Suppose $X_1, X_2, ...$ are iid normal RVs with mean 0 and variance $\sigma^2$. Define the process $$ Y_n = Y_{n-1} + X_n$$.
for integers $n \geq 1$ and with $Y_0 = 0$. I want to compute the auto-covariance of $Y_n$ and show that $\{Y_n\}$ is an independent increment process.
I wanted to proceed with the definition for the autocovariance and using the fact that $X_i$ are iid with mean 0 \begin{align} R_Y(m,k) &= E((Y_{m-1} - X_m)(Y_{k-1} - X_k)) \\ &=E(Y_{m-1}Y_{k-1} - Y_{m-1}X_n - Y_{n-1}X_m + X_m X_n) \\ &=E(Y_{m-1}Y_{k-1}) - E(Y_{m-1}X_n) - E(Y_{n-1}X_m) + E(X_m) E(X_n)\\ &=E(Y_{m-1}Y_{k-1}) - E(Y_{m-1}X_n) - E(Y_{n-1}X_m) \end{align}
Any hint on how to compute the left expectations? Also regarding showing the independent increments, the definition that I need to show is that for every $i < j$, $Y_i$ and $Y_{j} - Y_{i}$ are independent. I can't think of what definition of independence to use in this situation. Any idea?
This process is known in the econometric literature as a (non-stationary) AR(1) process or more probabilistically as a Random Walk.
First we notice, by recursion, that $Y_n = Y_0 + \sum_{i=1}^{n}X_i$. Hnece
$$ \mathbb{E}[Y_n] := Y_0 $$
For the autocovariance, we need to compute
$$ \mathbb{E}[(Y_n - Y_0)(Y_{n-j}-Y_0)] = \mathbb{E}\bigg[ \bigg( \sum_{i=1}^{n}X_i \bigg)\bigg( \sum_{k=1}^{n-j}X_k \bigg) \bigg] \\ = \mathbb{E}\bigg[ \sum_{i=1}^{n-j}X_{i}^{2} + \sum_{k \ne i}X_iX_k \bigg] = \text{by independence} \\ = \mathbb{E}\bigg[ \sum_{i=1}^{n-j}X_{i}^{2} \bigg] = (n-j) \sigma^2 $$
to prove the independence of the increments, we just need to notice that $Y_n -Y_{n-1} = X_n$. Thus the independence follows from the IID property of the $X_i$