For the result in quantum field theory about products of creation and annihilation operators, see
Wick's theorem .
In probability theory , Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis .
This theorem is also particularly important in particle physics , where it is known as Wick's theorem after the work of Wick (1950) .[1] Other applications include the analysis of portfolio returns,[2] quantum field theory[3] and generation of colored noise.[4]
Statement
If
(
X
1
,
…
,
X
n
)
{\displaystyle (X_{1},\dots ,X_{n})}
is a zero-mean multivariate normal random vector, then
E
[
X
1
X
2
⋯
X
n
]
=
∑
p
∈
P
n
2
∏
{
i
,
j
}
∈
p
E
[
X
i
X
j
]
=
∑
p
∈
P
n
2
∏
{
i
,
j
}
∈
p
Cov
(
X
i
,
X
j
)
,
{\displaystyle \operatorname {E} [\,X_{1}X_{2}\cdots X_{n}\,]=\sum _{p\in P_{n}^{2}}\prod _{\{i,j\}\in p}\operatorname {E} [\,X_{i}X_{j}\,]=\sum _{p\in P_{n}^{2}}\prod _{\{i,j\}\in p}\operatorname {Cov} (\,X_{i},X_{j}\,),}
where the sum is over all the pairings of
{
1
,
…
,
n
}
{\displaystyle \{1,\ldots ,n\}}
, i.e. all distinct ways of partitioning
{
1
,
…
,
n
}
{\displaystyle \{1,\ldots ,n\}}
into pairs
{
i
,
j
}
{\displaystyle \{i,j\}}
, and the product is over the pairs contained in
p
{\displaystyle p}
.
[5] [6]
In his original paper,[7] Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the
4
th
{\displaystyle 4^{\text{th}}}
order moments,[8] which takes the appearance
E
[
X
1
X
2
X
3
X
4
]
=
E
[
X
1
X
2
]
E
[
X
3
X
4
]
+
E
[
X
1
X
3
]
E
[
X
2
X
4
]
+
E
[
X
1
X
4
]
E
[
X
2
X
3
]
.
{\displaystyle \operatorname {E} [\,X_{1}X_{2}X_{3}X_{4}\,]=\operatorname {E} [X_{1}X_{2}]\,\operatorname {E} [X_{3}X_{4}]+\operatorname {E} [X_{1}X_{3}]\,\operatorname {E} [X_{2}X_{4}]+\operatorname {E} [X_{1}X_{4}]\,\operatorname {E} [X_{2}X_{3}].}
Odd case
If
n
=
2
m
+
1
{\displaystyle n=2m+1}
is odd, there does not exist any pairing of
{
1
,
…
,
2
m
+
1
}
{\displaystyle \{1,\ldots ,2m+1\}}
. Under this hypothesis, Isserlis' theorem implies that
E
[
X
1
X
2
⋯
X
2
m
+
1
]
=
0.
{\displaystyle \operatorname {E} [\,X_{1}X_{2}\cdots X_{2m+1}\,]=0.}
This also follows from the fact that
−
X
=
(
−
X
1
,
…
,
−
X
n
)
{\displaystyle -X=(-X_{1},\dots ,-X_{n})}
has the same distribution as
X
{\displaystyle X}
, which implies that
E
[
X
1
⋯
X
2
m
+
1
]
=
E
[
(
−
X
1
)
⋯
(
−
X
2
m
+
1
)
]
=
−
E
[
X
1
⋯
X
2
m
+
1
]
=
0
{\displaystyle \operatorname {E} [\,X_{1}\cdots X_{2m+1}\,]=\operatorname {E} [\,(-X_{1})\cdots (-X_{2m+1})\,]=-\operatorname {E} [\,X_{1}\cdots X_{2m+1}\,]=0}
.
Even case
If
n
=
2
m
{\displaystyle n=2m}
is even, there exist
(
2
m
)
!
/
(
2
m
m
!
)
=
(
2
m
−
1
)
!
!
{\displaystyle (2m)!/(2^{m}m!)=(2m-1)!!}
(see double factorial ) pair partitions of
{
1
,
…
,
2
m
}
{\displaystyle \{1,\ldots ,2m\}}
: this yields
(
2
m
)
!
/
(
2
m
m
!
)
=
(
2
m
−
1
)
!
!
{\displaystyle (2m)!/(2^{m}m!)=(2m-1)!!}
terms in the sum. For example, for
4
th
{\displaystyle 4^{\text{th}}}
order moments (i.e.
