Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions
For positive integers and p such that ≥ , let ℝᵈˣᵖ denote the set of × real matrices, ₚ be the identity matrix of order , and d,ₚ = { ∈ ℝᵈˣᵖ ∣ ′ = ₚ} be the Stiefel manifold in ℝᵈˣᵖ. Complete asymptotic expansions as → ∞ are obtained for the normalizing constants of the matrix Bingham and matri...
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| description | For positive integers and p such that ≥ , let ℝᵈˣᵖ denote the set of × real matrices, ₚ be the identity matrix of order , and d,ₚ = { ∈ ℝᵈˣᵖ ∣ ′ = ₚ} be the Stiefel manifold in ℝᵈˣᵖ. Complete asymptotic expansions as → ∞ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on d,ₚ. The accuracy of each truncated expansion is strictly increasing in ; also, for sufficiently large , the accuracy is strictly increasing in , the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both → ∞ and → ∞. Values of and arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as or increases. These results extend recently obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 094, 22 pages
Complete Asymptotic Expansions for the Normalizing
Constants of High-Dimensional Matrix Bingham
and Matrix Langevin Distributions
Armine BAGYAN and Donald RICHARDS
Department of Statistics, Pennsylvania State University, University Park, PA 16802, USA
E-mail: aub171@psu.edu, dsr11@psu.edu
Received February 28, 2024, in final form October 10, 2024; Published online October 21, 2024
https://doi.org/10.3842/SIGMA.2024.094
Abstract. For positive integers d and p such that d ≥ p, let Rd×p denote the set of d × p
real matrices, Ip be the identity matrix of order p, and Vd,p =
{
x ∈ Rd×p | x′x = Ip
}
be the
Stiefel manifold in Rd×p. Complete asymptotic expansions as d → ∞ are obtained for the
normalizing constants of the matrix Bingham and matrix Langevin probability distributions
on Vd,p. The accuracy of each truncated expansion is strictly increasing in d; also, for
sufficiently large d, the accuracy is strictly increasing in m, the number of terms in the
truncated expansion. Lower bounds are obtained for the truncated expansions when the
matrix parameters of the matrix Bingham distribution are positive definite and when the
matrix parameter of the matrix Langevin distribution is of full rank. These results are
applied to obtain the rates of convergence of the asymptotic expansions as both d → ∞
and p → ∞. Values of d and p arising in numerous data sets are used to illustrate the
rate of convergence of the truncated approximations as d or m increases. These results
extend recently-obtained asymptotic expansions for the normalizing constants of the high-
dimensional Bingham distributions.
Key words: Frobenius norm; generalized hypergeometric function of matrix argument;
Grassmann manifold; Hadamard’s inequality; hippocampus; neural spike activity; Stiefel
manifold; Super Chris (the rat); symmetric cone; zonal polynomial
2020 Mathematics Subject Classification: 60E05; 62H11; 62E20; 62R30
1 Introduction
For positive integers d and p such that d ≥ p let Rd×p denote the set of d×p real matrices, Ip be
the identity matrix of order p, and Vd,p =
{
x ∈ Rd×p | x′x = Ip
}
denote the Stiefel manifold. The
manifold Vd,p is endowed with a unique probability measure, denoted by dx, that is invariant
under the change of variables x 7→ HxK, where x ∈ Vd,p, and H and K are orthogonal matrices
of orders d× d and p× p, respectively (see Chikuse [4, p. 16], Muirhead [30, p. 70]).
