Let $x=\{x_1,x_2,...\}$ be a first-order autoregressive process, $AR(1)$. Since $x$ is an $AR(1)$ process, it is defined as follows:

$x_t=\mu+\phi x_{t-1}+\epsilon_t$

where $\mu$ and $\phi$ are constants, and $\epsilon_t$ is a $IID$ random variable distributed Normally with mean $0$ and variance $\sigma^2$.

Now consider another process, $y=\{y_1,y_2,...\}$ defined as follows:

$y_t = \left\{ \begin{array}{lr} x_t+y_{t-1} & : mod(t,a)\neq 1\\ x_t & : mod(t,a)= 1\\ \end{array} \right.\\$

where $a$ is a constant. How to calculate the mean and variance of process $y$?

  • $\begingroup$ What have you tried? $\endgroup$
    – Dasherman
    Sep 15 at 11:13
  • $\begingroup$ @Dasherman I was able to determine the mean of process y. Process y essentially is the combination of an AR(1) process, x, and a new process that is a cumsum of elements of process x. The mean of AR(1) process is pretty straightforward. The mean of the cumsum process is (a+2)/2 times the mean of AR(1) process. The overall mean of y is the weighted average of AR(1) and the cumsum processes. The weight of AR(1) process is 1/a and that of the cumsum process is (a-1)/a. I still do not know how to determine the variance of y. $\endgroup$ Sep 15 at 16:53

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