Limit theorems for the size of subpopulation of productive individuals

We consider n-type critical indecomposable Galton-Watson process with types of individuals 1, 2, . . ., n. Let ∇ {1, 2, . . . , n}, τ and t, τ t, be two discrete observation times and θ(t) be n-dimensional vector of nonnegative functions. We consider n-variate process X(τ, t) = (X¹(τ, t), . . . , X_n(τ, t)), where X _i(τ, t) is the number of type i individuals at time τ whose number of descendants at time t of types j, j ∈ is greater than corresponding level given by vector θ(t - τ). We study asymptotic behavior of X(τ, t) when τ, t τ in different cases of relationship between times τ and t for properly chosen level functions.