PART 3 Random Processes

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PART 3Random Processes
Huseyin Bilgekul Eeng571 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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Random Processes

• A RANDOM VARIABLE X, is a rule for assigning to
  every outcome, w, of an experiment a number
  X(w).
• Note X denotes a random variable and X(w)
  denotes a particular value.
• A RANDOM PROCESS X(t) is a rule for assigning to
  every w, a function X(t,w).
• Note for notational simplicity we often omit the
  dependence on w.

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Conceptual Representation of RP
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Ensemble of Sample Functions
The set of all possible functions is called the
ENSEMBLE.
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Random Processes

• 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
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Random Processes
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 continuous or 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
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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.

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DENSITY OF STOCHASTIC PROCESSES

• 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).

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DENSITY OF STOCHASTIC PROCESSES

• 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.

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MEAN AND VARIANCE OF RP

• 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

• 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

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HIGHER ORDER DENSITY OF RP

• 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
  .

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Autocorrelation function of RP

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

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Autocovariance Functions of RP
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Stationarity of Random Processes

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

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Stationarity of Random Processes
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Wide Sense Stationarity (WSS) of Random Processes
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Equality and Continuity of RP

• 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.

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Equality and Continuity of RP
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Mean Square Equality of RP

• Mean square equality

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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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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.

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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 ? ? ?.

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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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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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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 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.

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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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iid Random Stochastic Processes

• 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.

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Bernoulli Random Process
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Random walk process
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Random walk process

• Let ?0 denote the probability mass function of
  X0. The joint probability of X0, X1, ? Xn is

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Random walk process
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Random walk process

• 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, X1X(t1), X2X(t2), , XkX(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.

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The Brownian motion

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

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Graphical Distribution of the displacement of

• 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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Covariance and Correlation functions of