4
{\displaystyle 4}
random variables) there are three terms. For
6
th
{\displaystyle 6^{\text{th}}}
-order moments there are
3
×
5
=
15
{\displaystyle 3\times 5=15}
terms, and for
8
th
{\displaystyle 8^{\text{th}}}
-order moments there are
3
×
5
×
7
=
105
{\displaystyle 3\times 5\times 7=105}
terms.
Proof
Let
Σ
i
j
=
Cov
(
X
i
,
X
j
)
{\displaystyle \Sigma _{ij}=\operatorname {Cov} (X_{i},X_{j})}
be the covariance matrix, so that we have the zero-mean multivariate normal random vector
(
X
1
,
.
.
.
,
X
n
)
∼
N
(
0
,
Σ
)
{\displaystyle (X_{1},...,X_{n})\sim N(0,\Sigma )}
.
Using quadratic factorization
−
x
T
Σ
−
1
x
/
2
+
v
T
x
−
v
T
Σ
v
/
2
=
−
(
x
−
Σ
v
)
T
Σ
−
1
(
x
−
Σ
v
)
/
2
{\displaystyle -x^{T}\Sigma ^{-1}x/2+v^{T}x-v^{T}\Sigma v/2=-(x-\Sigma v)^{T}\Sigma ^{-1}(x-\Sigma v)/2}
, we get
1
(
2
π
)
n
det
Σ
∫
e
−
x
T
Σ
−
1
x
/
2
+
v
T
x
d
x
=
e
v
T
Σ
v
/
2
{\displaystyle {\frac {1}{\sqrt {(2\pi )^{n}\det \Sigma }}}\int e^{-x^{T}\Sigma ^{-1}x/2+v^{T}x}dx=e^{v^{T}\Sigma v/2}}
Differentiate under the integral sign with
∂
v
1
,
.
.
.
,
v
n
|
v
1
,
.
.
.
,
v
n
=
0
{\displaystyle \partial _{v_{1},...,v_{n}}|_{v_{1},...,v_{n}=0}}
to obtain
E
[
X
1
⋯
X
n
]
=
∂
v
1
,
.
.
.
,
v
n
|
v
1
,
.
.
.
,
v
n
=
0
e
v
T
Σ
v
/
2
{\displaystyle E[X_{1}\cdots X_{n}]=\partial _{v_{1},...,v_{n}}|_{v_{1},...,v_{n}=0}e^{v^{T}\Sigma v/2}}
.
That is, we need only find the coefficient of term
v
1
⋯
v
n
{\displaystyle v_{1}\cdots v_{n}}
in the Taylor expansion of
e
v
T
Σ
v
/
2
{\displaystyle e^{v^{T}\Sigma v/2}}
.
If
n
{\displaystyle n}
is odd, this is zero. So let
n
=
2
m
{\displaystyle n=2m}
, then we need only find the coefficient of term
v
1
⋯
v
n
{\displaystyle v_{1}\cdots v_{n}}
in the polynomial
1
m
!
(
v
T
Σ
v
/
2
)
m
{\displaystyle {\frac {1}{m!}}(v^{T}\Sigma v/2)^{m}}
.
Expand the polynomial and count, we obtain the formula.
◻
{\displaystyle \square }
Generalizations
Gaussian integration by parts
An equivalent formulation of the Wick's probability formula is the Gaussian integration by parts . If
(
X
1
,
…
X
n
)
{\displaystyle (X_{1},\dots X_{n})}
is a zero-mean multivariate normal random vector, then
E
(
X
1
f
(
X
1
,
…
,
X
n
)
)
=
∑
i
=
1
n
Cov
(
X
1
X
i
)
E
(
∂
X
i
f
(
X
1
,
…
,
X
n
)
)
.
{\displaystyle \operatorname {E} (X_{1}f(X_{1},\ldots ,X_{n}))=\sum _{i=1}^{n}\operatorname {Cov} (X_{1}X_{i})\operatorname {E} (\partial _{X_{i}}f(X_{1},\ldots ,X_{n})).}
The Wick's probability formula can be recovered by induction, considering the function
f
:
R
n
→
R
{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
defined by
f
(
x
1
,
…
,
x
n
)
=
x
2
…
x
n
{\displaystyle f(x_{1},\ldots ,x_{n})=x_{2}\ldots x_{n}}
. Among other things, this formulation is important in
Liouville Conformal Field Theory to obtain
conformal Ward's identities ,
BPZ equations [9] and to prove the
Fyodorov-Bouchaud formula .
