Discounted payments theorems for large deviations

Let Z(t) = Σ j=1N(t) Xj, t ≥ 0, be a stochastic process, where Xj are independent identically distributed random variables, and N(t) is non-negative integer-valued process with independent increments. Throughout, we assume that N(t) and Xj are independent. The paper considers normal approximation to the distribution of properly centered and normed random variable Zδ =∫0∞e- δt dZ(t), δ > 0, taking into consideration large deviations both in the Cramér zone and the power Linnik zones. Also, we obtain a nonuniform estimate in the Berry–Essen inequality.

Estimation of the convergence rate of two-dimensional random index vectors maxima

Nonlinearly normalized maxima of independent and identically distributed random vectors are pre-sented in this work. We’ve obtained nonuniform estimate of convergence in transfer theorem in case when normalization is nonlinear.

The estimation of the convergence rate for maxima of random variables vectors

Linearly normalized maxima of independent and identically distributed random vectors is presented in this work. We’ve obtained nonuniform estimate of convergence in case when normalization is linear. For clearness there is given an example is this paper. Transfer theorem was aplied.

The nonuniform estimate of the convergence rate of the k-th maxima

In this paper the nonuniform estimate of the convergence rate for the kth maxima of the independent identically distributed random variables is obtained.

Estimation of Convergence Rate in the Transfer Theorem for Maxima

Let ZN be a maximum of independent identically distributed random variables. In this paper, a nonuniform estimate of convergence rate in the transfer theorem max-scheme is obtained. Presented results make the estimates, given in [1] and [2], more precise.

Asymptotics of the Joint Distribution of Multivariate Extrema

Let Wn and Zn be a bivariate extrema of independent identically distributed bivariate random variables with a distribution function F. in this paper the nonuniform estimate of convergence rate of the joint distribution of the normalized and centralized minima and maxima is obtained.

Nonuniform estimate of maximum density convergence rate for independent random variables

The nonuniform estimate of convergence rate in the maximum density limit theorem of independent nonidentically distributed random variables is obtained. This result is generalization of the work presented in [1].