Post on 27-Mar-2015
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Random ProcessesRandom ProcessesIntroduction Introduction (2)(2)
Professor Ke-Sheng ChengProfessor Ke-Sheng ChengDepartment of Bioenvironmental Systems Department of Bioenvironmental Systems EngineeringEngineering
E-mail: rslab@ntu.edu.tw
Stochastic continuity
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)
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 .
Almost-sure convergence (Convergence with probability 1)
Mean-square convergence
Convergence in probability
Convergence in distribution
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.
Weak Law of Large Numbers (WLLN)
Strong Law of Large Numbers (SLLN)
The Central Limit Theorem
Venn diagram of relation of types of convergence
Note that even sure convergence may not imply mean square convergence.
Example
Ergodic Theorem
The Mean-Square Ergodic Theorem
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.
Mean-Ergodic Processes
Strong or Individual Ergodic Theorem
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.
)()()(
,,,),,,(
21
221121,,, 21
kXXX
kkkXXX
xFxFxF
xXxXxXPxxxFk
)()()(),,,( 2121,,, 21 kXXXkXXX xpxpxpxxxpk
Random walk process
Let 0 denote the probability mass
function of X0. The joint probability of
X0, X1, Xn is
)|()|()(
)()()(
)()()(
,,,
),,,(
10100
10100
101100
101100
1100
nn
nn
nnn
nnn
nn
xxPxxPx
xxfxxfx
xxPxxPxXP
xxxxxXP
xXxXxXP
)|(
)|()|()(
)|()|()|()(
),,,(
),,,,(
),,,|(
1
10100
110100
1100
111100
110011
nn
nn
nnnn
nn
nnnn
nnnn
xxP
xxPxxPx
xxPxxPxxPx
xXxXxXP
xXxXxXxXP
xXxXxXxXP
The property
is known as the Markov property.
A special case of random walk: the Brownian motion.
)|(),,,|( 1110011 nnnnnnnn xXxXPxXxXxXxXP
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:
Time series – AR random process
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.
)(tX
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 .)
0)0( X
]/[21)( tYYYtX
,, 21 YY
/t/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., .
]/[21)( tYYYtX
)(tX
tNtX ,0~)(
Graphical illustration of Distribution of X(t)
Time, t
PDF of X(t)
X(t)
If t and h are fixed and is sufficiently small then
httt
httt
tht
YYY
YYY
YYYYYYtXhtX
2
]/)[(2]/[1]/[
]/[21]/)[(21)()(
Distribution of the displacement
The random variable is normally distributed with mean 0 and variance h, i.e.
)()( tXhtX
)()( tXhtX
duh
u
hxtXhtXP
x
2exp
2
1)()(
2
Variance of is dependent on t, while variance of is not.
If , then ,
are independent random variables.
)(tX
)()( tXhtX
mttt 2210 )()( 12 tXtX
,),()( 34 tXtX )()( 122 mm tXtX
t
X
Covariance and Correlation functions of )(tX
t
YYYE
YYYYYYYYYE
YYYYYYE
htXtXEhtXtXCov
t
httttt
htt
2
21
2121
2
21
2121
)()()(),(
htt
t
htt
htXtXCov
htXtXCorrel
)(),(
)(),(