A stochastic process is a mathematical entity used to represent the evolution of a system over time governed by probabilistic laws, rather than deterministic rules. Unlike a single random variable, we define it fundamentally as a collection of random variables $\{X_n : n \in T\}$ indexed by time. In this lesson, we focus on the Simple Random Walk (SRW)—a discrete-time model simulating a gambler's fortune, starting at an initial value ($a$) and progressing through independent bets.
1. The Mechanics of a Simple Random Walk
We express the state of the walk at time $n$ as the sum of independent increments:
$$X_n = a + Z_1 + Z_2 + \dots + Z_n$$
where each $Z_i$ represents the outcome of a bet: $+1$ (win) with probability $p$, and $-1$ (loss) with probability $q = 1-p$.
Let $\{X_n\}$ be a simple random walk. If $k$ is an integer such that $-n \leq k \leq n$ and $n + k$ is even, then the probability of being at state $a+k$ after $n$ steps is:
$$P(X_n = a+k) = \binom{n}{\frac{n+k}{2}} p^{(n+k)/2} q^{(n-k)/2}$$
Crucial Pitfall: For all other values of $k$ (where $n+k$ is odd or $|k| > n$), $P(X_n = a + k) = 0$. This "parity check" ensures you can only reach specific states based on the number of steps taken.
2. Expectation and Fairness
The average trajectory of the process depends on the probability $p$. The expected value at time $n$ is given by:
$E(X_n) = a + n(2p - 1)$
- Fair Game ($p = 1/2$): The process is a Martingale. On average, the fortune remains constant: $E(X_{n+1} - X_n | X_n) = 0$.
- Subfair Game ($p < 1/2$): The process drifts downward toward ruin.
- Superfair Game ($p > 1/2$): The process drifts upward.
3. The Broader Landscape
While SRW deals with discrete sums, stochastic processes also encompass continuous models. For example, the Poisson Process ($N_t$) features independent increments where $P(N_t = k) = e^{-at} \frac{(at)^k}{k!}$. We also see these dynamics in target distributions for MCMC sampling, such as $f(y) = e^{-y^4}(1+|y|)^3$. These processes often utilize transition notation like $v_1 = v_0 A$.