The distribution and percentiles of channel capacity for multiple arrays

Abstract

We show that channel capacity of N-transmitter M-receiver antenna systems is approximately normal for both Raleigh fading and Ricean environments whether or not antennas are correlated. We give the distribution and percentiles of capacity as a power-series in $(MN)^{-1/2}$ when M or M/N is fixed, both for the case of fixed total power transmitted and also for the case, where total power transmitted increases with N.

Appendices

Appendix A

Here, we give some results for the complex normal, introduced by Wooding [20], and for the central and non-central complex Wishart. We write

$$\begin{aligned} \displaystyle \mathbf{Z} = \mathbf{X} + j\mathbf{Y} \sim \mathcal{CN}_M \left( \mathbf{0}, \mathbf{V} \right) \end{aligned}$$

if

$$\begin{aligned} \displaystyle \left( \begin{array}{l} \mathbf{X} \\ \mathbf{Y} \end{array}\right) \sim \displaystyle \mathcal{N}_{2M} \left( \mathbf{0}, \left( \begin{array}{ll} \displaystyle \mathbf{A} &{} \mathbf{B} \\ \displaystyle -\mathbf{B} &{} \mathbf{A} \end{array}\right) \right) \end{aligned}$$

with $\mathbf{A}$ and $\mathbf{B}$ both $M \times M$. This is sometimes referred to as the circular complex normal to distinguish it from the case of arbitrary cov$\left( {\mathbf{X} \atopwithdelims ()\mathbf{Y}} \right)$. So,

$$\begin{aligned}&\displaystyle \mathbf{V} = E\mathbf{Z} \mathbf{Z}^+ = 2 (\mathbf{A} - j\mathbf{B}), \\&\displaystyle \det \begin{pmatrix} \displaystyle \mathbf{A} &{} \mathbf{B} \\ \displaystyle -\mathbf{B} &{} \mathbf{A} \end{pmatrix} = \det {}^2 \left( \mathbf{A} - j\mathbf{B}\right) = 2^{-2M} \det {}^2 \mathbf{V}, \end{aligned}$$

and $\mathbf{Z}$ has density

$$\begin{aligned} \displaystyle f_\mathbf{Z} (\mathbf{z}) = \pi ^{-M} \left( \det \mathbf{V}\right) ^{-1} \exp \left( -\mathbf{z}^+ \mathbf{V}^{-1} \mathbf{z} \right) \end{aligned}$$

for $\mathbf{z}$ in $\mathcal {C}^M$. Also for $\mathbf{r}$, $\mathbf{s}$ in $\mathcal {C}^M$ with transposes $\mathbf{r}^T$, $\mathbf{s}^T$, if $\mathbf{t} = \mathbf{r}+j\mathbf{s}$, $T = \mathbf{r}^T \mathbf{X} + \mathbf{s}^T \mathbf{Y} = \left( \mathbf{t}^+ \mathbf{Z} + \mathbf{Z}^+ \mathbf{t} \right) / 2$ then

$$\begin{aligned} \displaystyle E\exp {T} = \exp \left( \mathbf{t}^+ \mathbf{V} \mathbf{t}/4\right) \end{aligned}$$

and so

$$\begin{aligned} \displaystyle E \exp \left( \mathbf{t}^+ \mathbf{Z} + \mathbf{Z}^+ \mathbf{t} \right) = \exp \left( \mathbf{t}^+ \mathbf{Vt} \right) . \end{aligned}$$

(A.1)

We write $\mathbf{Z} + {{\varvec{\mu }}} \sim \mathcal{CN}_M \left( {{\varvec{\mu }}}, \mathbf{V} \right)$. Turin [21] showed that for $\mathbf{Q}^+ = \mathbf{Q}$ in $\mathcal{C}^{M \times M}$ and t in $\mathcal C$,

$$\begin{aligned}&\displaystyle E \exp \left\{ \left( \mathbf{Z} + {{\varvec{\mu }}} \right) ^+ \mathbf{Q} \left( \mathbf{Z} + {{\varvec{\mu }}}\right) t \right\} \\&\quad = \det \left( \mathbf{I}_M - t \mathbf{QV} \right) ^{-1} \exp (-\gamma ), \end{aligned}$$

where

$$\begin{aligned} \displaystyle \gamma= & {} {{\varvec{\mu }}}^+ \mathbf{V}^{-1} \left[ \mathbf{I} - \left( \mathbf{I} - t \mathbf{VQ} \right) ^{-1} \right] \\ {{\varvec{\mu }}}= & {} t {{\varvec{\mu }}}^+ \left( \mathbf{Q}^{-1} - t \mathbf{V} \right) ^{-1} {{\varvec{\mu }}}. \end{aligned}$$

Although not stated, this requires that $\lambda _1 {\mathcal Re} (t) <1$, where $\lambda _1$ is the maximum eigenvalue of $\mathbf{V}^{1/2} \mathbf{Q} \mathbf{V}^{1/2}$.