In this article, we first consider the matrix Bingham probability distributions on Vd,p. For
non-zero symmetric matrices A ∈ Rp×p and Σ ∈ Rd×d, a random matrix X ∈ Vd,p is said to have
a (full) matrix Bingham distribution with parameters A and Σ if the probability density function
of X, with respect to the invariant measure dx, is
ϕ(x;A,Σ) = [Φd,p(A,Σ)]
−1 exp
(
trAx′Σx
)
, x ∈ Vd,p, (1.1)
where Φd,p(A,Σ) is the normalizing constant. The matrix Bingham family is now ubiquitous
in the statistical analysis of directional data and shape analysis, and the extant literature now
provides many properties of the family and its applications in fields including astronomy, biology,
This paper is a contribution to the Special Issue on Asymptotics and Applications of Special Functions in
Memory of Richard Paris. The full collection is available at https://www.emis.de/journals/SIGMA/Paris.html
mailto:aub171@psu.edu
mailto:dsr11@psu.edu
https://doi.org/10.3842/SIGMA.2024.094
https://www.emis.de/journals/SIGMA/Paris.html
2 A. Bagyan and D. Richards
computer vision, medicine, meteorology, physics and psychology; see, e.g., [3, 4, 7, 22, 23, 25,
24, 29, 33].
We also consider the matrix Langevin (or matrix von Mises–Fisher) family of probability
distributions on Vd,p. For non-zero B ∈ Rd×p, a random matrix X ∈ Vd,p is said to have a matrix
Langevin distribution with parameter B if the probability density function of X, with respect to
the invariant measure dx, is
ψ(x;B) = [Ψd,p(B)]−1 exp(trB′x), x ∈ Vd,p, (1.2)
where Ψd,p(B) is the normalizing constant. These distributions too are widely used in the
statistical analysis of shape and directional data; see, e.g., Chikuse [4], Mardia and Jupp [29],
Oualkacha and Rivest [31].
A major difficulty in statistical inference with the matrix Bingham and matrix Langevin
distributions is the calculation of the constants Φd,p(A,Σ) and Ψd,p(B). Some authors have
described this issue as the “main obstacle” to maximum likelihood estimation because the nor-
malizing constants, although known analytically, are difficult to compute in practice (Mantoux,
et al. [28, pp. 4, 11]). For that reason, substantial attention has been paid to the problem of
calculating those constants (see Kume and Wood [25], and Kume, Preston and Wood [24]).
In a recent article [1], we studied the Bingham distributions for the case in which p = 1. Under
a growth condition on Σ we obtained for Φd,1(1,Σ) a complete asymptotic series expansion in
negative powers of d, and we derived a bound for the tail of that series upon truncation after,
say, m terms. In [1], we used an explicit formula for the zonal polynomials indexed by partitions
with one part; however, as no explicit formula is known for the zonal polynomials in general,
a more powerful approach is needed for general p, and the purpose of the present article is to
provide such an approach.
In the present article, we obtain for all d and p complete asymptotic expansions of Φd,p(A,Σ)
and Ψd,p(B) in negative powers of d; these expansions are derived subject to growth conditions
on the Frobenius norms of Σ and B as d → ∞. We accomplish these results using detailed
properties of the zonal polynomials, including an integral representation due to James [18].
We also derive an infinite series bound and a closed-form bound for the remainder term in
each asymptotic expansion. These bounds, as they are valid for all d ≥ p, extend the results
in [1]. We show further that the new bounds decrease strictly to zero as d, or as m, the number
of terms in the truncated expansion, increases; equivalently, the accuracy of each truncated
asymptotic expansion is strictly increasing in d or m. We devote close attention to the closed-
form upper bound as it is likely to be more widely used than its infinite series counterpart; in
particular, we derive conditions on the values of the parameters such that the closed-form bound
is unimodal in m.
When the matrices A and Σ in (1.1) are positive definite, we obtain a lower bound on the
truncated expansion for Φd,p(A,Σ). Further, we derive a lower bound for the truncated expansion
of Ψd,p(B) when B is of full rank. These lower bounds can be expressed in closed-form, and
they decrease to zero as d→ ∞ or m→ ∞.
