# First and second moments of a piecewise time series

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 $$y$$?

• What have you tried? Sep 15 at 11:13
• @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. Sep 15 at 16:53