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If the log likelihood is approximately quadratic with constant Hessian, thenthe maximum likelihood estimator (MLE) is approximately normally distributed.No other assumptions are required. We do not need independent and identicallydistributed data. We do not need the law of large numbers (LLN) or the centrallimit theorem (CLT). We do not need sample size going to infinity or anythinggoing to infinity. Presented here is a combination of Le Cam style theoryinvolving 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 itsderivatives considered as random elements of a Polish space of continuousfunctions with the metric of uniform convergence on compact sets. We obtainresults for both one-step-Newton estimators and Newton-iterated-to-convergenceestimators.

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