1. Introduction

In multivariate statistical analysis, the distribution of the eigenvalues of a random matrix is widely applied to test various hypotheses. The distribution is also used in nuclear physics, since the behavior of highly excited energy levels in nuclear physics can be explained by considering the distribution of eigenvalues of some random matrices. For any n × n square matrix A_n with only real eigenvalues denoted by λ₁, λ₂, …, λ_n, the empirical spectral distribution function of A_n can be defined as where I_A denote the indicator function. If in the large n limit the limiting distribution exists, then this limit is referred to as the limiting spectral distribution of A_n. The limiting spectral properties of large random matrices have aroused great interests in statistics, finance, signal processing, and other disciplines. Interested readers can refer to Bai and Silverstein [1] to make a comprehensive review of this rapidly developing research field.

This paper analyzes the spectral property of the product of random matrices, which is useful for many problems in multivariate statistical analysis. For example, in multivariate discriminant analysis of variance analysis, and test the equality of two covariance matrices when the potential distribution is multivariate normal, it is interesting to consider the eigenvalues of a multivariate F matrix, where F is known to be the central multivariate F matrix when the two random matrices are distributed independent of each other as central Wishart matrices. Considerable works are conducted in the past on the theory of asymptotic distributions of certain functions of the eigenvalues of the multivariate F matrix, which is basically limited to the case where the number of variables is fixed and the sample size tends to infinity. However, many situations arise when the experimenter is confronted with the problem of inferring from the data with a very large number of variables. Here, we consider the general F matrix , which reduces to the central multivariate F matrix when T_n is the inverse of another independent covariance matrix. For the special case of T = I = identity, there are also many results available, see Grenader and Silverstein [2], Wachter [3], as well as Yin and Krishnaiah [4].

For random matrix , which is also a model often involved in the wireless communication, we focus on the limiting spectral distribution of X_n, when the column vectors of X_n are isotropic log-concave distributed in this paper. That is to say, the collected signal data is considered to be a whole vector, rather than composes of many unrelated signal components. As is well known, the isotropic logarithmic concave distribution has also been a widely studied distribution type in the fields of economics, statistics, and information theory in recent years [5]. By the way, the studies of randomness and matrices have always also been a topic of great concern for many other professional researchers (see [6–9]). In fact, any non-degenerate zero mean random vector can become an isotropic random vector after the proper linear transformation, such as the standard normal distribution random vector, Bernoulli type random vector with independent Bernoulli random variable components, and uniformly distributed random vector on convex set in R_n.

The research on the model starts from a class of important Hermitian random matrices with the form of , where matrix X_n = (X_ij)_n×N is a complex random matrix, and random variables X_ij’s are independent identically distributed (i.i.d.), where T_n is a n × n diagonal matrix and is independent of X_n, and its empirical spectral distribution , where H is a non-random probability distribution defined on [0, ∞). With regard to , the earliest researches are focused by Marčenko and Pastur [10] and Grenander and Silverstein [2], who proved that when n, N → ∞ and , the empirical spectral distribution of matrix weakly converges to a non-random probability distribution in probability. Then, Wachter [3], Jonsson [11], Yin [12] and Yin and Krishnaiah [4] conducted further research, and illustrated that the empirical spectral distribution converges weakly to a certain non-random probability distribution almost surely. It is noted that Yin and Krishnaiah [4] and Yin [12] established this result under the assumption that matrix T_n is not a diagonal matrix, but a Hermitian nonnegative definite matrix. Moreover, Yin [12] weakened the condition for X_n, while other results required that the element moment condition of X_n is at least higher than the second order. Except for Marčenko and Pastur [10], others only studied the case under the assumption that the element of X_n is real. More importantly, only Marčenko and Pastur [10] considered the case where H is arbitrary, while others require H to satisfy all moment conditions and adopt moment methods to prove the limit theorem.

