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PART 3Random Processes

Huseyin BilgekulEeng571 Probability and astochastic Processes

Department of Electrical and Electronic Engineering Eastern Mediterranean University

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Random Processes

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Kinds of Random Processes

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• A RANDOM VARIABLE X, is a rule for assigning to every outcome, of an experiment a number X(. – Note: X denotes a random variable and X( denotes

a particular value. • A RANDOM PROCESS X(t) is a rule for

assigning to every a function X(t, – Note: for notational simplicity we often omit the

dependence on .

Random Processes

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Conceptual Representation of RP

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The set of all possible functions is called the ENSEMBLE.

Ensemble of Sample Functions

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• A general Random or Stochastic Process can be described as: – Collection of time functions

(signals) corresponding to various outcomes of random experiments.

– Collection of random variables observed at different times.

• Examples of random processes in communications: – Channel noise, – Information generated by a source, – Interference.

t1 t2

Random Processes

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Let denote the random outcome of an experiment. To every such outcome suppose a waveform is assigned.The collection of such waveforms form a stochastic process. The set of and the time index t can be continuousor discrete (countably infinite or finite) as well.For fixed (the set of all experimental outcomes), is a specific time function.For fixed t,

is a random variable. The ensemble of all such realizations over time represents the stochastic

),( tX

}{ k

Si

),( 11 itXX

),( tX

t1

t2

t

),(n

tX

),(k

tX

),(2

tX

),(1

tX

),( tX

0

),( tX

Random Processes

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Random Process for a Continuous Sample Space

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Random Processes

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Wiener Process Sample Function

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Sample Sequence for Random Walk

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Sample Function of the Poisson Process

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Random Binary Waveform

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Autocorrelation Function of the Random Binary Signal

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Example

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Random Processes Introduction (1)

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Introduction

• A random process is a process (i.e., variation in time or one dimensional space) whose behavior is not completely predictable and can be characterized by statistical laws.

• Examples of random processes– Daily stream flow– Hourly rainfall of storm events– Stock index

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Random Variable• A random variable is a mapping function which assigns outcomes of a

random experiment to real numbers. Occurrence of the outcome follows certain probability distribution. Therefore, a random variable is completely characterized by its probability density function (PDF).

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STOCHASTIC PROCESS

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STOCHASTIC PROCESS

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STOCHASTIC PROCESS

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STOCHASTIC PROCESS

• The term “stochastic processes” appears mostly in statistical textbooks; however, the term “random processes” are frequently used in books of many engineering applications.

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STOCHASTIC PROC ESS

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DENSITY OF STOCHASTIC PROCESSES

• First-order densities of a random process A stochastic process is defined to be completely or totally characterized if the joint densities for the random variables are known for all times and all n. In general, a complete characterization is practically impossible, except in rare cases. As a result, it is desirable to define and work with various partial characterizations. Depending on the objectives of applications, a partial characterization often suffices to ensure the desired outputs.

)(),(),( 21 ntXtXtX

nttt ,,, 21

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• For a specific t, X(t) is a random variable with distribution .

• The function is defined as the first-order distribution of the random variable X(t). Its derivative with respect to x

is the first-order density of X(t).

])([),( xtXptxF

),( txF

xtxFtxf

),(),(

DENSITY OF STOCHASTIC PROCESSES

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• If the first-order densities defined for all time t, i.e. f(x,t), are all the same, then f(x,t) does not depend on t and we call the resulting density the first-order density of the random process ; otherwise, we have a family of first-order densities.

• The first-order densities (or distributions) are only a partial characterization of the random process as they do not contain information that specifies the joint densities of the random variables defined at two or more different times.

)(tX

DENSITY OF STOCHASTIC PROCESSES

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• Mean and variance of a random process The first-order density of a random process, f(x,t), gives the probability density of the random variables X(t) defined for all time t. The mean of a random process, mX(t), is thus a function of time specified by

ttttX dxtxfxXEtXEtm ),(][)]([)(

MEAN AND VARIANCE OF RP

• For the case where the mean of X(t) does not depend on t, we have

• The variance of a random process, also a function of time, is defined by

constant). (a )]([)( XX mtXEtm

2222 )]([][)]()([)( tmXEtmtXEt XtXX

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• Second-order densities of a random process For any pair of two random variables X(t1) and X(t2), we define the second-order densities of a random process as or .

