Limiting Behavior of High Order Correlations for Simple Random Sampling

Details Limiting Behavior of High Order Correlations for Simple Random Sampling Limiting Behavior of High Order Correlations for Simple Random Sampling

2011-12-28

arxiv.org/abs/1112.6059v1

Mathematics

Combinatorics

32 pages

Scientific paper

For N=1,2,..., let S_N be a simple random sample of size n=n_N from a population A_N of size N, where 0<=n<=N. Then with f_N=n/N, the sampling fraction, and 1_A the inclusion indicator that A is in S_N, for any H a subset of A_N of size k>= 0, the high order correlations Corr(k) = E (\prod_{A \in H} (1_A-f_N)) depend only on k, and if the sampling fraction f_N -> f as N -> infinity, then N^{k/2}Corr(k) -> [f(f-1)]^{k/2}EZ^k, k even and N^{(k+1)/2}Corr(k) -> [f(f-1)]^{(k-1)/2}(2f-1)(1/3)(k-1)EZ^{k+1}, k odd where Z is a standard normal random variable. This proves a conjecture given in [2].

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