The contents of the article are as follows. In Section 2, we obtain the asymptotic expansion
for Φd,p(A,Σ) and we remark on the properties of the expansion, and in Section 3, we obtain
a similar expansion for Ψd,p(B). In Section 4, we survey the range of values of d and p that
have appeared in the literature on analyses of data on Stiefel manifolds, describe the rate of
convergence to zero of each bound as d or m increases, and conclude by noting that our results
extend to more general classes of distributions on the Stiefel and Grassmann manifolds and to
the complex analogs of the matrix Bingham and matrix Langevin distributions. We provide
in Section 5.1 some necessary properties of the zonal polynomials, and all proofs are given in
Sections 5.2–5.4.
Complete Asymptotic Expansions for the Normalizing Constants 3
2 Complete asymptotics for the normalizing constant
of the high-dimensional matrix Bingham distribution
A partition κ = (κ1, . . . , κd) is a vector of integers such that κ1 ≥ · · · ≥ κd ≥ 0. The non-
zero κj are called the parts of κ. The length of κ, denoted by ℓ(κ), is the number of non-
zero κj , j = 1, . . . , d, and the weight of κ is |κ| = κ1 + · · ·+ κd.
Define the shifted factorial, (a)k = a(a+1)(a+2) · · · (a+k−1), where a ∈ R and k = 0, 1, 2, . . .;
in particular, (a)0 ≡ 1 since the defining product is empty. For a ∈ R and a partition
κ = (κ1, . . . , κd), the partitional shifted factorial is
(a)κ =
d∏
i=1
(
a− 1
2
(i− 1)
)
κi
. (2.1)
Assume that Rd×d
sym denote the space of real symmetric d × d matrices. For a partition
κ = (κ1, . . . , κd) and Σ ∈ Rd×d
sym, the zonal polynomial Cκ(Σ) is an eigenfunction of the Laplace–
Beltrami operator in the eigenvalues of Σ (see [17], [30, Chapter 7] and [35]). Further, Cκ(Σ) de-
pends only on the eigenvalues of Σ, and is symmetric in those eigenvalues.
Tables of the polynomials Cκ(Σ) for |κ| ≤ 12 were provided by James [16, pp. 498–499] and
Parkhurst and James [32]. Jiu and Koutschan [20] have provided software to calculate rapidly
the zonal polynomials to high values of |κ|, so the asymptotic expansions obtained in this article
can be calculated for practical values of m.
Let X ∈ Vd,p be a random matrix having the matrix Bingham distribution with the den-
sity function ϕ(x;A,Σ) as in (1.1). It is well known that the normalizing constant Φd,p(A,Σ)
can be expressed as a generalized hypergeometric function of two matrix arguments, denoted
by 0F0(A,Σ), or by the zonal polynomial expansion of that function,
Φd,p(A,Σ) = 0F0(A,Σ) =
∞∑
k=0
1
k!
∑
|κ|=k
Cκ(A)Cκ(Σ)
Cκ(Id)
, (2.2)
where the inner sum is over all partitions κ such that ℓ(κ) ≤ p. This formula for Φd,p(A,Σ) in
terms of 0F0(A,Σ) dates to De Waal [5]; cf. Bingham [2], Chikuse [4, p. 108]. By applying the
methods of [11, Theorem 6.3], we find that the series (2.2) converges absolutely for all (A,Σ)
and uniformly on compact regions.
For the special case in which A = Ip, on applying to (2.2) the identity (5.7), written in the
form Cκ(Ip)/Cκ(Id) = (p/2)κ/(d/2)κ, we obtain the normalizing constant in terms of a confluent
hypergeometric function of matrix argument,
Φd,p(Ip,Σ) =
∞∑
k=0
1
k!
∑
|κ|=k
(p/2)κ
(d/2)κ
Cκ(Σ) = 1F1
(
1
2
p;
1
2
d; Σ
)
,
as shown by various authors, e.g., Bingham [2], Chikuse [4, p. 33] and Muirhead [30, p. 288].
