BULLETIN Bull. Malaysian Math. Soc. (Second Series) 21 (1998) 37-46 of the MALAYSIAN MATHEMATICAL SOCIETY Limit Theorems for Exceedances of Sequence of Branching Processes 1 IBRAHIM RAHIMOV AND 2 HUSNA HASAN 1 Institute of Mathematics, Uzbek Academy of Sciences, 700143, Tashkent, Uzbekistan 2 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia 2 e-mail: husna@cs.usm.my Abstract. A problem of the first exceedance of a given level by the family of independent branching processes with and without immigration is considered. Using limit theorems for large deviations for processes with and without immigration limit theorems for the index of the first process exceeding some fixed or increasing levels in critical, subcritical and supercritical cases are proved. Asymptotic formulas for the expectation of the index are also obtained. 1. Introduction We consider an ordinary Bienaymé-Galton-Watson (BGW) process that can be defined as follows. Let } , 2 , 1 { }, , 2 , 1 , 0 { , 0 N i N k X ki be independent and identically distributed random variables taking values in the set . 0 N We define the process , ) (t X , N t by the following relation . ) ( , 1 ) 0 ( ) 1 ( 1 t X i ti X t X X The stochastic process X t () describes the evolution of a population of individuals who produce offspring independently. Now we consider the sequence of BGW branching processes X t i N i ( ), and define the “index” process , ) ( ) ( : ) ( t t X k min t v k for a given “level” function . ) (t Let f ts P ts k k i (,) () 0 be the generating function of the number of individuals (or particles) at time t. We consider the sequence of BGW branching processes X t i N i ( ), under the following assumptions: (a) X t i () are independent for any fixed t N 0 and , 1 ) 0 ( i X