Uniquely defined autoregression Let $\{w_t\}, t \in \mathbb{Z}$ be a random noise. Given a sequence $\{w_t\}$, does the autoregression
$x_t = x_{t-1} - 0.9 x_{t-2} + w_t$,  $ t \in \mathbb{Z}$
uniquely define a sequence $\{x_t\}$?
If not, is it enough to impose the value of the mean of $\{x_t\}$ to make this sequence uniquely defined?
 A: This is a two-term recursion.  Given arbitrary values for $x_0$ and $x_1,$ the formula
$$x_t = x_{t-1} -0.9 x_{t-2} + w_t$$
recursively determines $x_2, x_3, \ldots, x_n, \ldots$ and the inverse formula (found by solving this one for $x_{t-2}$ with $s=2-t$) 
$$x_{-s} = \frac{-x_{2-s} + x_{1-s} + w_{2-s}}{0.9}$$
recursively determines $x_{-1}, x_{-2}, \ldots, x_{-n}, \ldots.$
Both recursions make sense whether the $x_t$ and $w_t$ are interpreted as sequences of random variables or just sequences of numbers.
Note that since each formula expresses $x_t$ or $x_{-s}$ as a Real linear combination of the preceding $x$'s and $w$'s, it follows by induction that all terms in the sequence are real linear combinations of $x_0, x_1,$ and the $w$'s.  If you are thinking of these objects as numbers, with the $w$'s given, this shows that the set of all such sequences is a two-dimensional vector space.
In particular, imposing a single linear constraint on these sequences will yield a one-dimensional vector space.  Thus, the analog for random variables--such as fixing the means--still will not determine the sequence uniquely.
The generalization to longer-term recursions (AR$(k)$ models) should be clear.
