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Symmetric random walks, including those with step sizes drawn from a Gaussian process with mean zero, have expectation 0 at any time t. Since there's no drift t
by benmaraschino 13y ago
Symmetric random walks, including those with step sizes drawn from a Gaussian process with mean zero, have expectation 0 at any time t. Since there's no drift term, either walk won't be expected to slowly deviate from zero.
- deleted 13y ago[deleted]
- thetwiceler 13y agoThis is false. Think about it this way. A random walk is a martingale - you're expectation of where you'll be in the future is. So you're right that, before you start the walk, they have expectation of being at 0 at any time in the future. But the variance of your probability distribution of where you'll be at time t is linear in t. So say that your variance is v(t)=t. Then at t=1, there is a 32% chance that you'll be outside of the range (-1,1). As you can see, as t increases, you expected to drift further than further. So while the expectation of x(t) may be 0 for all time, the expectation of |x(t)| scales like sqrt(t) (the standard deviation of the distribution).
- benmaraschino 13y agoThanks for the clarification! I haven't studied stochastic processes in a while, so the distinction was lost on me; it definitely makes more sense to think of it as a martingale.
- mturmon 13y agoWhat you say is true. Another, related fact is that no matter how far from zero the process has strayed, eventually, with probability 1, it will return to zero.