Similar to [1, equation (2.4)], when d is large, our asymptotic approximation to Φd,p(A,Σ) is
the sum of the terms up to degree m− 1 of the series (2.2), i.e.,
Φd,p(A,Σ) ≈
m−1∑
k=0
1
k!
∑
|κ|=k
Cκ(A)Cκ(Σ)
Cκ(Id)
, (2.3)
m ≥ 2, and now we study the remainder series
Φd,p;m(A,Σ) =
∞∑
k=m
1
k!
∑
|κ|=k
Cκ(A)Cκ(Σ)
Cκ(Id)
. (2.4)
4 A. Bagyan and D. Richards
In the sequel, we will encounter the constants αp = (2π)−(p−1)/4pp1/4p and γ1 =
(√
3 + 1
)
/2,
arising in Lemmas 5.1 and 5.2, respectively. Also, if a1, . . . , ap are the eigenvalues of A, then
we define A+ = diag(|a1|, . . . , |ap|), the diagonal matrix with diagonal entries |a1|, . . . , |ap|. In the
following result, the proof of which is given in Section 5.3, we extend a result in [1] by deriving
an upper bound for Φd,p;m(A,Σ) in terms of the Frobenius norm, ∥Σ∥ =
[
tr
(
Σ2)]1/2.
Theorem 2.1. Let A ∈ Rp×p
sym, Σ ∈ Rd×d
sym, and suppose that ∥Σ∥ ≤ γ0d
r/2, where γ0 > 0 and
0 ≤ r < 1. Then for all m ≥ 2,
|Φd,p;m(A,Σ)| ≤ αp
∞∑
k=m
[
γ0γ1p
1/2(trA+)d
−(1−r)/2]k
(k!)1/2
(2.5)
≤ αp(4e/π)
1/4(e/m)(m/2)−(1/4)[γ0γ1p1/2(trA+)d
−(1−r)/2]m
× exp
(
cm
[
γ0γ1p
1/2(trA+)d
−(1−r)/2]2/2), (2.6)
where cm =
(
1 +m−1)−m
e. In particular, Φd,p;m(A,Σ) = O
(
d−(1−r)m/2) as d→ ∞.
Remark 2.2.
(i) The growth condition, ∥Σ∥ ≤ γ0d
r/2 with γ0 > 0 and r ∈ [0, 1), is the same as was assumed
in [1] for the Bingham distribution, i.e., the case p = 1. Further, the bounds (2.5) and (2.6)
provide a rate of convergence, viz., Φd,p;m(A,Σ) = O
(
d−m(1−r)/2) as d → ∞, that is the
same as we obtained in [1] for the case in which p = 1. On the other hand, unlike the
bounds in [1], which are valid only for sufficiently large d, the bounds (2.5) and (2.6) are
valid for all d ≥ p ≥ 1 and even for all r ∈ R.
(ii) The bounds (2.5) and (2.6) each are increasing in γ0 and in r; this is consistent with the
restriction ∥Σ∥ ≤ γ0d
r/2 representing a larger matrix region as γ0 or r increases.
(iii) Each bound is also strictly increasing in p or tr(A+) for fixed d, m, γ0, and r. This follows
from the fact that, in the kth summand in the bound in (2.5), the term αpp
k/2 is strictly
increasing in p; therefore, the bound itself is strictly increasing. A similar argument also
yields the same conclusion for the bound in (2.6).
(iv) It is evident that the bound in (2.5), as a function of d or m, decreases strictly to zero
as d→ ∞ or m → ∞. Equivalently, the accuracy of the truncated asymptotic expan-
sion (2.3) is strictly increasing in d or m. Further, it is evident that the closed-form bound
in (2.6) is strictly decreasing in d.
(v) The bound (2.5) on Φd,p;m(A,Σ) also allows us to obtain asymptotics as both d and p
tend to infinity. By (5.10), αp → (2π)−1/4 as p → ∞, and therefore we deduce that
Φd,p;m(A,Σ) = O(1) if (trA+)
2pd−(1−r) = O(1), i.e., if (trA+)
2p = O
(
d1−r).
