(2) A random process
X
(
t
) is statistically speciﬁed by its complete set of
n
th order probability distribution (or density) function
F
X
‡
x
1
,x
2
,
···
,x
n
;
t
1
,t
2
,
···
,t
n
·
for all
x
1
,x
2
,
···
,x
n
, and for all time
t
1
< t
2
<
···
< t
n
.
(3) The mean function, autocorrelation function, and autocovariance func
tion are deﬁned as:
Mean function:
μ
X
(
t
)
,
E
h
X
(
t
)
i
=
Z
∞
∞
xf
X
(
x
;
t
)
dx,
Autocorrelation function: for all
t
1
and
t
2
R
X
(
t
1
,t
2
)
,
E
h
X
(
t
1
)
X
*
(
t
2
)
i
=
Z
∞
∞
Z
∞
∞
x
1
x
*
2
f
X
(
x
2
,x
2
;
t
1
,t
2
)
dx
1
dx
2
,
Autocovariance function: for all
t
1
and
t
2
K
X
[
t
1
,t
2
]
,
E
h
(
X
(
t
1
)

μ
X
(
t
1
)
)(
X
(
t
2
)

μ
X
(
t
2
)
)
*
i
=
R
XX
(
t
1
,t
2
)

μ
X
(
t
1
)
μ
*
X
(
t
2
)
.
(4)
Deﬁnition (Independent Increments)
A random process is said to have
independent increments
if the set
of
n
random variables
X
(
t
1
)
,X
(
t
2
)

X
(
t
1
)
,
···
,X
(
t
n
)

X
(
t
n

1
)
are jointly independent for
t
1
< t
2
<
···
< t
n
and for all
n >
1.
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