The Brownian Motion, and Other Processes in Continuous Time

Abstract

As we have seen, a sequence of random variables can describe how some process evolves in time. For example, the gambler’s accumulated winnings at time n, that is, after the nth play of a game of chance, is given by the random variable Sn = Xl +...Xn, where the X’s are independent random variables representing the gambler’s winnings at each play of the game. For convenience, we can think of the subscript n of any such sequence of random variables as denoting the nth moment of time, and the value of the random variable Sn as the measurement made at that moment. The sequence therefore represents a “picture” of the evolution of the process at the times 1, 2 ,... .