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Cross-post.

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.

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Since the errors are iid, the process can be considered also as a Markov process (see Is AR(1) a Markov process?).

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.

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