Remark 2.3.
(i) The behavior of the closed-form bound (2.6), as a function ofm, is more intricate. We shall
prove in Section 5.3 that the bound (2.6) is strictly decreasing in m if d is sufficiently large,
precisely, if d ≥
[
γ0γ1p
1/2(trA+)
]2/(1−r)
.
(ii) For d <
[
γ0γ1p
1/2(trA+)
]2/(1−r)
, the bound (2.6), as a function of m, is unimodal. There-
fore, for smaller values of d, the bound (2.6) may increase for small values of m, but it
eventually attains a maximum value and then decreases strictly to zero thereafter. This
shows again that if d is small, then higher values of m may need to be chosen in order for
the approximation (2.3) to result in small errors, and the bounds (2.5) and (2.6) enable
us to ascertain suitable values of m in order for the approximation (2.3) to achieve any
required level of accuracy.
Complete Asymptotic Expansions for the Normalizing Constants 5
To close this section, we obtain a lower bound for the remainder term Φd,p;m(A,Σ) for the
case in which A and Σ are arbitrary positive definite matrices.
Theorem 2.4. Let A ∈ Rp×p
sym and Σ ∈ Rd×d
sym be positive definite, let σ(1) ≥ · · · ≥ σ(d) denote the
ordered eigenvalues of Σ, and set Nd = 1
2(d+ 2)(d− 1). Then for all m ≥ 1,
Φd,p;m(A,Σ) ≥ (2 +m)−mNd(1 +m)−(1+m)Nd
∞∑
k=m
(
1 +m
2 +m
)kNd (σ(p) trA)
k
k!
. (2.7)
Remark 2.5. Let W be a discrete random variable having a Poisson distribution with param-
eter µ =
(
1+m
2+m
)Ndσ(p) trA. Then (2.7) can be written as
Φd,p;m(A,Σ) ≥ (2 +m)mNd(1 +m)−(1+m)NdeµP(W ≥ m), (2.8)
and this result reveals several properties of the lower bound. First, by using the formula
P(W ≥ m) = 1− P(W < m), it is seen that the right-hand side of (2.8) can be expressed in
closed-form. Second, as m → ∞, it is simple to show that the right-hand side of (2.8) is
asymptotically equal to eNd+µm−NdP(W ≥ m), where W is Poisson-distributed with parame-
ter µ ≃ σ(p) trA, and this asymptotic lower bound converges to 0 as m→ ∞.
Third, suppose that µ is large. By the central limit theorem, (W − µ)/
√
µ is approximately
distributed as Z, a random variable having the standard normal distribution (with mean 0 and
standard deviation 1). Applying the usual correction for continuity that is used whenever, the
distribution of a discrete random variable is approximated by the distribution of a continuous
random variable, we obtain
P(W ≥ m) = P
(
W ≥ m− 1
2
)
= P
(
W − µ
√
µ
≥
m− 1
2 − µ
√
µ
)
≃ P
(
Z ≥
m− 1
2 − µ
√
µ
)
,
which can be evaluated numerically using the standard normal distribution function.
Fourth, it is evident that the lower bound in (2.7) decreases strictly to 0 as d→ ∞.
3 Complete asymptotics for the normalizing constant
of the high-dimensional matrix Langevin distribution
Let X ∈ Vd,p be a random matrix having the matrix Langevin distribution with the density
function ψ(x;A,Σ) given in (1.2). The normalizing constant Ψd,p(B) can be expressed as a gen-
eralized hypergeometric function of matrix argument,
Ψd,p(B) = 0F1
(
1
2
;
1
4
B′B
)
=
∞∑
k=0
1
k!
∑
|κ|=k
Cκ
(
1
4B
′B
)
(d/2)κ
,
cf. Chikuse [4, p. 31]. By applying [11, Theorem 6.3], we find that this series converges absolutely
for all B ∈ Rd×p and uniformly on compact regions.
