(n–1). In the general case of triangular arrays of identically distributed summands, the accuracy is at least of the order O(n–1/2). Nonuniform estimates are obtained for distribution functions and probabilities. The characteristic functionmethod is used.

The finiteness of the first meeting time between a Gaussian jump and a Brownian particles in the fluid

In this paper, we present more analysis about the finiteness of the first meeting time between Gaussian jump and Brownian particles in the fluid. Rather than using the uniform integrability conditions which are detailed in El-Hadidy [Mod. Phys. Lett. B 33(22) (2019) 1950256], we show the triangular arrays of the random sequence that represents the probable positions of the first meeting at any time [Formula: see text], to get the stronger moment conditions which are sufficient for the finiteness of this time.