A random walk on the set of all integers is a random process in which we move from a whole number to one of its neighboring values on every step; that means that on every step the value of our walk either increases or decreases by 1. These steps are independent of each other, which makes random walks a Markov process because it is not important how we got to the number at which the walk stands, only the value itself.
This kind of walks is very useful for modeling many practical problems. However, the most interesting things about them are their properties since we can, for example, observe the probability of the walk reaching a particular number after a fixed amount of steps taken or the probability of some value ever being reached by the walk at all. We can also search for the maximal or minimal value which the walk reaches.
Random walks on whole numbers with an unlimited number of steps or infinite walks for short are, however, the most useful in a practical sense. These types of walks are particularly interesting in their limits or in their probabilities of reaching a fixed number for one last time..
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