The 1-2-random walk : some bijections and a central limit theorem

A deeply researched topic in stochastic is the random walk. The most basic concept considers being at the starting point 0, and going 1 into positive or negative range with 50% probability each. Many problems have been solved for this stochastic process, for example the probability of returning to 0 or the convergence to the Brownian motion. However, this paper considers a modified random walk. Whenever the stochastic process is in non-negative range, the next step will be 1 up or 1 down with probability 1/2 again, and in negative range, the step length will be 2 instead of 1, increasing or decreasing the current value by 2, again with 50% probability each. In any case, there are no more restrictions and the step will be executed independently from previous steps. The problems solved for the classic random walks require other approaches in this modification, the 1-2-random walk. The first chapter of the paper is about basic concepts and briefly goes over the classic random walk. The convergence to the Brownian motion is shown with Donsker's theorem. In the second chapter, after a small detour to the 1-3-random walk (step length 3 in negative range), the 1-2-random walk is introduced. The task is: What is the probability of being in non-negative range for a very high amount of steps (tending to infinity)? For the 1-2-random walk, a bijection between the paths with length n ending in non-negative range and 3n-rectangles tiled into squares with length 1 or 2 is made. The third chapter attempts to find a convergence, like the Brownian motion for the classic random walk. For this purpose, stochastic differential equations are introduced. Some more bijections are made to be able to use Donsker's theorem again. At the end, another bijection considering the tiled 3n-rectangles is made, just to the paths in negative range this time, but they must have n+1 steps this time. Some ideas of solving the problem of the convergence of a generalized random walk are also stated.


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