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

2. Stochastic continuity

8. 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.

9. 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   .

12. Almost-sure convergence (Convergence with probability 1)

14. Mean-square convergence

15. Convergence in probability

17. Convergence in distribution

18. 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.

19. Weak Law of Large Numbers (WLLN)

20. Strong Law of Large Numbers (SLLN)

21. The Central Limit Theorem

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

23. Example

27. Ergodic Theorem

30. The Mean-Square Ergodic Theorem

31. 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.

32. Mean-Ergodic Processes

33. Strong or Individual Ergodic Theorem

36. 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() .

37. 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.

39. Random walk process

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

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

43. 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:

45. Time series – AR random process

46. 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.

47. 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 )

48. 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.,

49. Graphical illustration of Distribution of X(t)
Time, t
PDF of X(t)
X(t)

50. If t and h are fixed and  is sufficiently small then

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

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

53. t X

54. Covariance and Correlation functions of