• Nth-order densities of a random process The nth order density functions for at times

are given by or .

),;,( 2121 ttxxf ),( 21 xxf

)(tX

nttt ,,, 21

),,,;,,,( 2121 nn tttxxxf ),,,( 21 nxxxf

HIGHER ORDER DENSITY OF RP

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• Given two random variables X(t1) and X(t2), a measure of linear relationship between them is specified by E[X(t1)X(t2)]. For a random process, t1 and t2 go through all possible values, and therefore, E[X(t1)X(t2)] can change and is a function of t1 and t2. The autocorrelation function of a random process is thus defined by

),()()(),( 122121 ttRtXtXEttR

Autocorrelation function of RP

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Autocovariance Functions of RP

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• Strict-sense stationarity seldom holds for random processes, except for some Gaussian processes. Therefore, weaker forms of stationarity are needed.

nnnn tttxxxftttxxxf ,,,;,,,,,,;,,, 21212121

Stationarity of Random Processes

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Time, t

PDF of X(t)X(t)

Stationarity of Random Processes

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. allfor constant)( )( tmtXE

. and allfor ,),( 21121221 ttttRttRttR

Wide Sense Stationarity (WSS) of Random Processes

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• Equality

• Note that “x(t, i) = y(t, i) for every i” is not the same as “x(t, i) = y(t, i) with probability 1”.

Equality and Continuity of RP

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Equality and Continuity of RP

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• Mean square equality Mean Square Equality of RP

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Equality and Continuity of RP

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Random Processes Introduction (2)

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Stochastic Continuity

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Stochastic Continuity

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Stochastic Continuity

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Stochastic Continuity

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Stochastic Continuity

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Stochastic Continuity

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• 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.

Stochastic Convergence

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• 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 .

Sure Convergence (Convergence Everywhere)

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Stochastic Convergence

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Stochastic Convergence

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Almost-sure convergence (Convergence with probability 1)

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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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• 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.

Remarks

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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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Example

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Example

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Example

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Ergodic Theorem

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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.

The Mean-Square Ergodic Theorem

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Mean-Ergodic Process

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Strong or Individual Ergodic Theorem

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Strong or Individual Ergodic Theorem

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

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• 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

iid Random Stochastic Processes

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Bernoulli Random Process

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Random walk process

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• 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

xxPxxPxxxfxxfx

xxPxxPxXPxxxxxXP

xXxXxXP

Random walk process

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)|()|()|()(

)|()|()|()(),,,(

),,,,(),,,|(

1

10100

110100

1100

111100

110011

nn

nn

nnnn

nn

nnnn

nnnn

xxPxxPxxPx

xxPxxPxxPxxXxXxXP

xXxXxXxXPxXxXxXxXP

Random walk process

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The property

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

)|(),,,|( 1110011 nnnnnnnn xXxXPxXxXxXxXP

Random walk process

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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, 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:

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Gaussian process

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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.

)(tX

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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

.)

0)0( X

]/[21)( tYYYtX ,, 21 YY

/t/t

The Brownian motion

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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., .

]/[21)( tYYYtX )(tX

tNtX ,0~)(

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Graphical illustration of Distribution of X(t)

Time, t

PDF of X(t)X(t)

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• If t and h are fixed and is sufficiently small then

1 2 [( ) / ] 1 2 [ / ]

[ / ] 1 [ / ] 2 [( ) / ]

2

( ) ( )

t h t

t t t h

t t t h

X t h X t

Y Y Y Y Y Y

Y Y Y

Y Y Y

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Graphical Distribution of the displacement of

• The random variable is normally distributed with mean 0 and variance h, i.e.

)()( tXhtX

)()( tXhtX

duhu

hxtXhtXP

x

2exp

21)()(

2

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• 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

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t

X

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Covariance and Correlation functions of )(tX

t

YYYE

YYYYYYYYYE

YYYYYYE

htXtXEhtXtXCov

t

httttt

htt

2

21

2121

2

21

2121

)()()(),(

Cov ( ), ( )Correl ( ), ( )

=

X t X t hX t X t h

t t h

tt t h