214 Modeling and Inverse Problems in the Presence of Uncertainty
where R(t, s) = Cor{X(t), X(s)} and V(t, s) = Cov{X(t), X(s)}. In addition,
both R and V are non-negative definite in the sense that
l
X
j=1
l
X
k=1
c
j
R(t
j
, t
k
)c
k
≥ 0,
l
X
j=1
l
X
k=1
c
j
V(t
j
, t
k
)c
k
≥ 0 (7.7)
hold for any t
1
, t
2
, . . . , t
l
∈ T and for any c = (c
1
, c
2
, . . . , c
l
)
T
∈ R
l
. We note
that (7.7) can be easily established through the following equalities:
l
X
j=1
l
X
k=1
c
j
R(t
j
, t
k
)c
k
= c
T
E(XX
T
)c,
l
X
j=1
l
X
k=1
c
j
V(t
j
, t
k
)c
k
= c
T
E{(X − E(X))(X − E(X))
T
}c,
where X = (X(t
1
), X(t
2
), . . . X(t
l
))
T
.
Similarly, the m-time auto-correlation function of the stochastic process
{X(t) : t ∈ T} is defined by
Cor{X(t
1
), X(t
2
), . . . , X(t
m
)}
= E (X(t
1
)X(t
2
) ···X(t
m
)) ,
=
Z
R
m
m
Y
j=1
x
j
p
X
1
,...,X
m
(t
1
, x
1
, t
2
, x
2
, . . . , t
m
,