Standard deviation of absolute distance of a 1D random walk

Question

Given a 1D random walk (simple +1, -1 movements from the axis) I've seen proofs that the expected absolute distance tends to Sqrt(2*n/PI) and I've plotted graphs of 1D random walks along with this curve along and the associated standard deviation of the absolute distance. As expected most walks can be found in the area around the expected absolute distance curve up to 2 standard deviations from it.

Now take a walk that zooms steeply up or down towards the 2 standard deviation line from the expected absolute distance curve. From what little I understand of statistics very few random walks should be found above this line - 95% rule (and indeed my graphs confirm this). So doesn't this imply that such a walk should therefore most likely turn around and head back into the area closer to the expected absolute distance curve? Or at least continue on its original direction but at a less steep path thus following the root(N) curve closely. But this seems silly because it implies to some degree that you can predict the motion of the path (either a bounce or a flattening of the trajectory). Hence my confusion!

Answer 1

Great question. I have pondered nearly the same problem for a very long time..

Perhaps you would like to look at the Arc Sine Law.. Basically what "leads tends to stay in the lead".

Secondly regards your observation of standard deviation.. Using a simple coin toss..

100 tosses at 1 standard deviation would suggest that heads/tails = 45/ 55 on the 100th toss.

2 standard deviations would be heads /tails = 60/ 40. However that would equate to only 1 standard deviation by 400th toss. So yes one could argue that if heads is 2 standard deviations ahead at 100 toss then tails may "Predictably" make a comeback causing both to be within 1 standard deviation i.e. expected 68% of the time. Problem being your "out of the money" as an option trader might say within 300 tosses from your entry at toss 100.And that's were the arc sine law kicks you in the ***!!! intuitive thinking body parts!! and where many a trading account balance has miraculously disappeared.

I have come to the conclusion at present that

1. Yes if a particle moves above lets say 1 s/d then it may well return to oscillate within 1s/d (no drift & 1d walk assumed).

2. Yes it may zoom of to 3 s/d but oscillate around there till the money runs out. So making predictability worthless.

3. The above are really like stretching elastic bands.. Some stretch only a little, (limited excursion) , some keep stretching (arc sine) whilst others just stretch then contract then stretch over and over (68% i.e. oscillate around 1 s/d from n) .

The problem is one just does not know which elastic band ones playing with at any give time.. until AFTER THE FACT. Which is of course a little bit late and the markets have long ago had your cash and your left saying "IF Only"!!! L.O.L.

That's why I am now focused on the TYPE of Random Walk my equity is on rather than on solely its relationship to standard deviation at a given "N".

All the best with your thinking. and hope that my input gives you food for thought. Apologies in advance for the simplistic nature of my reply but I'm a chef by profession and math is only a very dear hobby of mine..