Central limit theorem for quadratic forms for sparse tables (Q1091690)

Decompose the sample space into \(k_ n\) disjoint subsets \(A_ t\), and let \(n_ t\) be the number of n observations in which \(A_ t\) occurs. Let \(k_ n\to +\infty\) with n. Here, n is either a large preassigned number (multinomial sampling), or a Poisson variable whose expectation tends to infinity. (Poisson sampling). The author studies the asymptotic behaviour of some weighted mean \(T_ n\) of general quadratic forms in \(n_ t-np_ t\), \(p_ t=P_ n(A_ t)\), \(1\leq t\leq k_ n\). For both types of sampling, \(T_ n-E(T_ n)\) is shown to be asymptotically normal. The author reduces the case of Poisson sampling to the multinomial case by showing that, under some conditions, \[ T_ n-E(T_ n)-[T^*_ n-E(T^*_ n)]\to 0\quad in\quad probability, \] where \(T_ n\) and \(T^*_ n\) are the weighted means for the two types of sampling.