[10]
Non-Gaussian random variables
For non-Gaussian random variables, the moment-cumulants formula[11] replaces the Wick's probability formula. If
(
X
1
,
…
X
n
)
{\displaystyle (X_{1},\dots X_{n})}
is a vector of random variables , then
E
(
X
1
…
X
n
)
=
∑
p
∈
P
n
∏
b
∈
p
κ
(
(
X
i
)
i
∈
b
)
,
{\displaystyle \operatorname {E} (X_{1}\ldots X_{n})=\sum _{p\in P_{n}}\prod _{b\in p}\kappa {\big (}(X_{i})_{i\in b}{\big )},}
where the sum is over all the
partitions of
{
1
,
…
,
n
}
{\displaystyle \{1,\ldots ,n\}}
, the product is over the blocks of
p
{\displaystyle p}
and
κ
(
(
X
i
)
i
∈
b
)
{\displaystyle \kappa {\big (}(X_{i})_{i\in b}{\big )}}
is the
joint cumulant of
(
X
i
)
i
∈
b
{\displaystyle (X_{i})_{i\in b}}
.
See also
References
^ Wick, G.C. (1950). "The evaluation of the collision matrix". Physical Review . 80 (2): 268–272. Bibcode :1950PhRv...80..268W . doi :10.1103/PhysRev.80.268 .
^ Repetowicz, Przemysław; Richmond, Peter (2005). "Statistical inference of multivariate distribution parameters for non-Gaussian distributed time series" (PDF) . Acta Physica Polonica B . 36 (9): 2785–2796. Bibcode :2005AcPPB..36.2785R .
^ Perez-Martin, S.; Robledo, L.M. (2007). "Generalized Wick's theorem for multiquasiparticle overlaps as a limit of Gaudin's theorem". Physical Review C . 76 (6): 064314. arXiv :0707.3365 . Bibcode :2007PhRvC..76f4314P . doi :10.1103/PhysRevC.76.064314 . S2CID 119627477 .
^ Bartosch, L. (2001). "Generation of colored noise" . International Journal of Modern Physics C . 12 (6): 851–855. Bibcode :2001IJMPC..12..851B . doi :10.1142/S0129183101002012 . S2CID 54500670 .
^ Janson, Svante (June 1997). Gaussian Hilbert Spaces . Cambridge Core . doi :10.1017/CBO9780511526169 . ISBN 9780521561280 . Retrieved 2019-11-30 .
^ Michalowicz, J.V.; Nichols, J.M.; Bucholtz, F.; Olson, C.C. (2009). "An Isserlis' theorem for mixed Gaussian variables: application to the auto-bispectral density". Journal of Statistical Physics . 136 (1): 89–102. Bibcode :2009JSP...136...89M . doi :10.1007/s10955-009-9768-3 . S2CID 119702133 .
^ Isserlis, L. (1918). "On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables" . Biometrika . 12 (1–2): 134–139. doi :10.1093/biomet/12.1-2.134 . JSTOR 2331932 .
^ Isserlis, L. (1916). "On Certain Probable Errors and Correlation Coefficients of Multiple Frequency Distributions with Skew Regression" . Biometrika . 11 (3): 185–190. doi :10.1093/biomet/11.3.185 . JSTOR 2331846 .
^ Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent (2019-11-01). "Local Conformal Structure of Liouville Quantum Gravity". Communications in Mathematical Physics . 371 (3): 1005–1069. arXiv :1512.01802 . Bibcode :2019CMaPh.371.1005K . doi :10.1007/s00220-018-3260-3 . ISSN 1432-0916 . S2CID 55282482 .
^ Remy, Guillaume (2020). "The Fyodorov–Bouchaud formula and Liouville conformal field theory". Duke Mathematical Journal . 169 . arXiv :1710.06897 . doi :10.1215/00127094-2019-0045 . S2CID 54777103 .
^ Leonov, V. P.; Shiryaev, A. N. (January 1959). "On a Method of Calculation of Semi-Invariants" . Theory of Probability & Its Applications . 4 (3): 319–329. doi :10.1137/1104031 .
Further reading
Koopmans, Lambert G. (1974). The spectral analysis of time series . San Diego, CA: Academic Press .