In this section, we will derive the general form of the CDF of the SCN of central semicorrelated Wishart matrix for arbitrary K and N. Then, we emphasize by considering the threedimensional case (K=3). Given (8) and the results of this section, P
_{d} can then be directly computed.
5.1 The general form
The CDF, F(x), with x≥1, of the SCN of the central semicorrelated Wishart matrix \( \mathbf {W} \sim \mathcal {CW}_{K}(N, \boldsymbol {\Sigma }_{K})\) with distinct correlation eigenvalues, denoted by σ
_{1}>σ
_{2}>⋯>σ
_{
K
}>0, is given by [30]
$$ F(x) = C \sum_{l=1}^{K}\int\limits_{0}^{\infty}\left \left[ \begin{array}{ll} \int\limits_{\lambda_{K}}^{x\lambda_{K}} \phi_{i}(u) \psi_{j}(u) \xi(u) du, & i\neq l \\ \phi_{i}(\lambda_{K})\psi_{j}(\lambda_{K})\xi(\lambda_{K}), & i=l \end{array} \right] \rightd\lambda_{K}, $$
(18)
where ϕ
_{
i
}(λ
_{
j
})= [ Φ(λ)]_{
i,j
} and ψ
_{
i
}(λ
_{
j
})= [ Ψ(λ)]_{
i,j
} with Φ(λ), Ψ(λ), ξ(λ
_{
i
}), and C are defined in [30, Table I].
If we have L equal correlation eigenvalues (say from p to q with q=p+L−1), then (18) is still valid by considering the parameters Φ(λ), Ψ(λ), ξ(λ
_{
i
}), and C that are defined in Theorem 1. This could be easily shown by comparing the joint distribution parameters in Theorem 1 and Table I of [30] and noting that they have the same general form used in deriving the SCN CDF in (18).
It is important to note that all the integrals inside the determinant in (18), \(I_{i,j} =\int _{\lambda _{K}}^{x\lambda _{K}}\phi _{i}(u) \psi _{j}(u) \xi (u) du\), still admit a closedform solution given, directly after integrating using [38, Eq. (3.351.1)], by
$$ I_{i,j} = \left\{\begin{array}{ll} \left[\gamma\left(l,\frac{x\lambda_{K}}{\sigma_{j}}\right)\gamma\left(l,\frac{\lambda_{K}}{\sigma_{j}}\right)\right]\cdot {\sigma_{j}^{l}} & i < p \text{or }i \geq q \\ \sum\limits_{k=1}^{qj}\left[(1)^{k+qj}\mathcal{L}(qj,k)\cdot\left[\gamma\left(l +k,\frac{x\lambda_{K}}{\sigma_{p}}\right)\gamma\left(l +k,\frac{\lambda_{K}}{\sigma_{p}}\right)\right]\right]\sigma_{p}^{l+jq} & p\leq i < q \\ \end{array}\right., $$
(19)
where γ(.,.) is the lower incomplete gamma function [38, Eq. (8.350.1)], the constant l=N−K+i, and 1≤i≤K.
5.2 SCN distribution for threedimensional systems
In the previous sections, we have considered an arbitrary number (L) of equal eigenvalues positioned anywhere in vector σ. Considering our assumptions that Ω
_{
K
} is a rank 1 matrix reveals, from Lemma 1, that the eigenvalues of the effective correlation matrix of the central semicorrelated Wishart matrix has (K−1)equal eigenvalues such that σ
_{1}>σ
_{2}=⋯=σ
_{
K
}>0.
Now, if we consider the case K=3, then \(\widehat {\mathbf {\Sigma }}_{K}\) has two equal eigenvalues (σ
_{1}>σ
_{2}=σ
_{3}>0). In this case, the CDF and PDF of the SCN are given in Theorem 2 and Corollary 1, respectively.
Theorem
2 (CDF of SCN of central semicorrelated Wishart matrix when Σ
_{3} has two equal eigenvalues).
The CDF of the SCN of a 3×3 central semicorrelated Wishart matrix \( \mathbf {W} \sim \mathcal {CW}_{K}(N, \mathbf {\Sigma }_{3})\) whose Σ
_{3} has two equal eigenvalues (σ
_{1}>σ
_{2}=σ
_{3}) is given by
$$ F_{1}(x) = C_{sc}[R_{1}(N,x)  G_{1}(N,x)], $$
(20)
where R
_{1}(N,x) is defined in (21), G
_{1}(N,x) in (22), S
_{1}(r,s,t,μ,ν,x) in (23), Δ
_{1}(r,s,t,μ,ν,ε,x) in (24), and C
_{
sc
} in (25).
$$ {} \begin{aligned} R_{i}(N,x) &= S_{i}\left(N3,N,N2,\sigma_{1}^{1},\sigma_{2}^{1},x\right)\\ &\quad+S_{i}\left(N2,N2,N1,\sigma_{1}^{1},\sigma_{2}^{1},x\right) \\ &\quad+S_{i}\left(N\,\,1,N\,\,1,N\,\,3,\sigma_{1}^{1},\sigma_{2}^{1},x\right), i=1,2 \end{aligned} $$
(21)
$$ {{} \begin{aligned} G_{i}(N,x) =\, &\Delta_{i}\left(N2,N1,N2,\sigma_{2}^{1},\sigma_{2}^{1},\sigma_{1}^{1},x\right) \\ &+\Delta_{i}\left(N2,N,N3,\sigma_{2}^{1},\sigma_{2}^{1},\sigma_{1}^{1},x\right)\\ &+\Delta_{i}\left(N3,N1,N1,\sigma_{2}^{1},\sigma_{2}^{1},\sigma_{1}^{1},x\right), \ \ \!\!i=\!1,2 \end{aligned}} $$
(22)
$$ {{}\begin{aligned} S_{i}(r,s,t,\mu,\nu,x) = \, &\Delta_{i}\left(r,s,t,\nu,\nu,\mu,x\right) + \Delta_{i}\left(t,r,s,\nu,\mu,\nu,x\right) \\ & \Delta_{i}\left(r,t,s,\nu,\mu,\nu,x\right) \Delta_{i}\left(r,s,t,\nu,\mu,\nu,x\right) \\ &+\Delta_{i}\left(t,s,r,\nu,\mu,\nu,x\right), \ \ i=1,2 \end{aligned}} $$
(23)
$$ \Delta_{1}\left(r,s,t,\mu,\nu,\varepsilon,x\right) = \mu^{2} \Delta_{3}\left(r,s,t,\nu,\varepsilon,\mu,x\right) $$
(24)
$$ C_{sc} = \frac{\left(\sigma_{2}  \sigma_{1}\right)^{2}}{(N1)!(N2)!(N3)!\sigma_{1}^{N2} \sigma_{2}^{2(N2)}} $$
(25)
with Δ
_{3} defined in (35) in Appendix 2, Theorem 3.
Proof.
The proof could be summarized as follows:

