ON FLUCTUATIONS IN COIN-TOSSING

In the classical coin-tossing game we have a sequence of independent random variables X_v, v = 1, 2,… each taking the values ± 1 with probability 1/2. We are interested in the signs of the partial sums . To eliminate the zeros of S_n we make the following convention: S_n is “positive” if S_n > 0 or if S_n = 0 but S_n−1 > 0; otherwise S_n is “negative.” The elegance of the results to be announced depends on this convention. Let N_n denote the number of “positive” terms among S₁, …, S_n. We shall confine ourselves to an even n, noting only that 0 ≤ N_n+1 − N_n ≤ 1. In the following r and m are positive integers and P(B∣A) is the conditional probability of B under the hypothesis A…