Strong Approximation of Brownian Motion
نویسنده
چکیده
Simple random walk and Brownian motion are two strongly interconnected mathematical concepts. They are widely involved in not only pure math, but also in many other scientific fields. In this paper I will first introduce and define some basic concepts of discrete-time random walk. Then I will construct Brownian Motion with some basic properties, and use a method called the strong approximation of Brownian Motion to show that the simple random walk indeed converges to standard Brownian motion. Finally I will give an application of this fact, and demonstrate how considering Brownian motion as a limit of the simple random walk can be convenient when looking at problems, since it is often easier to deal with Brownian motions instead of simple random walks.
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