Invariance principle for numbers of particles in cells of a general allocation scheme

Source abstract

Abstract Let η 1 , …, η N be a generalized allocation scheme of n particles over N cells defined by independent random variables ξ 1 , …, ξ N having power series distribution with parameter β . The expectation and variance of a random variable ξ i are denoted, respectively, by m ( β ) and σ 2 ( β ). We assume that n N = m ( β ) $\frac{n}{N}=m(\beta)$ . We consider the random processes X n , N ( t ) = ∑ i = 1 [ t N ] η i $X_{n,N}(t)=\sum\nolimits_{i=1}^{[tN]}\eta_i$ and Y n , N ( t ) = n − 1 / 2 ( X n , N ( t ) − [ t N ] n N ) $Y_{n,N}(t)=n^{-1/2}(X_{n,N}(t)-[tN]\frac{n}{N})$ , 0 ≤ t ≤ 1. We find conditions for the random processes σ − 1 ( β ) n N Y n , N $\sigma^{-1}(\beta)\sqrt{\frac{n}{N}}\,Y_{n,N}$ to converge in distribution as n , N → ∞ in the Skorokhod space to a Brownian bridge; we also find conditions for the random processes X n , N to converge in distribution (as N → ∞, with fixed n ) in the Skorokhod space to a random process nF n , where F n is an empirical process.