For $\mathbf{Z}_1, \ldots , \mathbf{Z}_N$ independent $\mathcal{CN}_M \left( \mathbf{0}, \mathbf{V} \right)$ the (central) complex Wishart is defined as

$$\begin{aligned} \displaystyle \mathbf{W}_N = \sum ^N_{n=1} \mathbf{Z}_n \mathbf{Z}_n^+. \end{aligned}$$

(A.2)

Goodman [22] proved

Theorem A. 1

Suppose $\mathbf{T}^+ = \mathbf{T}$ lies in $\mathcal{C}^{M \times M}$. Then

$$\begin{aligned}&\displaystyle E \exp \hbox {trace} \left( \mathbf{T} \mathbf{W}_N \right) \nonumber \\&\quad = \left\{ \det \left( \mathbf{I} - \mathbf{T} \mathbf{V}^{-1} \right) \right\} ^{-N}. \end{aligned}$$

(A.3)

Again, the condition $\lambda _1 < 1$ is implicit, where now $\lambda _1$ is the maximum eigenvalue of $\mathbf{V}^{-1/2} \mathbf{T} \mathbf{V}^{-1/2}$. He also gave parallels to real theory for complex multiple coherence, correlation and conditional coherence. His Sect 6 considers the case $\mathbf{X}(t) : \mathcal{R} \rightarrow \mathcal{R}^M$ a stationary Gaussian process with mean zero; define its Fourier transform $\mathbf{Z} (\omega ) : \mathcal{R} \rightarrow \mathcal{C}^M$ by

$$\begin{aligned} \displaystyle \mathbf{X} (t) = \int \exp \left( j \omega t \right) d\mathbf{Z}(\omega ). \end{aligned}$$

For any given $\omega$, $\mathbf{Z}(\omega ) \sim \mathcal{CN} \left( \mathbf{0}, \mathbf{V}(\omega ) \right)$ for a certain $\mathbf{V}(\omega )$. For $\omega _1 \ne \omega _2$, $\mathbf{Z} \left( \omega _1 \right)$ and $\mathbf{Z} \left( \omega _2 \right)$ are independent.

For z a scalar, set $\overline{z} = z^+$, the complex conjugate. From a version of (A.1), Reed [23] (using $\mathbf{H} = \mathbf{V}^T / 2$) proved

Theorem A. 2

We have

$$\begin{aligned}&\displaystyle E Z_{a_1} \cdots Z_{a_r} \overline{Z}_{b_1} \cdots \overline{Z}_{b_s} \nonumber \\&\quad = I(r = s) \sum ^{r!}_{p_1 \cdots p_r} V_{a_1 p_1} \cdots V_{a_r p_r} \end{aligned}$$

(A.4)

summed over all r! permutations $p_1 \cdots p_r$ of $b_1 \cdots b_r$.

He noted as a corollary that

$$\begin{aligned}&\displaystyle E|Z|^{2n} = n! \left( E|Z|^2\right) ^n,\\&\displaystyle E \left( Z_1 \overline{Z}_r \right) ^n = n! \left( EZ_1 \overline{Z}_2\right) ^n, \\&\displaystyle E Z_1 Z_2 \overline{Z}_3 \overline{Z}_4 = \sum ^2_{34} E Z_1 \overline{Z}_3 EZ_2 \overline{Z}_4. \end{aligned}$$

Setting

$$\begin{aligned} \displaystyle v_{ij} = V_{a_i b_j}, \ \displaystyle m_{1 \cdots r} = EZ_{a_1} \cdots Z_{a_r} \overline{Z}_{b_1} \cdots \overline{Z}_{b_r}, \end{aligned}$$

(A.4) gives

$$\begin{aligned} \displaystyle m_1= & {} v_{11}, \ \displaystyle m_{12} = v_{11} v_{12} + v_{12} v_{21}, \\ \displaystyle m_{123}= & {} v_{11} v_{22} v_{33} + \sum ^3_{123} v_{11} v_{23} v_{32} + \sum ^2_{23} v_{12} v_{23} v_{31}, \end{aligned}$$

and so on. Note that $|Z|^2 = \chi ^2_{2M} /2 = G_M$, where $G_M$ is a gamma random variable with mean M and density $x^{M-1} \exp (-x) / (M-1)!$ on $(0, \infty )$.

