Re: Is the best prediction of a random walk that it´s going to be the same as n

I would really like to know this: If a process, like the stock-market or the weather prognosis, is following a random walk (a Markov-process?) is it best to predict that the market or the wheather tomorrow (the next step) is going to be the same as today, and why??

Very Sincerely Mikael!!

First let's look at what a random walk is. A random walk is a stochastic process (so it takes random steps) that evolves in the following way. At each step, we take the current position, and add a random move that is independent of all the previous moves. For instance, I could take the following random walk on the 2 dimensional lattice. Start at (0,0), and at each step with equal probability move up (add (0,1) to my state) move to the right (add (1,0) to my state) move down (add (0,-1) to my state) or move left (add (-1,0) to my state). Each move that I make is independent of the previous moves, and at any given time I have no idea what move will be made next.

Now suppose that I'm at point X_t, and I want to say something about the next point X_t+1 in the walk. Well, one approach is to compute the expected value of X_t+1. This is the average value of the random point X_t+1, and can be computed by summing over the product of the possible values of X_t+1 times their probability of occurring. So E[X_t+1] = (¼)(X_t + (0,0)) + (¼)(X_t + (1,0)) + (¼)(X_t + (0,-1)) + (¼)(X_t + (-1,0)). When you add this all up, this means that E[X_t+1] = X_t. In this case, the expected value of the next point is just the current point.

But this is not always the case. Suppose that when we move right, we move two steps to the right, so that we add (2,0) to the point instead of (1,0). This is still a random walk, and we have E [X_t+1] = (¼)(X_t + (0,0)) + (¼)(X_t + (2,0)) + (¼)(X_t + (0,-1)) + (¼)(X_t + (-1,0)) = X_t + (½, 0). So here the expected value of the next point is not the previous point, and the previous point isn't as good a prediciton of the next value.

When making predictions, knowing the exact details of the random walk is very important!

Mark Huber