Similar to the approach in Section 2, we use as an asymptotic approximation to Ψd,p(B) the
truncated series,
Ψd,p(B) ≈
m−1∑
k=0
1
k!
∑
|κ|=k
Cκ
(
1
4B
′B
)
(d/2)κ
,
m ≥ 2, and we derive for the remainder series,
Ψd,p;m(B) =
∞∑
k=m
1
k!
∑
|κ|=k
Cκ
(
1
4B
′B
)
(d/2)κ
, (3.1)
6 A. Bagyan and D. Richards
an upper bound that is a complete asymptotic expansion in negative powers of d. The proof
of the following result is given in Section 5.4, and the constants αp = (2π)−(p−1)/4pp1/4p and
γ1 =
(√
3 + 1
)
/2 again appear in the proof.
Theorem 3.1. Let B ∈ Rd×p be such that ∥B∥ ≤ 2γ
1/2
0 dr/4, where γ0 > 0 and 0 ≤ r < 3. Then
for all m ≥ 2, Ψd,p;m(B) ≥ 0 and
Ψd,p;m(B) ≤ αp
∞∑
k=m
[
2γ0γ1p
5/2d−(3−r)/2]k
(k!)1/2
(3.2)
≤ αp(4e/π)
1/4(e/m)(m/2)−(1/4)[2γ0γ1p5/2d−(3−r)/2]m
× exp
(
cm
[
2γ0γ1p
5/2d−(3−r)/2]2/2). (3.3)
In particular, Ψd,p;m(B) = O
(
d−(3−r)m/2) as d→ ∞.
We note that each comment in Remark 2.2 has a direct analog for the matrix Langevin
distribution. In particular, the bound in (3.2) is strictly decreasing in d andm; strictly increasing
in γ0 and r; and strictly increasing in p for fixed values of all other parameters. Also, the bound
in (3.3) is strictly decreasing in d, and it is strictly decreasing in m if d is sufficiently large.
As regards asymptotics for d, p → ∞, by proceeding as in Remark 2.2, we deduce that
Ψd,p;m(B) = O(1) if p5/2d−(3−r)/2 = O(1), i.e., if p = O
(
d(3−r)/5).
In closing this section, we provide a lower bound for Ψd,p;m(B).
Theorem 3.2. Let B ∈ Rd×p such that B is of rank p. Let β(1) ≥ · · · ≥ β(p) denote the ordered
eigenvalues of 1
4B
′B, Np =
1
2(p+ 2)(p− 1), and µ =
(
1+m
2+m
)Np2pβ(p)e
−1. Then for all m ≥ 1,
Ψd,p;m(B) ≥
(1 +m)−Np(d+m)−m(2pβ(p))
m
m! 1F2(1;m+ 1, d+ 1 +m;µ). (3.4)
In proving this result, we will also deduce the weaker bound,
Ψd,p;m(B) ≥
(1 +m)−Np(d+m)−m(2pβ(p))
m
m!
,
a bound that is still noteworthy as it is a single-term lower bound, it is in closed-form, and it
converges to 0 as d→ ∞ or m→ ∞.
4 Implications for applications
The bounds derived in Sections 2 and 3 for the remainder terms in the corresponding asymptotic
expansions appear to be the first such explicit bounds available in the literature. Therefore,
we remark on the possible ramifications of those bounds for a variety of applications derived
from a survey of the literature on applications of the matrix Bingham and matrix Langevin
distributions.
The literature on such applications contains a broad range of reported values of d and p. The
case in which (d, p) = (3, 2) dates to analyses of data, drawn from astronomy and vectorcardiog-
raphy, as provided in the classic article of Jupp and Mardia [21]. The same value, (d, p) = (3, 2)
still appears in the recent literature, e.g., in data representing the orientation of two-dimensional
elliptical paths of objects in near-Earth orbit (Jauch, Hoff and Dunson [19]) and various graph
data sets (Mantoux et al. [28]).