Based on Yin’s results [12], Silverstein [13] studied the matrix by generalizing Yin’s assumptions [12], and extended the condition that “X_ij’s are real random variables” to complex cases, and studied the behavior when H is arbitrary. It was proved by Stieltjes transformation method that, almost surely, the empirical spectral distribution of converges weakly to a non-random probability distribution. Silverstein and Bai [14] found that when matrix T_n is an arbitrary diagonal matrix, the empirical spectral distribution of , where denotes a non-random probability distribution. Let be the Stieltjes transformation of , then is the solution of equation (1)

Moreover, for any , is the unique solution of Eq (1). Assuming that all elements of matrix X_n are i.i.d. random variables, Silverstein [13] investigated matrix based on the case that matrix T_n is a Hermitian non-negative definite matrix. Comparing the spectra of matrices and , we find that there is a difference of |n − N| zero eigenvalues between them. It is then easy to obtain the relationship between their empirical spectral distribution function, which is given by (2) where . Then, the relationship of Stieltjes transformation of their empirical spectral distribution is (3)

If there exist the limits of empirical spectral distributions and , they are denoted by and F, respectively. Then we calculate the limits of Eqs (2) and (3), respectively, and have and (4)

Then, if is conjectured, it can be obtained from Eqs (1) and (4) that the Stieltjes transformation m = m_F(z) of the limiting spectral distribution of matrix is the solution of equation (5) and the solution is unique on the set . In this paper, the limiting spectral distribution of such a matrix model is studied: suppose that the column vector X_⋅j of X_n, and we specify the joint distribution of column vectors of X_n. In particular, entries on the same column of X_n are correlated, in contrast with more common independence assumptions. Assume that X_⋅j is an n-dimensional random vector following isotropic log-concave distribution. In the following, we continue to focus on the case of matrix model with i.i.d. column vector X_⋅j of X_n being isotropic log-concave distributed. Here are the corresponding definitions.

Definition 1.1. A random real vector follows isotropic distribution, if it satisfies where 〈, 〉 is the ordinary inner product, and ‖⋅‖ is ordinary Euclidean norm. A random complex vector follows isotropic distribution, if follows isotropic distribution, and ReX is independent of ImX, where ReX and ImX represent the real part and the imaginary part of X, respectively.

Remark 1.1. According to the above definition, if X follows isotropic distribution, then EX = 0, Σ(X) = cov(X, X) = I_n. Any non-degenerate random vector can be transformed into a random vector obeying isotropic distribution by appropriate linear transformation. In fact, if the mean value of any random vector is 0, and its covariance matrix Σ(X) is reversible, then follows isotropic distribution.

Definition 1.2. A function is log-concave, if for any , Remark 1.2. If X is an n-dimensional continuous random variable with a log-concave distribution, and its density function is f(x), the marginal density function corresponding to any of its sub-vectors also follows a log-concave distribution. And if X follows the log-concave distribution, then any affine transformation of X follows the log-concave distribution. Log-concave distribution constitutes a large class of distributions, such as multidimensional uniform distribution, multidimensional normal distribution, multidimensional exponential distribution, etc.

Prior to giving the main results, we introduce some notations. implies that a random variable sequence {X_n} converges to the random variable X in distribution. Denote the trace of matrix A by tr(A). Let E be the expectation of matrix or vector, and C be a positive constant, and its value may be different in the context. Now, we present the main results.

Theorem 1.1. Suppose that

(a) X_n is an n × N complex random matrix, and the column vector X_⋅j (j = 1, ⋯, N) is a random vector with n-dimensional mutually independent isotropic log-concave distribution [15, 16].

(b) T_n is an n × n Hermitian non-negative definite matrix, and when n → ∞, its empirical spectral distribution , where H is any probability distribution defined on [0, ∞).

(c) T_n is independent of X_n.

(d) When n, N → ∞, .

Then, the empirical spectral distribution of matrix B_n converges weakly to a non-random distribution F almost surely, and the Stieltjes transformation m = m(z) of F satisfies the Eq (5) for any . And on , m = m(z) is the unique solution of Eq (5).

2. Proofs

In the proofs, we mainly use the continuity theorem of Stieltjes transform and the properties of isotropic log-concave random variables. Recall that, the Stieltjes transform of a distribution function F(x) is given by Then, we can write the Stieltjes transform of empirical spectral distribution as where I_n denotes the identity matrix of size n. For a sequence of functions of bounded variation {G_n}, whose Stieltjes transform is , where G_n(−∞) = 0 for all n, and a function of bounded variation G, where G(−∞) = 0, whose Stieltjes transform is m_G(z), then the continuity theorem of Stieltjes transformation shows that G_n converges vaguely to G when converges to m_G(z) uniformly for . Since the empirical spectral distribution sequence of random matrix is tight (Lytova and Pastur [17]), we can use the convergence of its corresponding Stieltjes transformation to obtain the weak convergence of empirical spectral distribution. In addition, according to the almost surely convergence of Stieltjes transformation, the almost surely convergence of empirical spectral distribution is obtained if the limiting spectral distribution is a deterministic probability density function.

