8.1 Stochastic Processes

Everything so far has concerned single random variables, or finite collections of them. A stochastic process is an indexed family, and the index is almost always time. The shift of view is what the rest of this section rests on, and the vocabulary is worth fixing before the particular processes appear.

Definition 8.1.1. A stochastic process is a collection \(\{X_t : t\in T\}\) of random variables defined on a common probability space and indexed by a set \(T\). The set \(T\) is the index set, usually interpreted as time; the set of values the \(X_t\) can take is the state space.

Note. Four combinations arise, and a process is classified by which it is:

\(-\): \(T\) countable and the state space countable — a Markov chain on a finite or countable set, for instance;
\(-\): \(T\) countable, state space continuous — a time series of real measurements;
\(-\): \(T\) an interval, state space countable — the Poisson process counting arrivals;
\(-\): \(T\) an interval, state space continuous — Brownian motion.

A process with \(T = \{0,1,2,\dots \}\) is called a discrete-time process; a process with \(T = [0,\infty )\) a continuous-time one.

Definition 8.1.2. A realisation, or sample path, of the process is the function \(t\mapsto X_t(\omega )\) obtained by fixing one outcome \(\omega \). The process is the collection of all such paths together with the probabilities attached to them.

Remark. The distinction between a process and its paths is the one that causes most trouble. A statement such as “\(X_t\rightarrow 0\)” is ambiguous until one says whether it concerns the paths, which is almost sure convergence, or the distributions, which is convergence in distribution — exactly the modes distinguished in the previous chapter, now applied along a time index. The four modes were not an abstract exercise; they are the language this section needs.

Properties a process may have

Definition 8.1.3. Let \(\{X_t\}\) be a stochastic process.

(i): It has independent increments if for \(t_0<t_1<\cdots <t_n\) the differences \(X_{t_1}-X_{t_0},\ \dots ,\ X_{t_n}-X_{t_{n-1}}\) are independent.
(ii): It has stationary increments if the distribution of \(X_{t+s}-X_{t}\) depends on \(s\) but not on \(t\).
(iii): It is strictly stationary if the joint distribution of \(\left (X_{t_1+h},\dots ,X_{t_n+h}\right )\) does not depend on \(h\).
(iv): It is weakly or second-order stationary if \(E(X_t)\) is constant and \(\Cov \left (X_t,X_{t+s}\right )\) depends on \(s\) alone.
(v): It has the Markov property if the future depends on the past only through the present: \[P\left (X_{t_{n+1}}\in A \mid X_{t_1},\dots ,X_{t_n}\right ) = P\left (X_{t_{n+1}}\in A \mid X_{t_n}\right ).\]

Note. Strict stationarity implies weak stationarity whenever the second moments exist, but not conversely — weak stationarity constrains only the first two moments and says nothing about the rest of the distribution. The one case where they coincide is the Gaussian process, because a joint normal distribution is determined by its mean vector and covariance matrix and by nothing else.

Some standard processes

Example 8.1.4 (White noise). A sequence \(\{\varepsilon _t\}\) of uncorrelated random variables with constant mean zero and constant variance \(\sigma ^{2}\). It is weakly stationary by construction, and it is the building block from which the processes below are assembled.

Example 8.1.5 (Random walk). Let \(\{\varepsilon _t\}\) be independent with mean zero and variance \(\sigma ^{2}\), and set \(X_0=0\), \(X_n = \sum _{i=1}^{n}\varepsilon _i\). Then \[E(X_n) = 0,\qquad \Var (X_n) = n\sigma ^{2},\] so the process has independent and stationary increments and the Markov property, but it is not stationary: its variance grows without bound. A random walk wanders away rather than settling, which is why differencing rather than averaging is the tool used on such series.

Example 8.1.6 (Counting process). A process \(\{N_t : t\geq 0\}\) taking non-negative integer values, with \(N_0=0\) and \(N_s\leq N_t\) whenever \(s\leq t\), so that \(N_t\) counts the events occurring in \([0,t]\). If it has independent and stationary increments and the number of events in an interval of length \(t\) is POI\((\lambda t)\), it is the Poisson process, which the next subsection treats in detail.

Remark. The three properties above — independent increments, stationary increments and the Markov property — are what make a process tractable, and the Poisson process has all three. That is the whole reason it occupies the place it does: not that arrivals in the world are especially Poisson, but that a process with those three properties and integer values is forced to be Poisson, so the assumptions are checkable even when the conclusion cannot be observed directly.

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