NONPARAMETRIC-ESTIMATION FOR GALTON-WATSON TYPE PROCESS

A stochastic process {X(n), n greater-than-or-equal-to 0} with X0 = 1 is said to be a Galton-Watson type process if {X(n)} is a non-negative valued Markov process such that E[e(-tXn +1)\X(n)] = e(-h(t)X(n)), t greater-than-or-equal-to 0, n greater-than-or-equal-to 0 where h(.) is the cumulant generating function of the random variable X1 with an infinitely divisible off-spring distribution. Here we study a nonparametric kernel-type estimator of h(.) based on the observations X1,...,X(n). Consistency property of this estimator is investigated.