Probability Theory 23 | Stochastic Processes [dark version]

Stochastic processes are framed as a clean way to model randomness that evolves over time: they are essentially random variables arranged across time points, all sharing the same underlying sample space. That simple structure matters because many real-world systems don’t just produce one random outcome—they change step by step, such as a growing bacteria population or a board game where a piece moves after each die roll. In this view, time can be discrete (indexed by natural numbers) or continuous (indexed by real numbers), depending on what the model needs.

A concrete coin-toss game makes the idea tangible. Consider repeatedly tossing a coin until two successive heads occur. The game can end early, later, or potentially never, so the random variable must be defined on the space of all possible infinite sequences of heads and tails. The sample space Ω is therefore the set of all sequences whose entries are either heads or tails, including sequences with infinitely many tosses.

The random variable Xₙ is then defined with three outcomes—0, 1, and 2—capturing the game’s progress after n tosses. Outcome 2 means the process has already achieved two consecutive heads within the first n tosses; once that happens, the game is effectively “absorbed,” and the value stays at 2 forever. Outcome 0 means no pair of successive heads has occurred yet and the nth toss is tails. Outcome 1 is the middle state: no consecutive heads have occurred yet, but the nth toss is heads—so the next toss could complete the target.

As n increases, the process “jumps” between these states with specific probabilities. From state 0, the next toss is heads with probability 1/2, moving the process to state 1; it is tails with probability 1/2, leaving the process in state 0. From state 1, a heads outcome with probability 1/2 advances to state 2 (two heads in a row), while a tails outcome with probability 1/2 resets back to state 0. From state 2, the probability of leaving is zero: the process remains at 2 with probability 1.

Before returning to the coin example, a formal definition is given. A stochastic process is built by choosing a time index set T (often T = ℕ or ℤ for discrete time, or T = ℝ for continuous time), fixing a sample space Ω, and then defining a random variable X_t for each time point t in T. Collecting these random variables into a single object—often written as (X_t)_{t∈T}—produces a “path” through state space as randomness unfolds over time. The coin game then becomes a simple illustration of how such time-indexed randomness can be represented as state transitions with clear probabilities, setting up the next step for deeper study in subsequent material.