Random walk: Need intuition behind the expected distance from origin versus expected deviation of the fraction of left/right steps.

random walk

It appears that in a 1D random walk the expected RMS distance from the origin is some positive value with any positive number of steps (N):

sqrt(N)/2

However, in this same walk, the expected deviation of the fraction of steps to any one direction tends towards zero. 

1/(2 * sqrt(N))

I cannot intuitively see why the former doesn't tend towards zero as well if the average steps to the left cancels with the steps to the right.

Does anyone have an intuitive explanation for this?

Best Answer

I think I intuitively see this more clearly now.

If one draws two parallel lines A and B with lengths equal to coin flips where heads adds a cm to A and tails adds a cm to B, then the more flips one takes, the larger the difference in lengths between A and B. This tends towards a positive value, much like the RMS distance from the origin.

However, if one takes the fraction of that difference (A-B)/(A+B) over expected difference, which is 0.5 for balanced flipping, then that fraction tends towards zero with more flips.

The excess tends to increase whereas the proportion of the excess over the total steps tends towards zero.

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