Let there be a set of observations $\{Y_0, Y_1, \dots, Y_n \}$ from a stochastic process $\{ Y_t\}_{t \in \mathbb{N}}$ where $Y_t \sim N( \theta^t \mu, 1)$, $Y_t$ is independent of $Y_s$ for all $s \ne t$ and $|\theta|<1, \mu \in \mathbb{R}$. I would like to find an estimator for the mean of $Y_m$ that we will call $\mu_m$, where $m<n$ that is consistent.
If I utilize the maximum likelihood framework and aim to maximize the log likelihood
\begin{align*} \sum_{i=0}^n\log \frac{1}{\sqrt{2\pi}} - \frac{1}{2}(Y_i - \theta^{-m+ i}\mu_m)^2 \end{align*}
I obtain the maximum likelihood estimator is given by $$ \hat{\mu}_m = \frac{Y_0 + Y_1 + \dots + Y_n}{\theta^{-m} + \theta^{-m+1}+ \dots+ \theta^{n-m}}$$ but it appears that denominator is finite as $n \rightarrow \infty$ so this estimator will not be consistent although it is unbiased. How would one find a consistent estimator for $\mu_m$? Is the maximum likelihood estimator salvageable in some way?
Any references to problems of this kind are also welcome.