From (A.3) with $N=1$ one can prove

Theorem A. 3

We have

$$\begin{aligned}&\displaystyle \kappa \left( Z_{a_1} \overline{Z}_{b_1}, \ldots , Z_{a_r} \overline{Z}_{b_r} \right) \nonumber \\&\quad = \sum ^{(r-1)!}_{C \left( p_1 \cdots p_r\right) } V_{a_1 p_1} \cdots V_{a_r p_r} \end{aligned}$$

(A.5)

summed over all $(r-1)!$ permutations $p_1 \cdot p_r$ of $b_1 \cdots b_r$ giving connected expressions. (By connected we disqualify breaking $1 \cdots r$ into two or more groups. For example, $V_{a_1 b_1} V_{a_2 b_2}$ breaks 12 into 1 and 2; $V_{a_1 b_2} V_{a_2 b_1} V_{a_3 b_4} V_{a_4 b_3}$ breaks 1234 into 12 and 34.

Here, the joint cumulants for complex random variables $U_1, U_2, \ldots$ are defined as for real random variables. For example, $\kappa \left( U_1, U_2 \right) = EU_1 U_2 - EU_1 EU_2$ not $EU_1 \overline{U}_2 - EU_1 E\overline{U}_2$. Setting $k_{1 \cdots r}$ equal to the left hand side of (A.5), this gives

$$\begin{aligned} \displaystyle k_1= & {} v_{11}, \ \displaystyle k_{12} = v_{12} v_{21}, \\ \displaystyle k_{123}= & {} v_{12} v_{23} v_{31} + v_{13} v_{32} v_{21}, \\ \displaystyle k_{1234}= & {} v_{12} \left( v_{23} v_{34} v_{41}\right. \\&\left. + v_{24} v_{43} v_{31}\right) \\&+ v_{13} \left( v_{34} v_{42} v_{21}\right. \\&\left. + v_{32} v_{24} v_{41}\right) \\&+ v_{14} \left( v_{43} v_{32} v_{21}\right. \\&\left. + v_{42} v_{23} v_{31}\right) . \end{aligned}$$

We now give a ‘brute force’ extension of Theorem A. 2 from $\mathbf{Z}$ to $\mathbf{H} = {{\varvec{\mu }}} + \mathbf{Z}$. We shall give the cumulants of $\mathbf{X} = \mathbf{H} \mathbf{H}^+$ for $\mathbf{H} = {{\varvec{\mu }}} + \mathbf{Z}$.

Theorem A. 4

Set

$$\begin{aligned} \displaystyle x_i = X_{a_i b_i} = H_{a_i} \overline{H}_{b_i} =\sum ^2_{j=0} x_{ij} \end{aligned}$$

for

$$\begin{aligned} \displaystyle x_{i0}= & {} \mu _{a_i} \overline{\mu }_{b_i}, \\ \displaystyle x_{i1}= & {} Z_{a_i} \overline{\mu }_{b_i} + \mu _{a_i} \overline{Z}_{b_i}, \\ \displaystyle x_{i2}= & {} Z_{a_i} \overline{Z}_{b_i}. \end{aligned}$$

Then

$$\begin{aligned}&\displaystyle Ex_1 = x_{10}+v_{11}, \nonumber \\&\displaystyle \kappa \left( x_1, \ldots , x_r \right) = \sum ^{2r}_{j=r} K_{rj} \end{aligned}$$

(A.6)

for $r \ge 2$, where

$$\begin{aligned} \small {K_{rj} = \left\{ \begin{array}{ll} \displaystyle \sum _{i_1 + \cdots + i_r = j} \kappa \left( x_{1i_1}, \ldots , x_{ri_r}\right) , &{} \hbox {if }j\hbox { is even,}\\ \displaystyle 0, &{} \hbox {if }j\hbox { is odd} \end{array} \right. } \end{aligned}$$

follows by (A.4) with $r \ne s$. So,

$$\begin{aligned} \displaystyle K_{r, 2r} = \kappa \left( x_{12}, \ldots , x_{r2} \right) , \end{aligned}$$

given by (A.5), and

$$\begin{aligned}&\displaystyle K_{r,r} = \kappa \left( x_{11}, \ldots , x_{r1}\right) , \\&\displaystyle K_{r, r+1} = \sum ^r_{1 \cdots r} \kappa \left( x_{{p_1}2}, x_{{p_2}1} x_{{p_3}1}, \ldots , x_{{p_r}1} \right) \end{aligned}$$

summed over all r permutations $p_1 \cdots p_r$ of $1 \cdots r$ giving distinct terms and so on for $K_{r, r+2}, \ldots , K_{r, 2r-1}$. In particular,

$$\begin{aligned}&\displaystyle K_{22} = \sum ^2_{12} \mu _{a_1} \overline{\mu }_{b_2} v_{21}, \end{aligned}$$

(A.7)

$$\begin{aligned}&\displaystyle K_{34} = \sum ^6_{123} \mu _{a_1} \overline{\mu }_{b_2} T_{123}, \end{aligned}$$

(A.8)

$$\begin{aligned}&\displaystyle K_{44} = \sum ^6 \overline{\mu }_{b_1} \overline{\mu }_{b_2} \mu _{a_3} \mu _{a_4} T_{1234} \end{aligned}$$

