如果
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W
W 是 Wishart 分布的随机矩阵
W
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W_m(n,\lambda I_m)
Wm(n,λIm)，则
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E\left [ \mathrm{det} W^k \right ]=\prod_{l=0}^{m-1}\frac{\Gamma(n-l+k)}{\Gamma(n-l)}
E[detWk]=l=0∏m−1Γ(n−l)Γ(n−l+k)
证明： Wishart 分布随机矩阵的特征值PDF为
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exp
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f(l) = \frac{\pi^{m\left ( m-1 \right )}}{\lambda ^{mn}\tilde{\Gamma} _m \left( {n} \right)\tilde{\Gamma} _m \left( {m} \right)} \exp\left(-\frac{1}{\lambda}\sum _{i=1}^{m} l_i \right) \prod\limits_{i} {l_i^{\left ( n-m\right )}} \prod\limits_{i < j} {\left( {l_i - l_j } \right)^2}
f(l)=λmnΓ~m(n)Γ~m(m)πm(m−1)exp(−λ1i=1∑mli)i∏li(n−m)i<j∏(li−lj)2
因此
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\begin{aligned} E\left [ \mathrm{det}W^k \right ]&=\int \prod_i l_i^k f(l)\\ &= \frac{\tilde{\Gamma} _m \left( {n+k} \right)}{\tilde{\Gamma} _m \left( {n} \right)}\int \frac{\pi^{m\left ( m-1 \right )}}{\lambda ^{mn}\tilde{\Gamma} _m \left( {n+k} \right)\tilde{\Gamma} _m \left( {m} \right)} \\&\hspace{6em}\cdot \exp\left(-\frac{1}{\lambda}\sum _{i=1}^{m} l_i \right) \prod\limits_{i} {l_i^{\left ( n+k-m\right )}} \prod\limits_{i < j} {\left( {l_i - l_j } \right)^2}\\ &=\frac{\tilde{\Gamma} _m \left( {n+k} \right)}{\tilde{\Gamma} _m \left( {n} \right)}=\prod_{l=0}^{m-1}\frac{\Gamma(n-l+k)}{\Gamma(n-l)}\end{aligned}
E[detWk]=∫i∏likf(l)=Γ~m(n)Γ~m(n+k)∫λmnΓ~m(n+k)Γ~m(m)πm(m−1)⋅exp(−λ1i=1∑mli)i∏li(n+k−m)i<j∏(li−lj)2=Γ~m(n)Γ~m(n+k)=l=0∏m−1Γ(n−l)Γ(n−l+k)
◊
◊
\Diamond\Diamond
◊◊
下面求
log
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\log_e \mathrm{det} W
logedetW 的期望：
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E\left [e^{\zeta \log_e \mathrm{det} W} \right ]=E\left [\mathrm{det} W^\zeta \right ]=\prod_{l=0}^{m-1}\frac {\Gamma(n-l+\zeta)}{\Gamma (n-l)}
E[eζlogedetW]=E[detWζ]=l=0∏m−1Γ(n−l)Γ(n−l+ζ)
所以
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\begin{aligned} E\left [\log_e \mathrm{det} W \right ]&=\frac{d}{d\zeta}E\left [e^{\zeta \log_e \mathrm{det} W} \right ]_{\zeta=0}\\&=\sum_{l=0}^{m-1}\frac{\Gamma'(n-l)}{\Gamma (n-l)}\\&=\sum_{l=0}^{m-1}\psi (n-l) \end{aligned}
E[logedetW]=dζdE[eζlogedetW]ζ=0=l=0∑m−1Γ(n−l)Γ′(n−l)=l=0∑m−1ψ(n−l)
同理可以得到
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\log_e \mathrm{det} W
logedetW 的方差：
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\begin{aligned} \mathrm{Var}\left [\log_e \mathrm{det} W \right ]&=\frac{d^2}{d\zeta^2}E\left [e^{\zeta \log_e \mathrm{det} W} \right ]_{\zeta=0}-\mathrm{E}^2\left [\log_e \mathrm{det} W \right ]\\&=\sum_{l=0}^{m-1}\dot{\psi }(n-l) \end{aligned}
Var[logedetW]=dζ2d2E[eζlogedetW]ζ=0−E2[logedetW]=l=0∑m−1ψ˙(n−l)
其中
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\psi (x)={d \over dx}\ln\Gamma(x)
ψ(x)=dxdlnΓ(x) 称为 Euler’s digamma function
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\psi (x+1)=\psi (x)+\frac{1}{x}\\ \psi (m)=\psi (m-l)+\frac{1}{m-1}=\psi (1)+\sum_{l=1}^{m-1}\frac{1}{l}\\ \psi (1)=-\gamma=-\lim\limits_{n\to +\infty}[(1+1/2+1/3+\cdots +1/n)-\ln n]\\ \dot{\psi} (m+1)=\dot\psi (m)-\frac{1}{m^2}
ψ(x+1)=ψ(x)+x1ψ(m)=ψ(m−l)+m−11=ψ(1)+l=1∑m−1l1ψ(1)=−γ=−n→+∞lim[(1+1/2+1/3+⋯+1/n)−lnn]ψ˙(m+1)=ψ˙(m)−m21
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\gamma=\lim\limits_{n\to +\infty}H_n-\ln n
γ=n→+∞limHn−lnn 是欧拉常数，
H
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H_n
Hn 是调和级数。
References:
Random matrix theory and wireless communications (Book Tulino) (Theorem 2.11).
D. Jonsson, “Some limit theorems for the eigenvalues of a sample covariance matrix,” J. Multivariate Analysis, vol. 12, pp. 1–38, 1982.
R. J. Muirhead, Aspects of multivariate statistical theory. New York, Wiley, 1982.
