Titel_SS06

Continuous Random Processes

A random process X()t is as mentioned a random function of time meaning that for any point
in time the value of X
t is a random variable. A realisation of a random process (e.g. water
level variation) is illustrated in Figure 2.14.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

1.50

2.00

2.50

3.00

3.50

Water level [m]

Time [days]

Figure 2.14: Realization of the water level variation as function of time.

In accordance with the definition of the mean value of a random variable the mean value of all
the possible realisation of the stochastic process at time t is given by:

XX()txfxt( , )

   

 dx (2.58)

The correlation between all possible realisations at two points in time and is described
through the so-called autocorrelation function

t 1 t 2
RXX(, )tt 12. The autocorrelation function is
defined by:

RXX(, )tt 12 EXt Xt() ( ) 1 2 xxf xx tt dxdx 12 XX( , ; , )1 212 1 2

      

 (2.59)

The auto-covariance function is defined as:

12  1 1 2 2 

112 2 12121

(, ) ( () ())( ( ) ( ))

( ( )) ( ( )) ( , ; , )

XX X X

XXXX

Ctt EXt t Xt t

x t x t f x x t t dx dx 2





      

 

 

(2.60)

For tt 12 t the auto-covariance function becomes the covariance function:

(^2) () (, ) (, )  (^2) ()
XXXtCttRttXX Xt (2.61)
where ( )X t is the standard deviation function.
The above definitions for the scalar processX( )t
t))T
may be extended to cover also vector valued
processes having covariance functions

. For i these become the auto-covariance functions and when
these are termed the cross-covariance functions. Finally the correlation function may be
defined as:

X( ) (tXtXt X 12 ( ), ( ),..., n(

CXXXijcovij(), ( )tXt 12 j

ij/

12

12

12

cov ( ), ( )

(), ( )

ij() ( )

ij

ij

XX

X tXt

Xt X t

tt

1





  (2.62)