I tried a question and intutively i can guess it's answer, but I am not sure if my logic is correct or not:
Let $\{X_1,X_2, \dots ,X_n\}$ be a sequence of independent and identically distributed (i.i.d.) zero-mean Gaussian random variables. Assume that Var[$X_i]=\sigma^2$ for all i. Define $Y_k = X_k − X_{k−1}$ for $k ≥ 2$ with $Y_1 = X_1$. Is $\{Y_k\}$ an independent Gaussian random sequence?
Here is my attempt:
The difference between two gaussian Random Variables is also gaussian, i.e. if $X_1 \sim N(\mu_1,\sigma_1^2), X_2 \sim N(\mu_2,\sigma_2^2)$then $X_1-X_2 \sim N(\mu_1 - \mu_2,\sigma_1^2+\sigma_2^2) $. Thus all the Random variables in the Sequence $\{Y_k\}$ will also be i.i.d.
Following the comments I have tried to prove that $Y_i$'s are also independent: To prove the independence of $Y_i$'s we need to show that Cov($Y_k,Y_{k+1})=0$.
So,
Cov($Y_k,Y_{k+1})= \mathbb E[(Y_k - \mathbb EY_k)(Y_{k+1}-\mathbb EY_{k+1})] =$
$\mathbb E[Y_k Y_{k+1}]=\mathbb E[X_{k}X_{k+1}-X_{k-1}X_{k+1}+X_{k}X_{k-1}-X_{k}^2]=0$
Please correct me if I am wrong.
Thanks in advance!