Would this process be a white noise sequence?
Consider the process $\{tY_t\}_{t = 1, . . . , 100}$, where $Y_t$ are independently, normally distributed with mean 0 and variance 1.
The process holds for the two first conditions which are:
- $$ E(Y_t) = 0 $$
- $$ Var (Y_t) = 1 $$
I am unsure about the third condition which states that:
- $$ E(Y_t,Y_s) = 0 $$
My intuitive guess would be that this condition will still hold as the sequence is independent.
No, the process is not white noise, because the variance is not constant, it depends on $t$. We have that $$ \operatorname{Var}[tY_t]=t^2. $$
However, the elements of this sequence are uncorrelated. Since $Y_t$ and $Y_s$ are independent, $$ \operatorname E[sX_stX_t]=st\operatorname EX_s\operatorname EX_t=0. $$ We do not need independence for the first equality, non-correlation is sufficient, but independence implies non-correlation.