Some density functions of a counting variable in the simple random walk

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Abstract
Extract Let 〈Xi 〉 be a sequence of completely independent and identically distributed random variables such that Xi = + 1, −1 with probabilities p (0 < p < 1) and q ≡ 1 − p respectively, and let 〈Sn〉 be the corresponding sequence of partial sums, i.e. . The Strong Law of Large Numbers asserts that, with probability one, where µ = E(Xi) = p - q, and therefore, with probability one, the inequality Sn > (µ + λ)n holds for only finitely many n-values, where λ is any positive number less than 2 to avoid triviality. Define the indicator variables Yn ≡ Yn (p, λ) by Yn = 1 if Sn > (µ + λ)n and Yn = 0 otherwise. The counting variable of interest here is N∞ ≡ ∞ (p, λ) and is defined by . Hence, P{N∞ < ∞} = 1, that is, N∞ is an honest random variable or N∞ has a proper distribution. The following theorem provides some exact density functions of N∞ for selected combinations of p and λ and thus elucidates the nature of chance fluctuations of sums of Bernoulli random variables with respect to the “finitely many” in the Strong Law of Large Numbers.

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