For larger values of d and p, a variety of applications have appeared in the literature. Hoff [12]
applied the matrix Bingham distributions with (d, p) = (270, 2) to analyze data on protein-
protein interaction networks.
Complete Asymptotic Expansions for the Normalizing Constants 7
Mantoux et al. [28] provided illustrative examples with low- and high-dimensional synthetic
data for (d, p) = (3, 2), (20, 5), (20, 10), and (40, 20); and with d = 21 and p ∈ {2, 5, 10}, they
also analyzed resting-state functional magnetic resonance imaging (rs-fMRI) data from the UK
Biobank repository of large-scale brain-imaging data set on tens of thousands of subjects (Sudlow
et al. [36]).
Holbrook, Vandenberg-Rodes and Shahbaba [13] considered Stiefel manifold data with values
of d ranging from 18 to 53. Their work included a study of data, collected during a memory
experiment on rodents, in which neural spike activity was recorded in the hippocampi of six
rats. Especially interesting is the analysis in [13] of the 53-dimensional data on Super Chris,
a particular rat whose neural spike data exhibited statistical features substantially distinct from
the neural spike data of his fellow rats; cf. Granados-Garcia et al. [10].
Jauch et al. [19] analyzed with (d, p) = (365, 3) a meteorological data that included average
daily temperatures for a collection of Canadian weather stations; for the case in which p ≥ 2,
this value of d was the largest that we encountered in the literature.
In summary, for p ≥ 2, we found applications in the literature with values of (d, p) where d
ranges between 3 and 365, and p ranges between 2 and 20. Further, for p = 1, values of d as
large as 62, 501 were noted in [1, Section 4].
The computation of the bounds in Theorems 2.1 and 3.1 is straightforward as it reduces to
calculating the function Rm(t), or the upper bound on Rm(t), in (5.36) and (5.47), respectively.
As we noted earlier, these bounds are (strictly) decreasing in d (and in m if t ≤ 1), increasing
in γ0 and r, and increasing in p. We reiterate that, in such calculations, the relationship between t
and d is that
t =
{
γ0γ1p
1/2(trA+)d
−(1−r)/2 (in Theorem 2.1),
2γ0γ1p
5/2d−(3−r)/2 (in Theorem 3.1),
equivalently,
d =
{(
γ0γ1p
1/2(trA+)/t
)2/(1−r)
(in Theorem 2.1),(
2γ0γ1p
5/2/t
)2/(3−r)
(in Theorem 3.1).
For small values of d, e.g., d = 3, the resulting values of t usually are large and the bounds also
can be large, indicating that larger choices for m may be needed. For large values of d, e.g.,
d = 365, the resulting values of t typically are small, and the resulting bounds in terms of Rm(t)
quickly become negligible even for small values of m.
In summary, we expect that the results obtained here will find broad usage as they enable
for the first time theoretical analyses and numerical calculations of bounds on the accuracy of
the resulting expansions. In particular, these results enable calculation of the given expansions
to any desired practical degree of accuracy.
5 Proofs
5.1 Properties of the zonal polynomials
James [18] obtained for the zonal polynomials Cκ(Σ) a crucial integral formula, as we now
explain. Let Or(d) =
{
H ∈ Rd×d | H ′H = Id
}
be the group of orthogonal matrices in Rd×d.
For H ∈ Or(d), denote by dH the Haar measure on Or(d), normalized to have total volume 1.
Also let detj(Σ) be the jth principal minor of Σ, j = 1, . . . , d. For each partition κ = (κ1, . . . , κd),
Cκ(Σ) = Cκ(Id)
∫
Or(d)
ℓ(κ)∏
j=1
[
detj
(
H ′ΣH
)]κj−κj+1 dH, (5.1)
8 A. Bagyan and D. Richards
where κd+1 ≡ 0 and
Cκ(Id) = 22|κ||κ|!(d/2)κ
∏
1≤i<j≤ℓ(κ)
(2κi − 2κj − i+ j)
ℓ(κ)∏
i=1
(2κi + ℓ(κ)− i)!
