Random Processes Introduction (2)

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Random ProcessesIntroduction (2)

• Professor Ke-Sheng Cheng
• Department of Bioenvironmental Systems
  Engineering
• E-mail rslab_at_ntu.edu.tw

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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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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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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.

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Mean-Ergodic Processes
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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.

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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

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• 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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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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• 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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Distribution of the displacement

• 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