Multivariate Gamma FunctionEdit
The multivariate gamma function is a cornerstone of multivariate analysis and random matrix theory. It generalizes the one-dimensional gamma function to the space of positive-definite matrices, providing a compact, closed-form normalization that appears wherever one integrates over matrix-valued random variables. In particular, the parameter p denotes the dimension of the matrices (p×p), and a is a complex parameter subject to a minimal real-part constraint. The function plays a central role in the densities of matrix-variate distributions and in the evaluation of multivariate integrals that arise in statistical theory.
For p = 1, the multivariate gamma function collapses to the ordinary gamma function, illustrating its role as a genuine generalization. Beyond this special case, Γ_p(a) admits a precise product representation that makes its structure transparent: it factors into p one-dimensional gamma terms with shifted arguments. Because of this, many properties of Γ_p(a) can be understood by assembling the familiar properties of the scalar gamma function. The multivariate gamma function also has a useful integral representation, which expresses it as an integral over real symmetric positive-definite matrices and connects it directly to matrix determinants and traces. These dual representations—product and integral—support a wide range of theoretical and computational techniques.
Definition
Let p be a positive integer and a a complex number with Re(a) > (p−1)/2. The real (or real-valued) multivariate gamma function Γp(a) is defined by the product formula Γ_p(a) = π^{p(p−1)/4} ∏{j=1}^{p} Γ(a − (j−1)/2), where Γ denotes the ordinary gamma function. This identity expresses Γ_p(a) entirely in terms of one-dimensional gamma functions and reveals how the dimension p contributes via the shifted arguments.
There is also an integral representation over the cone of real symmetric positive-definite p×p matrices. If S runs over all such matrices and dS denotes the Lebesgue measure on that space, then Γp(a) = ∫{S > 0} exp(−tr(S)) det(S)^{a − (p+1)/2} dS. Here det(S) is the determinant and tr(S) is the trace. The integral form makes explicit the connection with the geometry of positive-definite matrices and with determinants.
Gamma function is the scalar case, recovered when p = 1. The conditions Re(a) > (p−1)/2 guarantee convergence of the integral and the validity of the product formula.
Properties
Reduction to the scalar case: For p = 1, Γ_1(a) = Γ(a). This reflects the natural embedding of the multivariate case into the classical theory.
Recurrence in a: The multivariate gamma function satisfies Γp(a+1) = Γ_p(a) ∏{j=1}^{p} (a − (j−1)/2). This mirrors the familiar recurrence Γ(a+1) = a Γ(a) but incorporates the dimension-dependent shifts.
Relation to matrix-variate distributions: The normalization constants of matrix-variate densities—most prominently the Wishart distribution Wishart distribution and, by extension, the inverse-Wishart distribution—feature Γ_p(a) with a tied to the degrees of freedom and matrix dimension. For example, the real Wishart density with scale matrix Σ and degrees of freedom n uses Γ_p(n/2) in its denominator. This makes Γ_p(a) essential for computing probabilities, moments, and Bayesian posteriors in multivariate statistics.
Analytic properties: Γ_p(a) extends meromorphically to a larger domain in a, with poles that reflect the gamma factors in the product representation. Its logarithm is often used in numerical work because the product structure translates into sums of log-Γ terms, which improves numerical stability.
Special cases and extensions: When considering complex matrices or other algebraic settings, there are analogous multivariate gamma-type functions (for example, the complex multivariate gamma function). These variants preserve the core role of a dimension-dependent product of scalar gamma terms and appear in corresponding matrix-variate distributions and random-matrix contexts.
Computation and practical use
Numerical evaluation: In practice, Γp(a) is computed via the product formula, taking logs to maintain numerical stability: log Γ_p(a) = (p(p−1)/4) log π + ∑{j=1}^{p} log Γ(a − (j−1)/2). This avoids underflow or overflow when a or p is large.
Domain considerations: The requirement Re(a) > (p−1)/2 is essential for convergence of the integral representation and for the product formula to yield a finite positive value. When working with related distributions, this constraint translates into conditions on degrees of freedom or prior parameters.
Connections to functions of a matrix: Since Γ_p(a) can be viewed as a normalization constant for certain matrix-variate densities, its value directly influences the shape and normalization of those distributions. In turn, this affects statistical inference tasks such as posterior computations and marginal likelihoods in multivariate models.
Applications and context
Matrix-variate distributions: The most immediate arena for Γ_p(a) is the theory of matrix-variate distributions, including the matrix-variate normal family and the Wishart family. The latter describes sample covariance matrices and appears widely in multivariate statistics, signal processing, and econometrics. The normalization constants that arise in these densities are expressed through Γ_p(a). See Wishart distribution for the distribution’s role and properties.
Bayesian statistics and covariance estimation: In Bayesian analyses of multivariate normal models, prior distributions on covariance matrices or scale matrices frequently involve matrix-variate gamma-type normalizations. The multivariate gamma function thus enters the computation of priors, posteriors, and marginal likelihoods in models with p-dimensional covariance structures. See Inverse-Wishart distribution for a common conjugate prior setting.
Random matrix theory and multivariate analysis: Beyond classical statistics, Γ_p(a) appears in integrals over the space of positive-definite matrices that arise in random matrix theory, representation theory, and multivariate methods. Its product form provides a convenient bridge between high-dimensional integrals and products of one-dimensional gamma functions, simplifying both analysis and computation.
Related functions and generalizations: The multivariate gamma function has counterparts in complex and quaternionic settings and connects to broader families of special functions that appear in multivariate integration and spectral theory. See Complex multivariate gamma function for a common complex-analytic analogue and Matrix-variate distribution for a broader context of matrix-valued distributions.