(A.9)

for $T_{123} = \kappa \left( \overline{Z}_{b_1}, Z_{a_2}, Z_{a_3} \overline{Z}_{b_3} \right) = v_{23} v_{31}$ and $T_{1234} = \kappa \left( Z_{a_1}, Z_{a_2}, \overline{Z}_{b_3}, \overline{Z}_{b_4} \right) = 0$ by (A.5). Also,

$$\begin{aligned} \displaystyle K_{46} = \sum ^6_{1122} \sum ^2_{12} \mu _{a_1} \overline{\mu }_{b_2} \sum ^2_{34} v_{23} v_{34} v_{41}. \end{aligned}$$

(A.10)

This is enough to give $\kappa \left( H_{a_1} \overline{H}_{b_1}, \ldots , H_{a_r} \overline{H}_{b_r} \right)$ for $1 \le r \le 4$. Other values may be derived from (A.6) similarly.

Proof

Note that (A.7)–(A.9) follow from (A.4) and (A.5). To prove (A.10), note that

$$\begin{aligned} \displaystyle K_{46}= & {} k \left( 1^2 2^2\right) + k (12 12) + k (1221)\\&+k(2112) + k(2121) + k \left( 2^2 1^2\right) \\= & {} \sum ^6_{1122} k\left( 1^2 2^2\right) \end{aligned}$$

say, for $k \left( i_1 \cdots i_r \right) = \kappa \left( x_{1 i_1}, \ldots , x_{ri_r} \right)$. So,

$$\begin{aligned} \displaystyle k \left( 1^22^2\right)= & {} \kappa \left( x_{11}, x_{21}, x_{32}, x_{42}\right) \\= & {} \sum _{12}^2 \mu _{a_1}\overline{\mu }_{b_2}\kappa _{1234} \end{aligned}$$

for

$$\begin{aligned} \displaystyle \kappa _{1234}= & {} \kappa \left( \overline{Z}_{b_1}, Z_{a_2}, Z_{a_3} \overline{Z}_{b_3}, Z_{a_4} \overline{Z}_{b_4} \right) \\= & {} \mu _{1234} - \sum ^3 \mu _{12} \mu _{34} \end{aligned}$$

say,

$$\begin{aligned} \displaystyle \mu _{1234}= & {} E \left( Z_{a_2} \overline{Z}_{b_1} - v_{21} + v_{21} \right) \\&\quad \left( Z_{a_3} \overline{Z}_{b_3} - v_{33} \right) \\&\quad \left( Z_{a_4} \overline{Z}_{b_4} - v_{44} \right) \\= & {} \sum ^2_{34} v_{23} v_{34} v_{41} + v_{21} v_{34} v_{43} \end{aligned}$$

by (A.5), and $\mu _{12} = v_{21}$, $\mu _{34} = v_{34}v_{43}$, $\mu _{13} = \mu _{14} = 0$. So,

$$\begin{aligned} \displaystyle \kappa _{1234} = \sum ^2_{34} v_{23} v_{34} v_{41}, \end{aligned}$$

so that (A.10) holds. $\square$

Maiwald and Kraus [24] have obtained the first four non-central moments of $\mathbf{W}_N / N$ for $\mathbf{W}_N$ the complex Wishart of (A.2) by differentiating (A.3). We now give a simple method which gives the central and non-central moments of the Wishart up to 12th order.

Theorem A. 5

Suppose

$$\begin{aligned} \displaystyle S = \sum ^N_{n=1} X_n, \end{aligned}$$

where $X_n$ are independently-distributed as X in $\mathcal C$ with cumulants $\left\{ \kappa _r \right\}$ defined by

$$\begin{aligned} \displaystyle \ln E \exp (tX) = \sum ^\infty _{r=1} t^r \kappa _r / r! = K(t) \end{aligned}$$

(A.11)

say for t in $\mathcal{C}$. Then the non-central moments of S are

$$\begin{aligned} \displaystyle ES^r = \sum ^r_{i=1} N^i B_{ri} \end{aligned}$$

(A.12)

for $r \ge 1$, where

$$\begin{aligned} \displaystyle K(t)^i / i! = \sum ^\infty _{r=i} B_{ri} t^r / r!, \end{aligned}$$

