I have a process, $Y_t = X_t - X_{t-1}$.
$\{X_t\}$ is a stationary process with autoccovariance function $\gamma_{X}(k)$.
I wish to find the autocovariance function of the process $\{Y_t\}$ in terms of $\gamma_{X}(k)$.
I think I am correct in saying $\mathbb{E} (Y_t) = 0$.
So, $$\gamma_{Y}(k) = \mathbb{E}[\{Y_t - \mathbb{E}(Y_t)\}\{Y_{t+k} - \mathbb{E}(Y_{t+k})\}]$$ $$= \mathbb{E}(Y_tY_{t+k})$$ $$= \mathbb{E}[\{X_t -X_{t-1}\}\{X_{t+k} -X_{t+k-1}\}]$$
From here I have expanded, and can see we get different results for different $k$, however I'm not entirely sure that what I have done here is heading in the right direction.
Any comments would be appreciated,
Much thanks.
Edit: Continuing.
Let's say $\mathbb{E}(X_t) = \mu_X$
Then, $\gamma_X(k) = \mathbb{E}(X_tX_{t+k}) - \mu_X^2$
So, $$\mathbb{E}[\{X_t -X_{t-1}\}\{X_{t+k} -X_{t+k-1}\}]$$ $$= \mathbb{E}(X_tX_{t + k}) - \mathbb{E}(X_{t-1}X_{t + k}) - \mathbb{E}(X_tX_{t + k - 1}) + \mathbb{E}(X_{t-1}X_{t + k - 1})$$
$$= (\gamma_X(k) + \mu_X^2) - (\gamma_X(k+1) + \mu_X^2) - (\gamma_X(k-1) + \mu_X^2) + (\gamma_X(k) + \mu_X^2) $$
$$ = 2\gamma_X(k) -\gamma_X(k+1) - \gamma_X(k-1)$$