On formulas for moments of the Wishart distributions as weighted ...

1164 Yasuhide NUMATA and Satoshi KURIKI
Propsition 4.13 Let σ u,v = σ, δ u,v = δ. For W = (w u,v ) ∼ CW p (ν, Σ, ∆),
n∑
E[w˙1,¨1 w˙2,¨2 · · · w ∑
ṅ,¨n] = g lmn ν l σ m δ n−m ,
where g lmn is the number of E ⊂ K V such that len(E) = l, |E| = m and ∣ ∣ Ě ∣ ∣ = n − m.
m=0
Corollary 4.14 For the noncentral complex chi-square distribution χ 2 ν(δ) with ν degrees of freedom and
the noncentrality parameter δ, its n-th moment E(w n ) is given as follows:
n∑ ∑
E[w n ] = g lmn ν l δ n−m .
m=0
In the case where we add a new directed edge whose staring point is a fixed vertex, we have just one
choice of end-points that increase the number of cycles. Hence we obtain Lemma 4.15.
l
Lemma 4.15 Let 0 ≤ m ≤ n. Then the generating function G mn (t) of g lmn with respect to the number
l of cycles satisfies
G mn (t) = ∑ ( ) n ∏ m
g lmn t l = (t + n − i).
m
l≥0
i=1
We also obtain the following corollary, which is well-known expression for the noncentral chi-square
distribution (e.g. [5])
Corollary 4.16 For the n-th moment E(w n ) of the noncentral chi-square distribution χ 2 ν(δ) with ν degrees
of freedom and the noncentrality parameter δ,
n∑ ∑
n∑
n∑
( n ∏
E[w n ] = g lmn ν l δ n−m = G mn (ν)δ n−m = δ
m) m n−m (ν + n − i).
m=0
l
m=0
Remark 4.17 The numbers s n (m, l) defined by the following generating function are called the noncentral
Stirling numbers of the first kind:
∑
m∏
s n (m, l)t l = (t + n − i).
l
( If m = n, then s n (m, l) is the Stirling number of the first kind. Lemma 4.15 implies that g lmn =
n
)
m sn (m, l). Equivalently, we can explicitly describe the moments of the noncentral chi-square distribution
χ 2 ν(δ) with the noncentral Stirling numbers. Koutras pointed out that moments of some noncentral
distributions are described with the noncentral Stirling numbers of the first kind [7].
Next consider the real case.
i=1
l
m=0
Propsition 4.18 Let σ u,v = σ, δ u,v = δ. For W = (w u,v ) ∼ W p (ν, Σ, ∆),
n∑
E[w˙1,¨1 w˙2,¨2 · · · w ∑
ṅ,¨n] = g lmnν ′ l σ m δ n−m ,
where g ′ lmn is the number of E′ ⊂ K ′ V ′ such that len(E′ ) = l, |E ′ | = m and ∣ ∣ Ě ′∣ ∣ = n − m.
m=0
l
i=1