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detW和logdetW的期望

卿卿如梦 2022-02-28 阅读 23

如果 W W W 是 Wishart 分布的随机矩阵 W m ( n , λ I m ) W_m(n,\lambda I_m) Wm(n,λIm),则
E [ d e t W k ] = ∏ l = 0 m − 1 Γ ( n − l + k ) Γ ( n − l ) E\left [ \mathrm{det} W^k \right ]=\prod_{l=0}^{m-1}\frac{\Gamma(n-l+k)}{\Gamma(n-l)} E[detWk]=l=0m1Γ(nl)Γ(nl+k)
证明: Wishart 分布随机矩阵的特征值PDF为
f ( l ) = π m ( m − 1 ) λ m n Γ ~ m ( n ) Γ ~ m ( m ) exp ⁡ ( − 1 λ ∑ i = 1 m l i ) ∏ i l i ( n − m ) ∏ i < j ( l i − l j ) 2 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(m1)exp(λ1i=1mli)ili(nm)i<j(lilj)2
因此
E [ d e t W k ] = ∫ ∏ i l i k f ( l ) = Γ ~ m ( n + k ) Γ ~ m ( n ) ∫ π m ( m − 1 ) λ m n Γ ~ m ( n + k ) Γ ~ m ( m ) ⋅ exp ⁡ ( − 1 λ ∑ i = 1 m l i ) ∏ i l i ( n + k − m ) ∏ i < j ( l i − l j ) 2 = Γ ~ m ( n + k ) Γ ~ m ( n ) = ∏ l = 0 m − 1 Γ ( n − l + k ) Γ ( n − l ) \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]=ilikf(l)=Γ~m(n)Γ~m(n+k)λmnΓ~m(n+k)Γ~m(m)πm(m1)exp(λ1i=1mli)ili(n+km)i<j(lilj)2=Γ~m(n)Γ~m(n+k)=l=0m1Γ(nl)Γ(nl+k)
◊ ◊ \Diamond\Diamond
下面求 log ⁡ e d e t W \log_e \mathrm{det} W logedetW 的期望:
E [ e ζ log ⁡ e d e t W ] = E [ d e t W ζ ] = ∏ l = 0 m − 1 Γ ( n − l + ζ ) Γ ( n − l ) 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=0m1Γ(nl)Γ(nl+ζ)
所以
E [ log ⁡ e d e t W ] = d d ζ E [ e ζ log ⁡ e d e t W ] ζ = 0 = ∑ l = 0 m − 1 Γ ′ ( n − l ) Γ ( n − l ) = ∑ l = 0 m − 1 ψ ( n − l ) \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=0m1Γ(nl)Γ(nl)=l=0m1ψ(nl)
同理可以得到 log ⁡ e d e t W \log_e \mathrm{det} W logedetW 的方差:
V a r [ log ⁡ e d e t W ] = d 2 d ζ 2 E [ e ζ log ⁡ e d e t W ] ζ = 0 − E 2 [ log ⁡ e d e t W ] = ∑ l = 0 m − 1 ψ ˙ ( n − l ) \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=0E2[logedetW]=l=0m1ψ˙(nl)
其中 ψ ( x ) = d d x ln ⁡ Γ ( x ) \psi (x)={d \over dx}\ln\Gamma(x) ψ(x)=dxdlnΓ(x) 称为 Euler’s digamma function
ψ ( x + 1 ) = ψ ( x ) + 1 x ψ ( m ) = ψ ( m − l ) + 1 m − 1 = ψ ( 1 ) + ∑ l = 1 m − 1 1 l ψ ( 1 ) = − γ = − lim ⁡ n → + ∞ [ ( 1 + 1 / 2 + 1 / 3 + ⋯ + 1 / n ) − ln ⁡ n ] ψ ˙ ( m + 1 ) = ψ ˙ ( m ) − 1 m 2 \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(ml)+m11=ψ(1)+l=1m1l1ψ(1)=γ=n+lim[(1+1/2+1/3++1/n)lnn]ψ˙(m+1)=ψ˙(m)m21
γ = lim ⁡ n → + ∞ H n − ln ⁡ n \gamma=\lim\limits_{n\to +\infty}H_n-\ln n γ=n+limHnlnn 是欧拉常数, H n 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.

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