(A.13)

where $B_{ri} = B_{ri} \left( {{\varvec{\kappa }}} \right)$ are the exponential Bell polynomials in ${{\varvec{\kappa }}} = \left( \kappa _1, \kappa _2, \ldots \right)$ tabled on page 30 of Comtet [25] up to $r=12$. Putting $\kappa _1 = 0$ gives the central moment

$$\begin{aligned} \displaystyle \mu _r (S) = E \left( S - ES\right) ^r = \sum ^r_{1\le i \le r/2} N^i B_{ri0}, \end{aligned}$$

where $B_{ri0} = B_{ri}|_{\kappa _1 = 0}$ since $B_{ri0} = 0$ for $i > r/2$. For example,

$$\begin{aligned} \displaystyle ES^4 = \sum ^4_{i=1} N^i B_{4i}, \ \displaystyle \mu _4 (S) = \sum ^2_{i=1} N^i B_{4i0}, \end{aligned}$$

where $B_{41} = \kappa _4$, $B_{42} = 4 \kappa _1 \kappa _3 + 3 \kappa _2^2$, $B_{43} = 6 \kappa _1^2 \kappa _2$, $B_{44} = \kappa _1^4$, and $B_{410} = \kappa _4, B_{420} = 3 \kappa _2^2$. For $\mathbf{X}$ in $\mathcal{C}^p$ these become

$$\begin{aligned} \displaystyle E S_{a_1} \cdots S_{a_r} = \sum ^r_{i=1} N^i B_i^{a_i \cdots a_r}, \end{aligned}$$

and

$$\begin{aligned}&\displaystyle \mu \left( S_{a_1}, \ldots , S_{a_r} \right) = E \left( W_{a_1} - EW_{a_1} \right) \\&\quad \cdots \left( W_{a_r} - EW_{a_r} \right) = \sum _{1 \le i \le r/2} N^i B_{i0}^{a_1 \cdots a_r}, \end{aligned}$$

where $B_i^{a_1 \cdots a_r}$ and $B_{i0}^{a_1 \cdots a_r}$ can be written down immediately from $B_{ri}$ and $B_{ri0}$. For example,

$$\begin{aligned} \displaystyle ES_{a_1} \cdots S_{a_4} = \sum ^4_{i=1} N^i B_i^{a_1 \cdots a_4}, \end{aligned}$$

and

$$\begin{aligned} \displaystyle \mu \left( S_{a_1} \cdots S_{a_4} \right) = \sum ^2_{i=1} N^i B_{i0}^{a_1 \cdots a_4}, \end{aligned}$$

where

$$\begin{aligned}&\displaystyle B_1^{a_1 \cdots a_4} = B_{10}^{a_1 \cdots a_4} = \kappa ^{a_1 \cdots a_4}, \\&\displaystyle B_2^{a_1 \cdots a_4} = \sum ^4 \kappa ^{a_1} \kappa ^{a_2 a_3 a_4} + \sum ^3 \kappa ^{a_1 a_2} \kappa ^{a_3 a_4}, \\&\displaystyle B_{20}^{a_1 \cdots a_4} = \sum ^3 \kappa ^{a_1 a_2} \kappa ^{a_3 a_4},\\&\displaystyle B_3^{a_1 \cdots a_4} = \sum ^6 \kappa ^{a_1} \kappa ^{a_2} \kappa ^{a_3 a_4}, \\&\displaystyle B_4^{a_1 \cdots a_4} = \kappa ^{a_1} \cdots \kappa ^{a_4}, \end{aligned}$$

where $\displaystyle \sum ^M$ sums over all M permutations of indices $1 \cdots 4$ giving distinct terms, and

$$\begin{aligned} \displaystyle \kappa ^{a_1 \cdots a_r} = \kappa \left( X_{a_1}, \ldots , X_{a_r} \right) \end{aligned}$$

for $1 \le a_i \le p$, $i=1, \ldots , r$. For $\mathbf{X}$ in $\mathcal{C}^{p \times q}$ these become

$$\begin{aligned} \displaystyle ES_{a_1 b_1} \cdots S_{a_r b_r} = \sum ^r_{i=1} N^i B_i^{a_1 b_1 \cdots a_r b_r}, \end{aligned}$$

(A.14)

and

$$\begin{aligned}&\displaystyle \mu \left( S_{a_1 b_1}, \ldots , S_{a_r b_r} \right) \nonumber \\&\quad = \sum _{1 \le i \le r/2} N^i B_{i0}^{a_1 b_1 \cdots a_r b_r}, \end{aligned}$$

(A.15)

where $B_i^{a_1 b_1 \cdots a_r b_r}$ is $B_i^{a_1 \cdots a_r}$ with $a_i$ replaced by $\left( a_i b_i \right)$ and similarly for $B_{i0}^{a_1 b_1 \cdots a_r b_r}$, these being given in terms of

$$\begin{aligned} \displaystyle \kappa ^{a_1 b_1 \cdots a_r b_r} = \kappa \left( X_{a_1 b_1}, \ldots , X_{a_r b_r} \right) \end{aligned}$$

for $1 \le a_i \le p$, $1 \le b_i \le q$ and $i = 1, \ldots , r$. For example,

$$\begin{aligned} \displaystyle ES_{a_1 b_1} \cdots S_{a_4 b_4} = \sum ^4_{i=1} N^i B_i^{a_1 b_1 \cdots a_4 b_4} \end{aligned}$$

