Random Processes
Introduction
Remark 1. For an arbitrary index set \(T\), and a real-valued function \(x \in \R^T\), the projection operator \(\pi_t:\R^T\to\R\) maps \(x\in\R^T\) to \(\pi_t(x) = x_t\).
Definition 1 (Random process). Let \((\Omega, \sF, P)\) be a probability space. For an arbitrary index set \(T\) and state space \(\sX \subseteq \R\), a map \(X : \Omega \to \sX^T\) is called a random process if the projections \(X_t:\Omega\to\sX\) defined by \(\omega \mapsto X_t(\omega) \triangleq (\pi_t\circ X)(\omega)\) are random variables on the given probability space.
Definition 2. For each outcome \(\omega \in \Omega\), we have a function \(X(\omega): T \mapsto \sX\) called the sample path or the sample function of the process \(X\).
Remark 2. A random process \(X\) defined on probability space \((\Omega, \sF, P)\) with index set \(T\) and state space \(\sX\subseteq\R\), can be thought of as
a map \(X: \Omega \times T \to \sX\),
a map \(X: T \to \sX^\Omega\), i.e. a collection of random variables \(X_t: \Omega \to \sX\) for each time \(t \in T\),
a map \(X: \Omega \to \sX^T\), i.e. a collection of sample functions \(X(\omega): T \to \sX\) for each random outcome \(\omega \in \Omega\).
Classification
State space \(\sX\) can be countable or uncountable, corresponding to discrete or continuous valued process. If the index set \(T \subseteq \R\) is countable, the stochastic process is called discrete-time stochastic process or random sequence. When the index set \(T\) is uncountable, it is called continuous-time stochastic process. The index set \(T\) doesn’t have to be time, if the index set is space, and then the stochastic process is spatial process. When \(T = \R^n \times [0, \infty)\), stochastic process \(X\) is a spatio-temporal process.
Example 3. We list some examples of each such stochastic process.
Discrete random sequence: brand switching, discrete time queues, number of people at bank each day.
Continuous random sequence: stock prices, currency exchange rates, waiting time in queue of \(n\)th arrival, workload at arrivals in time sharing computer systems.
Discrete random process: counting processes, population sampled at birth-death instants, number of people in queues.
Continuous random process: water level in a dam, waiting time till service in a queue, location of a mobile node in a network.
Measurability
For random process \(X:\Omega\to \sX^T\) defined on the probability space \((\Omega, \sF, P)\), the projections \(X_t \triangleq \pi_t\circ X\) are \(\sF\)-measurable random variables. Therefore, the set of outcomes \(A_{X_t}(x) \triangleq X_t^{-1}(-\infty, x] \in \sF\) for all \(t \in T\) and \(x \in \R\).
Definition 4. A random map \(X:\Omega\to\sX^T\) is called \(\sF\)-measurable and hence a random process, if the set of outcomes \(A_{X_t}(x) = X_t^{-1}(-\infty, x]\in \sF\) for all \(t \in T\) and \(x\in \R\).
Definition 5. The event space generated by a random process \(X: \Omega\to\sX^T\) defined on a probability space \((\Omega,\sF,P)\) is given by
Definition 6. For a random process \(X:\Omega \to \sX^T\) defined on the probability space \((\Omega,\sF,P)\), we define the projection of \(X\) onto components \(S \subseteq T\) as the random vector \(X_S: \Omega \to \sX^S\), where \(X_S \triangleq (X_s: s \in S)\).
Remark 3. Recall that \(\pi_t^{-1}(-\infty, x] = \bigtimes_{s \in T}(-\infty, x_s]\) where \(x_s = x\) for \(s = t\) and \(x_s = \infty\) for all \(s\neq t\). The \(\sF\)-measurability of process \(X\) implies that for any countable set \(S \subseteq T\), we have \(A_{X_S}(x_S) \triangleq \cap_{s \in S}A_{X_s}(x_s) \in \sF\) for \(x_S \in \sX^S\).
