# Expected value for an autoregressive process

Consider the following stochastic process:

$$Y_i=c\cdot Y_{i-1}+\varepsilon_i,$$ where $$Y_0\sim \mathcal N(0,\sigma^2)$$ is independent of the white noise $$\varepsilon_i \overset{\text{iid}}\sim \mathcal{N}(0,\sigma^2).$$

(Here iid means independent and identical and $$\mathcal N$$ denotes the Normal distribution. Also, $$\sigma>0$$ is a parameter.)

What can be said about the conditional expectation $$\mathbb E(Y_{i+1}\mid Y_1,Y_2,Y_3,\dots,Y_i)$$? What about the conditional Variance $$\mathbb V(Y_{i+1}\mid Y_1,Y_2,Y_3,\dots,Y_i)$$?

My attempt: We can solve the recurrence: For any $$n\geq 1$$, we have

$$Y_{i+n}=c^n Y_i + E,$$

where $$E=\sum_{i=1}^n c^{n-i} \varepsilon_i \sim\mathcal N\left(0,\sigma^2\frac{1-c^{2n}}{1-c^2}\right).$$

Now I'm stuck.

It follows that $$Y_{i+1}$$ conditional on $$Y_1, Y_2, \ldots, Y_i$$ is the same in distribution as $$Y_{i+1}$$ conditional on only $$Y_i$$. Therefore, conditional expectation and variance can be found like so: $$\mathbb E \left( Y_{i+1} | Y_1, Y_2, \ldots, Y_i \right) = \mathbb E \left( Y_{i+1} | Y_i \right) = \mathbb E \left(c Y_i + \varepsilon_{i+1} | Y_i \right) = c Y_i$$
$$\mathbb V \left( Y_{i+1} | Y_1, Y_2, \ldots, Y_i \right) = \mathbb V \left( Y_{i+1} | Y_i \right) = \mathbb V \left( c Y_i + \varepsilon_{i+1} | Y_i \right) = \sigma^2.$$
Regarding your attempt, you're on the road to determine the conditional expectation and variance of $$Y_{i+n}$$ conditional on $$Y_i$$, which doesn't seem to be what you are looking for based on your initial two questions.