Asymptotical behavior of some statistical estimators in the smooth case. I. Study of the likelihood ratio

The paper considers properties of likelihood ratio determined by (1.1). We prove that the distributions in functional space $\mathbf C_0$, generated by the processes $Z_n(\theta)$ $(-\infty<\theta<\infty)$ tend to the distribution in $\mathbf C_0$, generated by the process $Z(\theta)$ defined by (2.1), provided conditions I–IV of section 1 are satisfied. As a consequence, we have asymptotical normality of the maximum likelihood estimator without assumptions of continuity of $\log f(x,\theta)$ and existence of $f''_{\theta\theta}(x,\theta)$. We deduce several other consequences of this result useful in the second part of the paper.