Now, we give the proof of Theorem 1.1. Recall that and , and let m_n(z) and be Stieltjes transformations of empirical spectral distribution and , respectively. Let , where , and X_⋅j represents j-th column of X_n. Then, the proof of Theorem 1.1 is obtained by the following lemmas.

Lemma 2.1. For any fixed , let , then Proof. For any , Since (B_n − zI) is invertible, we have (6) For any integer 1 ≤ j ≤ N, we define From we know that Moreover, we have (7) From (3), we know that (8) Then combining (6), (7) and (8), we obtain that The proof of Lemma 2.1 is completed.

Lemma 2.2. For n × n matrices B and , where B and B + qq* are invertible, then (9) Proof. Let q*(B + qq*)⁻¹ ≡ t*, then q* = t*B + t*qq*. Multiply B⁻¹ on both sides of Eq (9), and then we have (10) Multiply q on both sides of Eq (10), and we get that Because q is arbitrary, we obtain that 1 + q*B⁻¹q ≠ 0. Thus, (11) From Eq (10), we obtain that (12) Then, combining (11) and (12), we have Namely, The Eq (9) is proved.

Lemma 2.3. For matrix , we have where and Proof. Define , then (13) By Lemmas 2.1 and 2.2 and Eq (13), we have Take the trace of two sides of the above equation, and divide by n, then we get that where The proof of Lemma 2.3 is completed.

Lemma 2.4. For matrix , we have Proof. By Lemma 2.3, we have

For any z = u + iv, and any j, we have Thus, Then for d_j, we have where A ∨ B = max{A, B}. Here we use Lemma 3.3 and the following facts from lemma 2.3 [13] and Eq (3.10) [18]; that is,

when ,

1. , where the spectral norm ‖T_n‖ can be controlled by a sufficiently large number τ₀, i.e., ‖T_n‖≤τ₀, and this result is obtained by using a similar discussion from Silverstein and Bai [14].
2. .

Hence, when n → ∞, The proof is completed.

Lemma 2.5. Consider that the subsequence of sequence {m_n(z)} converges to m. Let be the limit with respect to . Then, the following function can be found such that Proof. This lemma is to obtain that is tight. Define By Lemma 2.8 [19], we get that δ is positive, a.s. So there exists m ∈ D_c such that , where . If , and , from the fact that We obtain that function f(τ) is bounded and Then the lemma is proved.

Consequently, it can be obtained that Combined with Eq (4), we get that m satisfies Eq (5). Since m is the only solution satisfying Eq (5) [19], we have that m_n(z) → m. By the results from Lemmas 2.1-2.5, the empirical spectral distribution of matrix B_n converges weakly to F almost surely, and the Stieltjes transformation of F satisfies Eq (5), and thus Theorem 1.1 is proved.

3. Appendix

In recent years, many studies have been conducted on the properties of random variables with isotropic distributions following log-concave distribution. We now introduce some related properties that not only help to understand the family of isotropic log-concave distributions, but also play an important role in the proofs. First, a result proved by Klartag [20] on large deviation is presented.

Lemma 3.1. Let n ≥ 1 be a positive integer, and X be a random vector following isotropic log-concave distribution in , then where C, k > 0 are constants.

Based on this result, Pastur and Shcherbina [21] obtained the following concentration inequality about Euclidean norm.

Lemma 3.2. Let n ≥ 1 be a positive integer, and X be a random vector following isotropic log-concave distribution in , then where C₁, k > 0 are constants.

Pajor [15] used the above results to draw the following important conclusion, which is also a significant fact used in the proof.

Lemma 3.3. Let M be a complex random matrix whose spectral norm is ‖M‖ ≤ 1. If (or ) follows an isotropic log-concave distribution, then where b, k > 0 are constants.

4. Conclusions

This paper investigates the spectral properties of general F matrix . There has been a lot of work on the spectral properties of this matrix model, but the existing results always assume X_n being a matrix composed of n × N independent identically distributed elements. In this paper, we study that the column vector of X_n is a whole vector, not composed of unrelated independent components. Suppose that the column vectors of X_n follow isotropic log-concave distribution, we establish that the empirical spectral distribution of B_n converges to a non-random distribution probability F almost surely. Specifically, we prove that the Stieltjes transformation of F satisfies a deterministic form of equation and is the unique solution of the equation.