and

$$\begin{aligned} \displaystyle \mu \left( S_{a_1 b_1}, \ldots , S_{a_4 b_4} \right) = \sum ^2_{i=1} N^i B_{i0}^{a_1 b_1 \cdots a_4 b_4}, \end{aligned}$$

where

$$\begin{aligned} \displaystyle B_1^{a_1 b_1 \cdots a_4 b_4}= & {} B_{10}^{a_1 b_1 \cdots a_4 b_4} = \kappa ^{a_1 b_1 \cdots a_4 b_4}, \\ \displaystyle B_2^{a_1 b_1 \cdots a_4 b_4}= & {} \sum ^4 \kappa ^{a_1 b_1} \kappa ^{a_2 b_2, a_3 b_3, a_4 b_4} \\&+ B_{20}^{a_1 b_1 \cdots a_4 b_4}, \\ \displaystyle B_{20}^{a_1 b_1 \cdots a_4 b_4}= & {} \sum ^3 \kappa ^{a_1 b_1} \kappa ^{a_2 b_2} \kappa ^{a_3 b_3, a_4 b_4}, \\ \displaystyle B_4^{a_1 b_1 \cdots a_4 b_4}= & {} \kappa ^{a_1 b_1} \cdots \kappa ^{a_4 b_4}. \end{aligned}$$

Now consider the weighted version:

$$\begin{aligned} \displaystyle S_P = \sum ^N_{n=1} P_n X_n, \end{aligned}$$

where $\left\{ P_n \right\}$ are constants in $\mathcal C$ and $\left\{ X_n \right\}$ are independent copies of X in $\mathcal C$ with cumulants $\left\{ \kappa _r \right\}$. Then

$$\begin{aligned} \displaystyle K_{S_P} (t)= & {} \sum ^N_{n=1} K_X \left( P_n t\right) \\= & {} N \sum ^\infty _{r=1} \kappa _r P_{rN} t^r / r! = N K(t) \end{aligned}$$

say, so

$$\begin{aligned} \displaystyle ES_P^r = \sum ^r_{i=1} N^i B_{ri} ({{\varvec{\alpha }}}) \end{aligned}$$

for $r \ge 1$, where $\alpha _r = P_{rN} \kappa _r$. Similarly, for $\mathbf{X}$ in $\mathcal{C}^{p \times q}$, (A.14)–(A.15) hold for

$$\begin{aligned} \displaystyle \mathbf{S} = \mathbf{S}_P \hbox { with } \kappa ^{a_1 b_1 \cdots a_r b_r} \hbox { multiplied by } P_{rN}. \end{aligned}$$

(A.16)

To apply (A.14), (A.15) to $\mathbf{S}$ the central complex Wishart $\mathbf{W}_N$ of (A.2), put $p = q = M$ and substitute $\kappa ^{a_1 b_1 \cdots a_r b_r}$ of (A.5). Finally, for the non-central Wishart defined by

$$\begin{aligned} \displaystyle \mathbf{W}_N = \sum ^N_{n=1} \mathbf{H}_n \mathbf{H}_n^+, \end{aligned}$$

where $\mathbf{H}_1, \ldots , \mathbf{H}_N$ are independent $\mathcal{CN}_M \left( {{\varvec{\mu }}}, \mathbf{V}\right)$, the moments of $\mathbf{W}_N$ are given by (A.14), (A.15) with $p=q=M$ and $\kappa ^{a_1 b_1 \cdots a_r b_r}$ of (A.6).

Proof

It follows from (A.11) that

$$\begin{aligned} \displaystyle E \exp (t S) = \exp \left\{ N K(t) \right\} = \sum ^\infty _{i=0} N^i K(t)^i / i!. \end{aligned}$$

So, (A.12) follows from (A.13) since $B_{r0} = 0$ for $r \ne 0$. $\square$

Appendix B

Theorem B. 1

Suppose that $\mathbf{A}$ in $\mathcal{C}^{M \times M}$ is a non-singular matrix, ${{\varvec{\tau }}}$ lies in $\mathcal{C}^M$ and $\rho$ lies in $\mathcal C$. Set

$$\begin{aligned}&\displaystyle \mathbf{B} = \mathbf{A} + {{\varvec{\tau }}} {{\varvec{\tau }}}^+,\\&\displaystyle f_r = {{\varvec{\tau }}}^+ \mathbf{A}^{-r} {{\varvec{\tau }}}, \ \displaystyle \Delta = 1 + \rho f_1, \ \displaystyle d = \rho / \Delta . \end{aligned}$$

Then

$$\begin{aligned}&\displaystyle \mathbf{B}^{-1} = \mathbf{A}^{-1} - d \mathbf{A}^{-1} {{\varvec{\tau }}} {{\varvec{\tau }}}^+ \mathbf{A}^{-1}, \end{aligned}$$

