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Random Processes Introduction (2) Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering

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Stochastic continuity

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Stochastic Convergence A random sequence or a discrete-time random process is a sequence of random variables {X 1 ( ), X 2 ( ), …, X n ( ),…} = {X n ( )},. For a specific, {X n ( )} is a sequence of numbers that might or might not converge. The notion of convergence of a random sequence can be given several interpretations.

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Sure convergence (convergence everywhere) The sequence of random variables {X n ( )} converges surely to the random variable X( ) if the sequence of functions X n ( ) converges to X( ) as n for all, i.e., X n ( ) X( ) as n for all.

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Almost-sure convergence (Convergence with probability 1)

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Mean-square convergence

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Convergence in probability

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Convergence in distribution

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Remarks Convergence with probability one applies to the individual realizations of the random process. Convergence in probability does not. The weak law of large numbers is an example of convergence in probability. The strong law of large numbers is an example of convergence with probability 1. The central limit theorem is an example of convergence in distribution.

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Weak Law of Large Numbers (WLLN)

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Strong Law of Large Numbers (SLLN)

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The Central Limit Theorem

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Venn diagram of relation of types of convergence Note that even sure convergence may not imply mean square convergence.

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Example

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Ergodic Theorem

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The Mean-Square Ergodic Theorem

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The above theorem shows that one can expect a sample average to converge to a constant in mean square sense if and only if the average of the means converges and if the memory dies out asymptotically, that is, if the covariance decreases as the lag increases.

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Mean-Ergodic Processes

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Strong or Individual Ergodic Theorem

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Examples of Stochastic Processes iid random process A discrete time random process {X(t), t = 1, 2, …} is said to be independent and identically distributed (iid) if any finite number, say k, of random variables X(t 1 ), X(t 2 ), …, X(t k ) are mutually independent and have a common cumulative distribution function F X ( ).

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The joint cdf for X(t 1 ), X(t 2 ), …, X(t k ) is given by It also yields where p(x) represents the common probability mass function.

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Random walk process

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Let 0 denote the probability mass function of X 0. The joint probability of X 0, X 1, X n is

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The property is known as the Markov property. A special case of random walk: the Brownian motion.

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Gaussian process A random process {X(t)} is said to be a Gaussian random process if all finite collections of the random process, X 1 =X(t 1 ), X 2 =X(t 2 ), …, X k =X(t k ), are jointly Gaussian random variables for all k, and all choices of t 1, t 2, …, t k. Joint pdf of jointly Gaussian random variables X 1, X 2, …, X k :

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Time series – AR random process

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The Brownian motion (one-dimensional, also known as random walk) Consider a particle randomly moves on a real line. Suppose at small time intervals the particle jumps a small distance randomly and equally likely to the left or to the right. Let be the position of the particle on the real line at time t.

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Assume the initial position of the particle is at the origin, i.e. Position of the particle at time t can be expressed as where are independent random variables, each having probability 1/2 of equating 1 and 1. ( represents the largest integer not exceeding.)

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Distribution of X (t) Let the step length equal, then For fixed t, if is small then the distribution of is approximately normal with mean 0 and variance t, i.e.,.

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Graphical illustration of Distribution of X (t) Time, t PDF of X(t) X(t)X(t)

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If t and h are fixed and is sufficiently small then

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Distribution of the displacement The random variable is normally distributed with mean 0 and variance h, i.e.

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Variance of is dependent on t, while variance of is not. If, then, are independent random variables.

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t X

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Covariance and Correlation functions of

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