Operator in probability theory
In probability theory , for a probability measure P on a Hilbert space H with inner product
⟨ ⟨ -->
⋅ ⋅ -->
,
⋅ ⋅ -->
⟩ ⟩ -->
{\displaystyle \langle \cdot ,\cdot \rangle }
, the covariance of P is the bilinear form Cov: H × H → R given by
C
o
v
(
x
,
y
)
=
∫ ∫ -->
H
⟨ ⟨ -->
x
,
z
⟩ ⟩ -->
⟨ ⟨ -->
y
,
z
⟩ ⟩ -->
d
P
(
z
)
{\displaystyle \mathrm {Cov} (x,y)=\int _{H}\langle x,z\rangle \langle y,z\rangle \,\mathrm {d} \mathbf {P} (z)}
for all x and y in H . The covariance operator C is then defined by
C
o
v
(
x
,
y
)
=
⟨ ⟨ -->
C
x
,
y
⟩ ⟩ -->
{\displaystyle \mathrm {Cov} (x,y)=\langle Cx,y\rangle }
(from the Riesz representation theorem , such operator exists if Cov is bounded ). Since Cov is symmetric in its arguments, the covariance operator is
self-adjoint . When P is a centred Gaussian measure , C is also a nuclear operator . In particular, it is a compact operator of trace class , that is, it has finite trace .
Even more generally, for a probability measure P on a Banach space B , the covariance of P is the bilinear form on the algebraic dual B # , defined by
C
o
v
(
x
,
y
)
=
∫ ∫ -->
B
⟨ ⟨ -->
x
,
z
⟩ ⟩ -->
⟨ ⟨ -->
y
,
z
⟩ ⟩ -->
d
P
(
z
)
{\displaystyle \mathrm {Cov} (x,y)=\int _{B}\langle x,z\rangle \langle y,z\rangle \,\mathrm {d} \mathbf {P} (z)}
where
⟨ ⟨ -->
x
,
z
⟩ ⟩ -->
{\displaystyle \langle x,z\rangle }
is now the value of the linear functional x on the element z .
Quite similarly, the covariance function of a function-valued random element (in special cases is called random process or random field ) z is
C
o
v
(
x
,
y
)
=
∫ ∫ -->
z
(
x
)
z
(
y
)
d
P
(
z
)
=
E
(
z
(
x
)
z
(
y
)
)
{\displaystyle \mathrm {Cov} (x,y)=\int z(x)z(y)\,\mathrm {d} \mathbf {P} (z)=E(z(x)z(y))}
where z (x ) is now the value of the function z at the point x , i.e., the value of the linear functional
u
↦ ↦ -->
u
(
x
)
{\displaystyle u\mapsto u(x)}
evaluated at z .
See also
Further reading
References
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