Limiting process of symmetric simple random walk

Abstract

In the present era, the study of stochastic process is getting more and more significant, in case that nearly nothing can be examined in a static sight and most of them are in an evolving procedure. In this article, we focus on one of the most common and most frequently used model, symmetric simple random walk, and discussed its property in R. Firstly, we review the basic terminology in the study of probability theory and stochastic process, clarify the basic concepts as distribution function, expectation, variance, independence, etc.; meanwhile, introduce two 0-1 laws. Then, we summarize the laws of large numbers and central limit theorem, in the order of definition of convergence, weak law, then strong laws; after that, the characteristic functions, the concept of independent and identical distribution, then comes to the <i>Lévy</i> central limit theorem. These parts are all for the preparation of the main conclusion: a limiting process of a symmetric simple random walk is a Brownian process. In this part, we start from the construction of symmetric simple random walk, then create a path to the limiting statue, and prove the final conclusion with the use of the concepts introduced previously; and finally, discuss several properties of Brownian motion; the limiting process of random walk.