What a stochastic process models
Section of Stochastic processes, part of Stochastic processes
What a stochastic process models
| Type | Section |
| Taught in | Stochastic processes |
| Part | Stochastic processes |
Definition 1 (Stochastic process) A stochastic process on a probability space (\Omega,\mathcal{F},\mathrm{P}), with state space S and index set T, is a family \{X_t:t\in T\} of S-valued random variables. If T=\{0,1,2,\ldots\}, time is discrete. If T=[0,\infty), time is continuous.
One outcome \omega produces the function t\mapsto X_t(\omega). That function is a sample path. The distribution of the single variable X_t is a marginal distribution. The joint distributions of (X_{t_1},\ldots,X_{t_k}), for every finite list of times, are the finite-dimensional distributions. The marginals do not determine those joints, so they do not determine the dependence across time.
Example 1 (One bit, or a fresh bit every time) Let B be a fair Bernoulli variable, so \mathbb{P}(B=1)=\mathbb{P}(B=0)=1/2.
Question. Compare the process X_n=B for every n with a process Y_n of independent fair Bernoulli variables.
Solution. At each fixed time both processes are Bernoulli(1/2). The joints disagree. The repeated bit satisfies \mathbb{P}(X_1=X_2)=1. The independent bits satisfy
\mathbb{P}(Y_1=Y_2)=\mathbb{P}(Y_1=Y_2=0)+\mathbb{P}(Y_1=Y_2=1)=\tfrac12\cdot\tfrac12+\tfrac12\cdot\tfrac12=\tfrac12.
A model for repeated observations has to say how the observations are coupled. Matching every marginal is not enough.
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See also
Stochastic process · Stationary and independent increments · State, clock, and assumptions · Toolkit for stochastic processes · Modelling checklist for a stochastic process · What a model can identify · Markov’s inequality · Chebyshev’s inequality