Estimates for distributions of sums stopped at Markov time

In the present paper asymptotically right estimates for moments and large deviation probabilities for sums of a random number of random variables are derived. A relatively simple method of obtaining the required estimates based on the properties of stopping times and close to known proofs of the Wald identity is suggested. The demand for such estimates arises in a variety of problems of probability theory. In our opinion they are of independent interest and may be useful in a diversity of applications (see, for instance, [3], [4]).It should be pointed out that deriving inequalities for distributions of maximum of partial sums A. N. Kolmogorov [1] used essentially the ideas of stopping times. In their paper A. N. Kolmogorov and Yu. V. Prokhorov [2] proposed a proof of the Wald identity, permitting an easily removed of the special propositions of Wald, who considered sums of independent and identically distributed summands stopped at the time of the first output exit from the interval. The same ideas form the basis of the considerations below.