Random Walk v Distribution

Question

Is there a way to tell if a sequence of random numbers comes from a 'random walk' or a from a random sample from a distribution? In other words, suppose I generate 1000 values using the usual random walk formula [xi+1 = xi + RANDOMNORMAL value]. OR I select 1000 values using a typical Normal Random Number Generator. Can I distinguish one sequence as a random walk and the other as coming from a distribution, without knowing anything about how the random walk steps were generated OR what distribution function was used for the sample of random numbers?

Answer 1

Broadly speaking, a random walk will have much higher correlations between close by times than an iid sequence. In particular, you can consider $X_n$ given by sampling from a Brownian motion at integer times, vs. an iid sequence $Y_n$ of $N(0,1)$ random variables. In the latter case, $E[Y_m Y_n]=0$ for $m \neq n$. In the former case, $E[X_m X_n]=\min \{ m,n \}$. If all you have is one trajectory then technically you cannot robustly say anything about statistics. Nevertheless, if you plot the autocorrelation with some fixed window, for example if you plot $X_n X_{n+r}$ and $Y_n Y_{n+r}$ for some fixed positive integer $r$, you should find that the former oscillates around zero while the latter fluctuates but generally drifts upward.