(B.1)

$$\begin{aligned}&\displaystyle \det \mathbf{B} = \Delta \det \mathbf{A}, \end{aligned}$$

(B.2)

and, for $r = 1,2,\ldots$,

$$\begin{aligned} \displaystyle \gamma _r = \hbox {trace } \mathbf{B}^{-r} \end{aligned}$$

is given in terms of

$$\begin{aligned} \displaystyle a_r = \hbox {trace } \mathbf{A}^{-r} \end{aligned}$$

by

$$\begin{aligned} \displaystyle \gamma _r= & {} \displaystyle a_r + \sum ^r_{i=1} \begin{pmatrix} r \\ i \end{pmatrix} (-d)^i f_2^{i-1} f_{r+2-i} \end{aligned}$$

(B.3)

$$\begin{aligned}= & {} \displaystyle a_r - f_2^{-1} f_{r+2} + f_2^{-1} {{\varvec{\tau }}}^+ \mathbf{A}^{-1} \nonumber \\&\quad \left( \mathbf{A}^{-1} - df_2 \mathbf{I}_M \right) ^r \mathbf{A}^{-1} {{\varvec{\tau }}}. \end{aligned}$$

(B.4)

So,

$$\begin{aligned} \displaystyle \gamma _1= & {} a_1 - f_2 d, \\ \displaystyle \gamma _2= & {} a_2 - 2 f_3 d + f_2^2 d^2, \\ \displaystyle \gamma _3= & {} a_3 - 3f_4 d + 3f_2 f_3 d^2 - f_2^3 d^3,\\ \displaystyle \gamma _4= & {} a_4 - 4 f_5 d + 6f_2 f_4 d^2 \\&- 4f_2^2 f_3 d^3 + f_2^4 d^4, \\ \displaystyle \gamma _5= & {} a_5 - 5f_6 d + 10f_2 f_5 d^2 \\&- 10 f_2^2 f_4 d^3 + 5 f_2^3 f_3 - f_2^5 d^5, \\ \displaystyle \gamma _6= & {} a_6 - 6f_7 d + 15f_2 f_6 d^2 \\&- 20 f_2^2 f_5 d^3 + 15 f_2^3 f_4 d^4 \\&- 6f_2^4 f_3 d^5 + f_2^6 d^6. \end{aligned}$$

Also $e_r = {{\varvec{\tau }}}^+ \mathbf{B}^{-r} {{\varvec{\tau }}}$ is given by

$$\begin{aligned} \displaystyle e_r= & {} \Delta ^{-r} f_r \hbox { for } r = 1, 2, \\ \displaystyle e_3= & {} \Delta ^{-2} \left( f_3 - f_2^2 d\right) , \\ \displaystyle e_4= & {} \Delta ^{-2} \left( f_4 - 2f_2 f_3 d + f_2^3 d^2\right) , \\ \displaystyle e_5= & {} \Delta ^{-2} \left\{ f_5 - \left( 2f_2 f_4 + f_3^2 \right) d\right. \\&\left. + 3f_2^2 f_3 d^2 - f_2^4 d^3 \right\} , \\ \displaystyle e_6= & {} \Delta ^{-2} \left\{ f_6 - \left( 2f_2 f_5 + 2f_3 f_4\right) d\right. \\&+ \left( 3f_2^2 f_4 + 3f_2 f_3^2 \right) d^2 \\&\left. - 4f_2^3 f_3 d^3 + f_2^5 d^4 \right\} . \end{aligned}$$

For $r \ge 2$ the general formula is

$$\begin{aligned} \displaystyle e_r = \Delta ^{-2} \sum _{i=0}^{r-2} (-d)^i \left[ \widehat{B}_{i+r, i+1} (\mathbf{f}) \right] _{f_1 = 0}, \end{aligned}$$

(B.5)

where for $\mathbf{f} = \left( f_1, f_2, \ldots \right)$, $\widehat{B}_{ri} (\mathbf{f})$ is the ordinary Bell polynomial tabled on page 309 of Comtet [25], and defined by

$$\begin{aligned} \displaystyle \left( \sum ^\infty _{r=1} z^r f_r \right) ^i = \sum ^\infty _{r=i} z^r \widehat{B}_{ri} (\mathbf{f}) \end{aligned}$$

for z in $\mathcal{C}$.