Considering (18) and setting K=3.

Substituting the parameters from Theorem 1.

Expanding the summation and using (19).

Expanding the determinant and integrating using Eqs. (3.351.1) and (3.351.3) of [38].
Then, the result comes after simplification.
Corollary
1.
The PDF of the SCN of a 3×3 central semicorrelated Wishart matrix \( \mathbf {W} \sim \mathcal {CW}_{K}(N, \mathbf {\Sigma }_{3})\) whose Σ
_{3} has two equal eigenvalues (σ
_{1}>σ
_{2}=σ
_{3}) is given by
$$ f_{1}(x) = C_{sc}\cdot \left[R_{2}(N,x)G_{2}(N,x)\right], $$
(26)
where R
_{2}(N,x) is defined in (21), G
_{2}(N,x) in (22), S
_{2}(r,s,t,μ,ν,x) in (23), C
_{
sc
} in (25), and Δ
_{2}(r,s,t,μ,ν,ε,x) in (27) below.
$$ \Delta_{2}\left(r,s,t,\mu,\nu,\varepsilon,x\right) = \mu^{2} \Delta_{4}\left(r,s,t,\nu,\varepsilon,\mu,x\right) $$
(27)
with Δ
_{4} defined in (38) in Appendix 2, Corollary 2.
Proof.
Obtained by differentiating Eq. (20).
For the sake of completeness, the reader could find, in Appendix 2, the CDF and PDF of the SCN of the central semicorrelated Wishart matrix when all the correlation eigenvalues are distinct.