I am given the Stochastic process $Y_n$, where $n \in Z$ defined by:
$ Y_n = X_n - X_{n-1}$
where $X_n$ is a process with independent and identically distributed geometric variables $X_n \sim G(p)$
a) Calculate the mean and correlation of $X_n$
b) Calculate the mean and correlation of $Y_n$
c) Is $Y_n$ stationary?
d) Calculate the variance $var(Y_1 + Y_2)$
a)
the mean of $X_n$ is:
$E(X_n) = \frac{1}{p}$ because $X_n$ is a variable of a geometric distribution
the correlation is:
$r_X(n+k,n) = E(X_{n+k} X_n)$
Now considering:
$var(X_n) = \frac{1-p}{p^2}$
so:
$ r_X(n+k,n) = E(X_{n+k} X_n) = \frac{1-p}{p^2} \delta(k)$
b)
the mean of $Y_n$ is:
$E(Y_n) = E(X_n - X_{n-1}) = \frac{1}{p} - \frac{1}{p} = 0 $
the correlation can be calculated using the formula $r_Y(n+k,n) = r_Y(k,0)$ so:
$r_Y(n+k,n) = E(Y_{n+k} Y_n) = E((X_{n+k} - X_{n+k-1})(X_n - X_{n-1}) ) = E(X_{n+k}X_n - X_{n+k}X_{n-1} - X_{n+k-1}X_n + X_{n+k-1}X_{n-1}) = $
$r_X(k) - r_X(k+1) - r_X(k-1) + r_X(k) =$
$2\frac{1-p}{p^2} \delta(k) -\frac{1-p}{p^2} \delta(k+1) -\frac{1-p}{p^2} \delta(k-1)$
where the result does not depend from $n$ so the process $Y_n$ is stationary.
d)
$var(Y_1 + Y_2) = var(X_1 - X_0 + X_2 - X_1) = var(X_2 - X_0) = var(X_2) + var(X_0) = 2\frac{1-p}{p^2}$
because $X_n$ are independent.
I have many doubts about this exercise especially the calculation of the two correlations, so I would like to know if it is solved in the right way