= Stochastic process
{title2=$(X_t)_{t\in T}$}
{wiki}
A stochastic process describes uncertainty about a whole history: not just the <probability distribution> of a measurement now, but how measurements at different times fit together. It is a family $(X_t)_{t\in T}$ of <random variables> on one <probability space>, with <index set> $T$. Time is the usual index: $T=\{0,1,2,\ldots\}$ for successive observations or $T=[0,\infty)$ for continuous time. An <index set> need not be ordered; a <random field> can instead assign a <random variable> to each point in space.
There are two complementary views. Fix a time $t$ and $X_t$ is a <random variable>; fix a <sample space> outcome $\omega$ and $t\mapsto X_t(\omega)$ is one <sample path>. A model specifies the <probability distribution> of these histories. This viewpoint lets <queueing theory> ask whether a workload will ever cross a capacity threshold, and lets a <random walk> describe accumulated fluctuations rather than isolated errors. The common <probability space> and the distinction between a time coordinate and a <sample path> are developed in https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/b9342e718ec4c58d98ec97dcfcb0d56e_MIT6_436JF18_lec20.pdf[MIT, The basics of stochastic processes, section 1].
A concrete construction starts with a <Bernoulli process> of independent fair bits $B_1,B_2,\ldots$. Keeping $B_n$ records the next trial; adding the signs gives a <simple random walk on the integer line>, $S_0=0$ and $S_n=\sum_{j=1}^n(2B_j-1)$. Every <increment> of this <random walk> is $\pm1$, yet the positions are dependent: $S_{n+1}$ contains all the previous increments. Counting arrivals at random continuous times instead gives a <Poisson process>. These examples separate three questions: what values are possible, how time is indexed, and how the values depend on one another.
\Image[/stochastic-process-paths.png]
{title=Three kinds of random history: fair bits, their accumulated signs, and Poisson arrival counts}
For the fair <simple random walk on the integer line>, the <expected value> is $\mathbb E S_n=0$ and the <variance> is $\operatorname{Var}(S_n)=n$. At time four, returning to zero requires exactly two positive steps, so $\mathbb P(S_4=0)=\binom42/2^4=3/8$. A rate-two <Poisson process> instead has $\mathbb P(N_1=0)=e^{-2}\approx0.1353$. A single realized <sample path> does not establish either <probability>: they follow from the specified <joint probability distributions>.
The most revealing feature is often dependence. Two <stochastic processes> can have identical <marginal distributions> at every time while one forgets everything at the next step and the other repeats its first value forever. Their <finite-dimensional distributions> and <covariance functions> distinguish them. That extra structure is what makes a <stochastic process> useful for prediction.
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