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Given that $a_t \sim WN(2, 0.5)$, I have generated the process defined by

$$Y_1 = a_1$$

$$Y_t = \theta Y_{t - 1} + a_t$$

to be: $$Y_t = \theta^{t - 1}a_1 + \theta^{t - 2}a_2 + \cdots + \theta a_{t - 1} + a_t$$

I am then asked to determine the range of the parameter $\theta$ for which the representation of $Y_t$ in terms of $a_t$ exists.

Is the idea to write $Y_t$ as: $$\sum\limits_{i = 0}^{t - 1} \theta^ia_{t - i}$$

and treat as a finite Geometric Series?

I have also found the mean of this series to be: $$E[Y_t] = 2\sum\limits_{i = 1}^t \theta^{t - i}$$

But in finding the variance, I am confused on how to find $E[Y_t^2]$ and computing $(E[Y_t])^2$ seems difficult given the form I have.

Any help would be greatly appreciated.

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  • $\begingroup$ Is there any information regarding $Y_t$ is a stationary process or not? $\endgroup$ Commented Feb 19, 2021 at 8:51

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I'm fairly new to stack exchange so I am not able to comment... hopefully someone will edit this for clarity.

To solve for the variance of Y(t) you need to represent it as a geometric series, then use the property of variance, namely Var(aX) = a^2 Var(X)

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