Let $X$ be a random variable having uniform distribution on the segment $[-\theta, \theta]$. I construct the following estimator for unknown parameter $\theta$. $$ \hat{\theta}(x_1,\ldots,x_n) = \frac{n+1}{n-1}\max\{x_1,\ldots,x_n\} $$
By direct methods it can be shown that this estimator is unbiased and consistent.
My question. Is this estimator strongly consistent?
Assuming that $x_1,x_2,\dots, $ as usual are realizations of i.i.d. copies of $X$, the answer is Yes, the sequence $\hat\theta(x_1,\dots,x_n)$ is converging to $\theta$ almost surely.
For any $\varepsilon >0$, the probability that $X_n$ lies between $\theta-\varepsilon$ and $\theta$ is $P(\theta-\varepsilon<X_n\leq\theta)=\varepsilon/(2\theta) > 0.$ By the second Borel-Cantelli lemma", the probability that there exists some $n$ for which $|X_n -\theta|<\varepsilon$ is therefore 1. Thus the simpler sequence $M_1,M_2,\dots$ of the maxima, $m_n = \max(x_1,\dots,x_n)$, converges almost surely to $\theta$.
Now $$ \begin{align} P(\lim_{n\to\infty}\hat\theta = \theta)&=P(\lim_{n\to\infty}M_n\cdot\frac{n-1}{n+1} = \theta)\\ &=P(\lim_{n\to\infty}M_n\cdot\underbrace{\lim_{n\to\infty}\frac{n-1}{n+1}}_{=1} = \theta) \\ &=P(\lim_{n\to\infty}M_n=\theta)=1. \end{align} $$
probability that