The Probability of Return to the Origin

We had asked for the probability with which a random walk in the line will pass through a given position. If this position is the origin, we are asking for the probability of returning to the origin. Let us make an attempt at answering this question.

Suppose x is a point in the integer lattice on the real line (fancy language for x is an integer) and suppose n is the number of steps in a random walk. We begin by considering n-step random walks.

• Compute the number N_n,x of n-step random walks that are at location x after n steps. That is, calculate the number n-step random walks with $S_{n}(\omega) = x$. (It may help to write n = p + q where p is the number of +1's appearing in the walk and q is the number of -1's appearing in the walk up to this point.)

• Find the probability p_n,x for an infinite random walk $\omega$ to satisfy $S_{n}(\omega) = x$. The case where x = 0 is of particular use in this argument, so let us give it a separate label, u_n. Notice, n must be even in this case. Why?

• As in the gambler's ruin problem, it will be useful to keep track of the first return to the origin. Let f_2k denote the probablity of the first return occurring at step 2k, that is f_2k is the probability of $S_{1}(\omega) \neq 0, \ldots, S_{2k-1}(\omega) \neq 0, \mbox{ and } S_{2k}(\omega) = 0$.
  • Show that the probability that a random walk returns to the origin is
    
    $$f_{2} + f_{4} + \ldots + f_{2k} + \ldots$$
    
    

  • Show that the u's and the f's are related by
    
    $$u_{2k} = f_{2}u_{2k-2} + f_{4}u_{2k-4} + \ldots + f_{2k}u_{0}$$
    
    
    This will not be immediately useful to us, but we will be able to use it further on in the discussion of random walks.

In order to proceed with the calculation of the probability of return to the origin, we must express the f's in a simple fashion. This will be accomplished in several steps. First, it will be useful to visualize a random walk as a piecewise-linear graph of a function in the plane. The graph is obtained by connecting the points $(i,S_{i}(\omega))$.

• Show that for $y, \ y' >0$, the number of random walks from (x,y) to (x',y'), x' > x, that touch or cross the x-axis is the same as the number of random walks from (x,-y) to (x',y'). (We are concerned only with the number segments of random walks of length x' - x steps here.)

  Hint: Think geometrically. Draw a generic picture for this situation as a piecewise-linear plot in the plane and show that each random walk of the first type can be altered to produce a random walk of the second type.

• Show that the number of n-step random walks with $S_{n}(\omega) = x$ and $S_{i}(\omega) > 0$ for i > 1 is the same as the number of n-step random walks with $S_{n}(\omega) = x$, $S_{1}(\omega) = 1$, and $S_{i}(\omega) > 0$ for i > 1. Use this observation and the previous geometric fact to compute the number of n-step random walks from the origin to (n,x) with $S_{i}(\omega) > 0$.

• Show that the probability that $S_{2i}(\omega) \neq 0$, $i = 1, \ldots, n$ is twice the probability that $S_{2i}(\omega) > 0$, $i = 1, \ldots, n$.

• Show that the probability of $\omega$ satisfying
  
  $$S_{1}(\omega) > 0, \ldots, S_{2k}(\omega) > 0$$
  
  
  is the same as the sum over all r >0 of the probabilities that
  
  $$S_{1}(\omega) > 0, \ldots, S_{2k-1}(\omega) > 0, S_{2k}(\omega) = 2r$$
  
  

• Show that the previous sum of probablities is equal to u_2k. What does this say in terms of probabilities of random walks.

• Use this to express f_2k in terms of the u's.

• What is the probability that an infinite step random walk will return to the origin?