Local central limit theorems, the high-order correlations of rejective sampling and logistic likelihood asymptotics

Local Central Limit Theorems , the High - Order Correlations of Rejective Sampling and Logistic Likelihood

Let I1, . . . , In be independent but not necessarily identically distributed Bernoulli random variables, and let Xn = ∑n j=1 Ij . For ν in a bounded region, a local central limit theorem expansion of P(Xn = EXn + ν) is developed to any given degree. By conditioning, this expansion provides information on the high-order correlation structure of dependent, weighted sampling schemes of a populati...

Local Central Limit Theorems, the High Order Correlations of Rejective Sampling, and Applications to Conditional Logistic Likelihood Asymptotics

Local Central Limit Theorems , the High Order Correlations of Rejective Sampling , and Applications to Conditional Logistic Likelihood

Let I1, . . . , In be independent but not necessarily identically distributed Bernoulli random variables, and let Xn = ∑n j=1 Ij . For ν in a bounded region, a local central limit theorem expansion of PI (Xn = EI Xn + ν) is developed to any given degree. By conditioning, this expansion provides information on the high order correlation structure of dependent, weighted sampling schemes of a popu...

LOCAL CENTRAL LIMIT THEOREMS, THE HIGH-ORDER CORRELATIONS OF REJECTIVE SAMPLING AND LOGISTIC LIKELIHOOD ASYMPTOTICS BY RICHARD ARRATIA, LARRY GOLDSTEIN1 AND BRYAN LANGHOLZ1 University of Southern California

Let I1, . . . , In be independent but not necessarily identically distributed Bernoulli random variables, and let Xn = ∑nj=1 Ij . For ν in a bounded region, a local central limit theorem expansion of P(Xn = EXn + ν) is developed to any given degree. By conditioning, this expansion provides information on the high-order correlation structure of dependent, weighted sampling schemes of a populatio...

Central and Local Limit Theorems in Markov Dependent Random Variables

We consider an irreducible and aperiodic Markov chain {kn}n=0 over the finite state space E = {1, . . . , p} with positive regular transition matrix P = {pij} and additive component {Un} such that {Sn} = {(kn, Un)} is also a Markov chain over the state space E1 = E × R. We prove a central and a local limit theorem for this chain when the probability density functions of {Sn}, conditional on the...