Asymptotics of Maximum Likelihood without the LLN or CLT or Sample Size Going to Infinity

Abstract: If the log likelihood is approximately quadratic with constant Hessian, then the maximum likelihood estimator (MLE) is approximately normally distributed. No other assumptions are required. We do not need independent and identically distributed data. We do not need the law of large numbers (LLN) or the central limit theorem (CLT). We do not need sample size going to infinity or anything going to infinity. Presented here is a combination of Le Cam style theory involving local asymptotic normality (LAN) and local asymptotic mixed normality (LAMN) and Cram\'er style theory involving derivatives and Fisher information. The main tool is convergence in law of the log likelihood function and its derivatives considered as random elements of a Polish space of continuous functions with the metric of uniform convergence on compact sets. We obtain results for both one-step-Newton estimators and Newton-iterated-to-convergence estimators.