On the Simple Symmetric Random Walk and its Maximal Function

Abstract

For the simple symmetric one-dimensional random walk S_n we derive closed form and asymptotic formulas for E max(S_n,0), E|S_n|, E M_n, EM_n², cov(S_n,M_n) and the correlation coefficient of S_n and M_n, where M_n = max {0, S₁,…,S_n}, using elementary material in Feller's classic textbook.

KEY WORDS:

• Covariance
• Correlation coefficient