Proof

Note that (B.1) holds by equation (3) of Henderson and Searle [19], and (B.2) follows. Set $\mathbf{a} = \mathbf{A}^{-1}$ and $\mathbf{b} = \mathbf{A}^{-1} {{\varvec{\tau }}} {{\varvec{\tau }}}^+ \mathbf{A}^{-1}$, so

$$\begin{aligned} \displaystyle \mathbf{B}^{-1}= & {} \mathbf{a} - d \mathbf{b},\nonumber \\ \displaystyle \left( \mathbf{a} - d\mathbf{b}\right) ^r \nonumber \\= & {} \mathbf{a}^r - d \sum ^r \mathbf{a}^{r-1} \mathbf{b} \nonumber \\&+ d^2 \sum ^{{r \atopwithdelims ()2}} \mathbf{a}^{r-2} \mathbf{b}^2 - \cdots , \end{aligned}$$

(B.6)

where $\displaystyle \sum ^m \mathbf{a}^i \mathbf{b}^j$ sums over all m permutations of the $i+j$ elements of $\mathbf{a}^i \mathbf{b}^j$ giving distinct terms. So,

$$\begin{aligned}&\displaystyle \hbox {trace } \left( \mathbf{a}-d\mathbf{b}\right) ^r = \sum ^r_{i=0} \begin{pmatrix} r \\ i \end{pmatrix} (-d)^i \\&\hbox {trace } \left( \mathbf{a}^{r-i} \mathbf{b}^i \right) . \end{aligned}$$

Set ${{\varvec{\tau }}}_r = \mathbf{A}^{-r} {{\varvec{\tau }}}$ so that

$$\begin{aligned} \displaystyle {{\varvec{\tau }}}_r^+ {{\varvec{\tau }}}_s f_{r +s}, \displaystyle \hbox {trace } \left( \mathbf{a}^j \mathbf{b}^i \right) = {{\varvec{\tau }}}_1^+ \mathbf{a}^j {{\varvec{\tau }}}_1 f_2^{i-1} \end{aligned}$$

for $i \ge 1$, and (B.3), (B.4) follow.

Set

$$\begin{aligned} \displaystyle \mathbf{S}_r^m = \sum _r^m {{\varvec{\tau }}}_i {{\varvec{\tau }}}_j^+ \end{aligned}$$

summed over $\left\{ i + j = r, i> 0, j > 0 \right\}$, where m is the number of terms: $\mathbf{S}_2^1 = {{\varvec{\tau }}}_1 {{\varvec{\tau }}}_1^+$, $\mathbf{S}_3^2 = {{\varvec{\tau }}}_1 {{\varvec{\tau }}}_2^+ + {{\varvec{\tau }}}_2 {{\varvec{\tau }}}_1^+$, $\mathbf{S}_4^3 = {{\varvec{\tau }}}_1 {{\varvec{\tau }}}_3^+ + {{\varvec{\tau }}}_2 {{\varvec{\tau }}}_2^+ + {{\varvec{\tau }}}_3 {{\varvec{\tau }}}_1^+$, and so on. By (B.6),

$$\begin{aligned} \displaystyle \mathbf{B}^{-2}= & {} \mathbf{a}^2 - d \mathbf{S}_3^2 + d^2 f_2 \mathbf{S}_2^1, \\ \displaystyle \mathbf{B}^{-3}= & {} \mathbf{a}^3 - d \mathbf{S}_4^3 + d^2 \left( f_3 \mathbf{S}_2^1 + f_2 \mathbf{S}_3^2 \right) - d^3 f_2^2 \mathbf{S}_2^1, \\ \displaystyle \mathbf{B}^{-4}= & {} \mathbf{a}^4 - d \mathbf{S}_5^4 + d^2 \left( f_4 \mathbf{S}_2^1 + f_3 \mathbf{S}_3^2 + f_2 \mathbf{S}_4^3 \right) \\&- d^3 \left( 2 f_2 f_3 \mathbf{S}_2^1 + f_2^2 \mathbf{S}_3^2 \right) + d^4 f_2^3 \mathbf{S}_2^1, \\ \displaystyle \mathbf{B}^{-5}= & {} \mathbf{a}^5 - d \mathbf{S}_6^5 + d^2 \sum ^4_{i=1} f_{5-i} \mathbf{S}^i_{i+1} \\&- d^3 \left( 2f_2 f_4 \mathbf{S}_2^1 + 2 f_2 f_3 \mathbf{S}_3^2 + f_2^2 \mathbf{S}_4^3 \right) \\&+d^4 \left( 3f_2^2 f_3 \mathbf{S}_2^1 + f_2^3 \mathbf{S}_3^2 \right) \\&- d^5 f_2^4 \mathbf{S}_2^1, \end{aligned}$$

and so on. Note (B.5) and the expressions for $e_1, \ldots , e_6$ above follow. $\square$

As a special case we have

$$\begin{aligned} \displaystyle {{\varvec{\tau }}}^+ \left( \mathbf{I}_M + {{\varvec{\tau }}} {{\varvec{\tau }}}^+ \right) ^{-r} {{\varvec{\tau }}} = \left| {{\varvec{\tau }}} \right| ^2 \left( 1 + \left| {{\varvec{\tau }}} \right| ^2 \right) ^{-r}. \end{aligned}$$

(B.7)