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 nth 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