## Random Walk (33/365)

One of the annoying things, for me, when studying probability are the examples furnished in textbooks because they almost always have to do with physics or gambling or finance. There is nothing wrong with physics examples except for the fact that they too easily fit into the probabilistic framework. So, I can’t generalize them to more abstract situtations. Gambling and financial examples hold no interest for me whatsoever. What I really want are suprising examples.

Let me give an example. As I mentioned, I am currently working out problems on random walks, martingales, and markov chains. Take the basic random walk which we construct as follows. Let the sample space be $\Omega = \{ \omega : \omega = (\omega_1, \dots, \omega_n), \omega_i = \pm 1 \}$ and $p(\omega) = p^{v(\omega)} (1-p)^{n- v(\omega)}$ where $v(\omega) = (\sum \omega_i + n) / 2$.

Now, we consider the sequence of these random variables

$\displaystyle S_0(\omega) = 0 \\ S_k(\omega) = \omega_1 + \dots + \omega_k = S_{k-1}(\omega) + \omega_k$

Here are some standard ways of interpreting this sequence of random variables

1. Given a 2-dimensional grid, $S_k$ represents the position after $k$ steps (starting from $(0,0)$) taken by either going up one or going to the right one.

2. Consider a gambling game with two players. If $\omega_i = 1$ lets say player A gains one dollar from player B and when $\omega_i = -1$ player B gains one dollar from player A. Suppose player A and B start with $d_A$ dollars and $d_B$ dollars. $S_k$ therefore represents the amount of money won by player $A$ after $k$ turns. If $S_k = d_B$ then player B has lost all his money. So we here can ask questions like what is the probability that player A or player B will be ruined (i.e. loses all his money).

This is all well and good and the examples generalize to more general random walks but I want to consider some completely left-field examples. I can’t guarantee they will lead anywhere and may be utterly rubbish but I think it will be an interesting exercise. See you next time!

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