# Proving that a random walk using a maximum likelihood estimator can diverge to infinity

Consider a sequence of continuous random variables $\{X_n\}^\infty_{n=1}$ that are independent and identically distributed under the probability density function $f_\theta (x)$, where $\theta \in [\ell ,u]$ is an unknown but deterministic parameter, $\ell < 0 <u$ and $\theta \neq 0$.Consider the random sequence:

$$S_n = \sum_{i=2}^n \log\frac{f_{\hat{\theta}_{1:i-1}}(X_i)}{f_0(X_i)}$$

where $\hat{\theta}_{1:i-1}$ is the maximum likelihood estimate of the parameter $\theta$ using the observations $\{X_n\}^{i-1}_{n=1}$, and $S_0 \triangleq S_1 \triangleq 0$. Let $\tau =\inf\{n \geq 2 : S_n \leq 0\}$. I am trying to show that $S_n$ may diverge to infinity before becoming non-positive, with non-zero probability, i.e., $$\mathbb{P}(\tau = \infty) >0.$$ My intuition says that this should hold due to the fact that $\hat{\theta}_{1:i-1}$ converges to $\theta$ almost surely as $i \rightarrow \infty$. Do you think this result should hold for the case that the maximum likelihood estimator is used to estimate the parameter $\theta$. I would really appreciate if anyone could provide any results that could be useful to show this claim! Thanks!

Is your question whether this can happen, or whether it must happen? It can happen, it doesn't have to happen. Your maximum likelihood estimators $\hat\theta_{1:i-1}$ don't even have converge to $\theta$... without some conditions on how $f_{\theta}$ depends on $\theta$, you can't say very much for sure. (For instance, if $f_{\theta}$ is actually independent of $\theta$, then $S_n$ is identically zero and $\tau = 2$.)
• My question is whether the event that $S_n$ diverges to infinity before becoming negative is of non-zero probability. I guess a better question would be if there are some further assumptions on the family of distributions $f_\theta$ such that this would hold. For example, we could assume that we are dealing with an exponential family of distributions. – SpawnKilleR May 17 '18 at 2:40
• For convenience, let's assume that the ML estimator converges to $\theta$, which I think is true for exponential families under mild conditions. Would such a structure lead to the claim being true? – SpawnKilleR May 17 '18 at 8:16