Sentence examples for for random sums from inspiring English sources

However, the established convergence rates in limit theorems for random sums of this paper are perfect illustrations for powerful applications of the Trotter-distance method in studies of limit theorems for random sums.

Lin and Stoyanov (2002) and Gut (2003) studied the moment problem for random sums of independently identically distributed (i.i.d).i.d

Then Theorem 3.2 will state the weak law of large numbers for random sums in following form: S N n N n → P 0 as  n → + ∞.

The main purpose of this paper is to establish some estimates for the rates of convergence in limit theorems for random sums of independent identically distributed random variables via Trotter-distance.

It is worth pointing out that the mathematical tools have been used in the study of limit theorems for random sums review to date, including characteristic function, positive linear operators and probability metrics.

The statements in Remark 3.1 and Remark 3.2 are considerably weaker than the Kolmogorov strong Law of Large Numbers for random sums, S N n N n → a. s.

The operator method used in this paper is quite elementary and it also could be applied for the probability distributions of random sums S N n = ∑ k = 1 N n X n k in the Poisson approximation, where N n, n = 1, 2, …  , are positive integer-valued random variables, independent of all X n k, k = 1, 2, …, n ; n = 1, 2, …  .

Yet another underpinning for the presence of signal summation is provided by considering the accompanying distributions of random sums.

(For N n = 0 we set S N n = S 0 = 0.) In 1948 Robbins [1] gave sufficient conditions for the validity of the central limit theorem for normalized random sums of (1).

We remark that Baltrūnas et al. [7] obtained an important equivalently precise large deviations result for the random sums of nonnegative subexponential r.v.s.

This paper investigates some precise large deviations for the random sums of the differences between two sequences of independent and identically distributed random variables, where the minuend random variables have subexponential tails, and the subtrahend random variables have finite second moments.