# Prove a general random walk is not stationary

Here is my attempt. $$X_t$$ is a general random walk is when we have a sequence of independent and identically distributed random variables $$Y_1,Y_2,...$$ such that $$X_0=0,$$ $$X_1=Y_1$$, $$X_2=Y_1+Y_2$$ and so on which then we have $$X_t=X_{t-1}+Y_t$$.

If we let $$Y_i=a$$ where $$a$$ is any constant, then we have $$X_t=at$$. Then $$E(X_t)=aE(t)$$ which we can see it depends on $$t$$ so the mean is not constant so it is not even weakly stationary.

Does this work?

You are asked to prove that $$\{X_n\}$$ cannot be stationary, so considering a special case is not good enough. You have to assume that $$Y_i$$'s are not $$0$$ random variables. Suppose $$\{X_n\}$$ is stationary. Then the distribution of $$X_n$$ does not depend on $$n$$. In particular $$Y_1$$ has same distribution as $$Y_1+Y_2$$. But $$Y_1$$ and $$Y_2$$ are i.i.d. This implies $$Y_1=Y_2=0$$ almost surely. [There are many ways of proving this. One way is to use characteristic functions. I will give more details if needed].