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Random Processes Introduction (2). Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering E-mail: rslab@ntu.edu.tw. Stochastic continuity . Stochastic Convergence.

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random processes introduction 2

Random ProcessesIntroduction (2)

Professor Ke-Sheng Cheng

Department of Bioenvironmental Systems Engineering

E-mail: rslab@ntu.edu.tw

stochastic convergence
Stochastic Convergence
  • A random sequence or a discrete-time random process is a sequence of random variables {X1(), X2(), …, Xn(),…} = {Xn()}, .
  • For a specific , {Xn()} is a sequence of numbers that might or might not converge. The notion of convergence of a random sequence can be given several interpretations.
sure convergence convergence everywhere
Sure convergence (convergence everywhere)
  • The sequence of random variables {Xn()} converges surely to the random variable X() if the sequence of functions Xn() converges to X() as n  for all , i.e.,

Xn() X() as n  for all .

  • 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.
venn diagram of relation of types of convergence
Venn diagram of relation of types of convergence

Note that even sure convergence may not imply mean square convergence.

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.
examples of stochastic processes
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(t1), X(t2), …, X(tk) are mutually independent and have a common cumulative distribution function FX() .

The joint cdf for X(t1), X(t2), …, X(tk) is given by
  • It also yields

where p(x) represents the common probability mass function.

Let 0 denote the probability mass function of X0. The joint probability of X0, X1, Xn is
The property

is known as the Markov property.

A special case of random walk: the Brownian motion.

gaussian process
Gaussian process
  • A random process {X(t)} is said to be a Gaussian random process if all finite collections of the random process, X1=X(t1), X2=X(t2), …, Xk=X(tk), are jointly Gaussian random variables for all k, and all choices of t1, t2, …, tk.
  • Joint pdf of jointly Gaussian random variables X1, X2, …, Xk:
the brownian motion one dimensional also known as random walk
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.
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 .)

distribution of x t
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., .
distribution of the displacement
Distribution of the displacement
  • The random variable is normally distributed with mean 0 and variance h, i.e.
Variance of is dependent on t, while variance of is not.
  • If , then ,

are independent random variables.