Translated from Russian Journal LIMIT THEOREMS FOR THE TOTAL NUMBER OF DESCENDANTS FOR THE (]ALTON-WATSON BRANCHING PROCESS*

Abstract. The main results of the present paper deal with the asymptotic behavior of the con-ditional distribution for the whole number of descendants Sn of a single particle in the Galton-Watson process with respect to the condition that the process degenerates at time n and the expectation for the number of particles generated by one particle tends to 1 as n o. Key words, the Galton-Watson branching processes, processes close to critical ones, degener-acy, asymptotic behavior of the total number of descendants 1. Introduction. The Galton-Watso