The Polyas Random Walk Theorem

Abstract

Since there are a variety of uncertainties and probabilities in the actual world, mathematicians and other researchers have found numerous answers to these common problems. They have also developed theories, such as the probability theory, to help them organize their significant discoveries. A drunk person will eventually find their way home, but a drunk bird might never find it, as said by Shizuo Kakutani. A random walk, in its most basic definition, is just a path on a network or lattice where each step is decided at random using some probability distribution, such a coin or fair dice. A motion, such as going left or right on a number line or up or down on a plane, would correspond to each outcome of tossing a coin, heads or tails. Recurrent random walks are those that end up back at the beginning after a predetermined number of steps. Any non-recurrent random walk is transitory.