Download Random Processes - Basic Definitions

Random Processes - Basic Definitions
Guy Lebanon
January 6, 2006
We started some time ago with random variables X : Ω → R and moved to random vectors (X1 , . . . , Xn )
which are a vector of random variables. In a similar way, random processes are an infinite collection of
random variables (of infinite dimensional random vectors). Discrete-time random processes are a collection
of RV indexed by a countable set, for example the natural numbers X = {Xn : n = 1, 2, . . .} or the integers
X = {Xn : n ∈ Z}. Continuous-time processes are a collection of RV indexed by a non-countable set that
is a subset of the real-line, for example X = {Xt : t ∈ R, t ≥ 0}. We will use the index n for discrete-time
processes and t for continuous-time processes. When we want to treat both cases we will use the index t.
Regardless of whether the process X are discrete-time or continuous-time the RVs Xn or Xt can be either
discrete or continuous RVs.
If we look at the random processes as a function of the index n (for discrete-time) or t (for continuoustime) and hold ω ∈ Ω fixed we get a discrete-time or a continuous-time signal (or function) Xω , e.g.
Xω (t) = Xt (ω), called the realization or sample path of the processes at ω ∈ Ω. In this view, random
processes are probabilistic models on continuous or discrete signals. Just like vector-RV we can define events
{X ∈ A} = {ω ∈ Ω : Xω ∈ A}
for A being a set of (discrete-time or continuous-time) functions. The probability P (X ∈ A) is then defined
as the probability of that event. The processes can be independent (meaning the collection of RV defining
the process are independent) or not.
Example (independent discrete-time discrete-valued processes): let Ω = (0,
and P be a uniform disP1)
∞
tribution over it. Denoting b1 , b2 , . . . as the decimal expansion of ω i.e. ω = i=1 bi 10−i , bi ∈ {0, 1, . . . , 9}
we
Q∞define the processes
Q∞ X 1by Xn (ω) = bn , Xω = (b1 , b2 , . . .) and for example, P (X = (1, 1, 1, 1, . . .)) =
P
(X
=
1)
=
n
n=1
n=1 10 = 0.
Example (dependent continuous-time continuous-valued processes): For the same sample space and P ,
consider the process defined by Xt (ω) = ω sin(2πt). The realization path is a sinusoid with random amplitude. This is a dependent process (Xω (t1 ) and Xω (t2 ) are dependent RVs) with continuous-time and
continuous RVs.
Just as we define the marginal distribution of a random vector, we can define the finite-dimensional
marginal distribution of a random process, for example,
P ((X1 , . . . , Xk ) ∈ A) = P ({ω ∈ Ω : (X1 (ω), . . . , Xk (ω)) ∈ A}).
The notion of pdf and pmf are difficult to translate to random processes due to its infinite dimensionality.
Instead, we specify them for the finite-dimensional marginals of the processes.
Theorem 1 (Kolmogorov). Let I ⊂ R be an index set (think of this as the range of the index n or t). A
collection of cdfs FXc1 ,...,Xck (xc1 , . . . , xck ) for all choices of c1 , . . . , ck ∈ I and for all orders k = 1, 2, . . . that
is consistent in the sense that
FXc1 ,...,Xck (xc1 , . . . , xck ) = FXc1 ,...,Xck ,Xck+1 (xc1 , . . . , xck , ∞)
uniquely defines a random process for which the cdfs above are the marginal cdfs.
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In a similar way, a collection of pdfs fXc1 ,...,Xck (xc1 , . . . , xck ) for all choices of c1 , . . . , ck and for all orders
k uniquely defines a continuous-valued process if the pdf are consistent in the sense that
Z ∞
fXc1 ,...,Xck ,Xck+1 (xc1 , . . . , xck , xck+1 ) dxck+1 = fXc1 ,...,Xck (xc1 , . . . , xck ).
−∞
A collection of pmfs pXc1 ,...,Xck (xc1 , . . . , xck ) for all choices of c1 , . . . , ck and for all orders k uniquely defines
a discrete-valued process if the pmfs are consistent in the sense
X
pXc1 ,...,Xck ,Xck+1 (xc1 , . . . , xck , xck+1 ) = pXc1 ,...,Xck (xc1 , . . . , xck ).
xck+1
The standard relation between these functions (cdf,pmf,pdf) behaves as described earlier in the course for
finite dimensional random vectors).
Example: The discrete-time process described above can be defined by the collection of k-order pmfs
P (Xc1 = xc1 , . . . , Xck = xc1 ) =
1
10k
for all k and for all c1 , . . . , ck ∈ {0, 1, . . . , 9}. Note that the pmfs are consistent with each other (in the sense
described above).
Definition 1. A process is said to have independent increments if for any k and for all choice of indices
t1 < · · · < tk the random variables
Xt2 − Xt1 , . . . , Xtk − Xtk−1
are independent (this definition holds for both discrete-time and continuous-time).
Definition 2. A process is said to be a Markov process if for any k and for all choice of indices t1 < · · · < tk
fXtk |Xt1 =xt1 ,...,Xtk−1 =xtk−1 (xtk ) = fXtk |Xtk−1 =xtk−1 (xtk ) for continuous-time processes
(1)
pXtk |Xt1 =xt1 ,...,Xtk−1 =xtk−1 (xtk ) = pXtk |Xtk−1 =xtk−1 (xtk ) for discrete-time processes
(2)
A process with independent increments is necessarily Markov but not vice-verse.
Recall that for vector RV the mean and covariance are vectors and matrices. The analogue for processes
become functions. Note however, the inappropriate (but conventional) definition of autocorrelation in light
of our previous definition of correlation.
• The mean of a random process X is the function mX (t) = E(Xt )
• The autocorrelation of a random process X is the function RX (t1 , t2 ) = E(Xt1 Xt2 )
• The auto-covariance of a random process X is the function CX (t1 , t2 ) = E((Xt1 − mX (t1 ))(Xt2 −
mX (t2 )). By the results we have for covariance we also have CX (t1 , t2 ) = RX (t1 , t2 ) − mX (t1 )mX (t2 ).
• The variance at time t of the process is CX (t, t)
Example: For a RV Y define a process Xt = Y cos(2πt). The mean is mX (t) = E(Y cos(2πt)) = E(Y ) cos(2πt)
and the auto-correlation is RX (t1 , t2 ) = E(Y cos(2πt1 )Y cos(2πt2 )) = E(Y 2 ) cos(2πt1 ) cos(2πt2 ) and the
autocovariance is
CX (t1 , t2 ) = RX (t1 , t2 ) − mX (t1 )mX (t2 ) = (E(Y 2 ) − (E(Y ))2 cos(2πt1 ) cos(2πt2 )
If we have two processes X , Y we have similar definitions for their interaction:
• The two processes are said to be independent if all the finite dimensional marginals of the two processes
(Xt1 , . . . , Xtk ), (Yt01 , . . . , Yt0l ) are two vector RVs (independent from each other).
• The cross correlation is RX ,Y (t1 , t2 ) = E(Xt1 Yt2 ). It the cross correlation is identically zero the two
processes are said to be orthogonal.
• The cross-covariance of the two processes is CX ,Y (t1 , t2 ) = E((Xt1 − mX (t1 ))(Yt2 − mY (t2 )).
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