Remark 4. We can define \(A_X(x) \triangleq \cap_{t \in T}A_{X_t}(x_t)\) for any \(x \in \R^T\). However, \(A_X(x)\) is guaranteed to be an event only when \(S \triangleq \set{t \in T: \pi_t(x) < \infty}\) is a countable set. In this case,
Example 7 (Bernoulli sequence). Consider a sample space \(\set{H,T}^\N\). We define a mapping \(X: \Omega \to \set{0,1}^\N\) such that \(X_n(\omega) = \SetIn{H}(\omega_n) = \SetIn{\omega_n = H}\). The map \(X\) is an \(\sF\)-measurable random sequence, if each \(X_n: \Omega \to \set{0,1}\) is a bi-variate \(\sF\)-measurable random variable on the probability space \((\Omega,\sF,P)\). Therefore, the event space \(\sF\) must contain the event space generated by sequence of events \(E\in\sF^\N\) defined by \(E_n \triangleq \set{\omega \in \Omega: X_n(\omega)=1} = \set{\omega \in \Omega: \omega_n = H} \in \sF\) for all \(n\in \N\). That is,
Distribution
Definition 8. For a random process \(X: \Omega \to \sX^T\) defined on the probability space \((\Omega,\sF, P)\), we define a finite dimensional distribution \(F_{X_S}: \R^S \to [0,1]\) for a finite \(S \subseteq T\) by
Example 9. Consider a probability space \((\Omega,\sF, P)\) defined by the sample space \(\Omega = \set{H,T}^\N\), the event space \(\sF \triangleq \sigma(E)\) where \(E_n = \set{\omega \in \Omega: \omega_n = H}\) for \(n\in\N\), and the probability measure \(P: \sF \to [0,1]\) defined by Let \(X: \Omega\to\set{0,1}^\N\) defined as \(X_n(\omega) = \Ind{E_n}(\omega)\) for all outcomes \(\omega \in \Omega\) and \(n \in \N\). For this random sequence, we can obtain the finite dimensional distribution \(F_{X_S}: \R^S \to [0,1]\) for any finite \(S \subseteq T\) and \(x \in \R^S\) in terms of \(I_0(x) \triangleq \set{i \in S: x_i < 0}\) and \(I_1(x) \triangleq \set{i \in S: x_i \in [0,1)}\), as
To define a measure on a random process, we can either put a measure on subsets of sample paths \((X(\omega) \in \R^T: \omega \in \Omega)\), or equip the collection of random variables \((X_t \in \R^\Omega: t \in T)\) with a joint measure. Either way, we are interested in identifying the joint distribution \(F: \R^T \to [0,1]\). To this end, for any \(x \in \R^T\), we need to know First of all, we don’t know whether \(A_X(x)\) is an event when \(T\) is uncountable. Though, we can verify that \(A_X(x) \in \sF\) for \(x \in \R^T\) such that \(\set{t \in T: x_t < \infty}\) is countable. Second, even for a simple independent process with countably infinite \(T\), any function of the above form would be zero if \(x_t\) is finite for all \(t \in T\). Therefore, we only look at the values of \(F_X(x)\) for \(x \in \R^T\) where \(\set{t \in T: x_t < \infty}\) is finite. That is, for any finite set \(S \subseteq T\), we focus on the events \(A_S(x_S)\) and their probabilities. However, these are precisely the finite dimensional distributions. Set of all finite dimensional distributions of the stochastic process \(X: \Omega \to \sX^T\) characterizes its distribution completely.
Example 10. Consider a probability space \((\Omega,\sF, P)\) defined by the sample space \(\Omega = \set{H,T}^\N\) and the event space \(\sF \triangleq \sigma(E)\) where \(E_n = \set{\omega \in \Omega: \omega_n = H}\) for all \(n\in\N\). Let \(X: \Omega\to\set{0,1}^\N\) defined as \(X_n(\omega) = \Ind{E_n}(\omega)\) for all outcomes \(\omega \in \Omega\) and \(n \in \N\). For this random sequence, if we are given the finite dimensional distribution \(F_{X_S}: \R^S \to [0,1]\) for any finite \(S \subseteq T\) and \(x \in \R^S\) in terms of sets \(I_0(x) \triangleq \set{i \in S: x_i < 0}\) and \(I_1(x) \triangleq \set{i \in S: x_i \in [0,1)}\), as defined in Eq. [eqn:FDDBernoulli]. Then, we can find the probability measure \(P: \sF \to [0,1]\) is given by
Independence
Definition 11. A random process is independent if the collection of event spaces \((\sigma(X_t): t \in T)\) is independent. That is, for all \(x_S \in \R^S\), we have That is, independence of a random process is equivalent to factorization of any finite dimensional distribution function into product of individual marginal distribution functions.
Example 12. Consider a probability space \((\Omega,\sF, P)\) defined by the sample space \(\Omega = \set{H,T}^\N\), the event space \(\sF \triangleq \sigma(E)\) where \(E_n = \set{\omega \in \Omega: \omega_n = H}\) for all \(n\in\N\), and the probability measure \(P: \sF \to [0,1]\) defined by Then, we observe that the random sequence \(X: \Omega\to\set{0,1}^\N\) defined by \(X_n(\omega) \triangleq \Ind{E_n}(\omega)\) for all outcomes \(\omega \in \Omega\) and \(n \in \N\), is independent.
Definition 13. Two stochastic processes \(X: \Omega \to \sX^{T_1},Y:\Omega\to \sY^{T_2}\) are independent, if the corresponding event spaces \(\sigma(X), \sigma(Y)\) are independent. That is, for any \(x \in \R^{S_1}, y \in \R^{S_2}\) for finite \(S_1 \subseteq T_1, S_2\subseteq T_2\), the events \(A_{S_1}(x) \triangleq \cap_{s\in S_1}X_s^{-1}(-\infty, x_s]\) and \(B_{S_2}(y) \triangleq \cap_{s \in S_2}Y_s^{-1}(-\infty, y_s]\) are independent. That is, the joint finite dimensional distribution of \(X\) and \(Y\) factorizes, and