On Convergence in Probability to Brownian Motion

Abstract

Several methods for embedding discrete martingales in Brownian motion by means of stopping times have been presented. Various conditions on the increments of the martingales are sufficient to insure that the trajectories of the embedded process and the Brownian motion are close. This paper will characterize all discrete stochastic processes, which can be constructed on some probability space supporting a Brownian motion, in such a way that the constructed process and the Brownian motion are close in probability, under suitable normalization. These are exactly the processes $\{S_j | j = 0, 1, \cdots\}$ such that, for any $\varepsilon > 0$ and $M = 0, 1, \cdots$ the conditional probability that the $(M + 1)$st change in size of at least 1 completed by the trajectory $0, S_1/n, S_2/n, \cdots$ is in $\lbrack 1, 1 + \varepsilon \rbrack$ (or $\lbrack -(1 + \varepsilon), - 1 \rbrack)$, given the process up to the $M$th such change, converges in probability to $\frac{1}{2}$ as $n \rightarrow \infty$. It is not required that any moments exist, or even that $E(S_{j+1} |S_1, \cdots, S_j) = S_j, \text{a.s.}$ In proving the main result, a new technique for constructing discrete processes from Brownian motion is presented.