Moments of Randomly Stopped Sums
1. Introduction. Let (Ω, F, P) be a probability space, let x 1 x 2, … be a sequence of random variables on Ω, and let F n be the σ-algebra generated by x 1, …, x n with F 0 = (φ,Ω). A stopping variable (of the sequence x 1, x2, …) is a random variable t on Ω with positive integer values such that the event [t = n] e F n for every n ≧ l.Let \( {S_n} = \Sigma _i^n{x_i};{\text{ then }}{S_t} = {S_{t(\omega )}}(\omega ) = \Sigma _1^t{x_i} \) is a randomly stopped sum. We shall always assume that 
 
$$E|{x_n}| < \infty , E({x_{n + 1}}|{\Im _n}) = 0, (n \geqslant 1).$$ 
 
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