. (5.2)
The formula (5.1) was first given by James [18, equation (10.4)], further applications of it
were given in [26] and [34], and the formula was extended to a more general setting in [11,
Theorem 4.8].
By (5.1), Cκ(Σ) is homogeneous of degree |κ|; moreover Cκ(Σ) depends only on the ℓ(κ) non-
zero parts of κ, so that Cκ(Σ) is unchanged if zeros are appended to κ. Since any symmetric
matrix can be diagonalized by an orthogonal transformation, then it also follows from (5.1)
that Cκ(Σ) is a symmetric function of the eigenvalues of Σ.
The importance of the normalization (5.2) is that the zonal polynomials satisfy the identity,
(tr Σ)k =
∑
|κ|=k
Cκ(Σ), (5.3)
for all k = 0, 1, 2, . . .; see [16, p. 479, equation (21)], [30, p. 228, equation (iii)], [35, equa-
tion (35.4.6)]. Also noteworthy is the special case of (5.2) in which κ = (k), a partition with
one part
C(k)(Id) =
(d/2)k
(1/2)k
. (5.4)
The (seemingly formidable) expression for Cκ(Id) in (5.2) arises from the representation
theory of Sk, the group of permutations on k symbols, and we comment on the connection
as follows. It is a classical result that the collection of irreducible representations of Sk is in
one-to-one correspondence with the set of partitions of k (see Ledermann [27, Chapter 4]). For
a given partition κ = (κ1, . . . , κd) we use the standard notation χ[2κ](1) for the dimension of the
irreducible representation, corresponding to the partition 2κ = (2κ1, . . . , 2κd), of the symmetric
group S2|κ| on 2|κ| symbols. James [16, pp. 478–479, equations (18) and (20)] proved that
Cκ(Id) =
22|κ||κ|!(d/2)κ
(2|κ|)!
χ[2κ](1) =
(d/2)κ
(1/2)|κ|
χ[2κ](1), (5.5)
and then (5.2) is derived from (5.5) by applying Frobenius’ famous formula for χ[2κ](1) (Leder-
mann [27, p. 123]).
Another consequence of (5.5) is that
Cκ(Id)
(d/2)κ
=
χ[2κ](1)
(1/2)|κ|
. (5.6)
Since the dimension of an irreducible representation does not change if zeros are appended to
the corresponding partition, then it follows that, for any partition κ such that ℓ(κ) ≤ p, the
right-hand side of (5.6) remains unchanged if d is replaced by p; hence
Cκ(Id)
(d/2)κ
=
Cκ(Ip)
(p/2)κ
. (5.7)
We remark that the identity (5.7) can be verified directly, albeit laboriously, from (5.2). Further,
(5.7) has been noted explicitly in the literature, e.g., by Chikuse [4, p. 30] and by Edelman et
al. [8, p. 259]. Also, if ℓ(κ) = 1, then (5.7) reduces to (5.4).
Complete Asymptotic Expansions for the Normalizing Constants 9
The zonal polynomials also have a combinatorial formulation. Recall that the set of all
partitions is endowed with the lexicographic ordering : For partitions κ = (κ1, . . . , κd) and
λ = (λ1, . . . , λd) such that |κ| = |λ|, we say that λ is less than κ, written λ < κ, if there
exists i ∈ {1, . . . , d− 1} such that (λ1, . . . , λi−1) = (κ1, . . . , κi−1) and λi < κi.
For Σ ∈ Rd×d
sym, denote by σ1, . . . , σd the eigenvalues of Σ. For each partition κ = (κ1, . . . , κℓ)
of length ℓ, the monomial symmetric function of Σ corresponding to κ is
Mκ(Σ) =
∑
(j1,...,jℓ)
σ
κ1
j1
· · ·σκℓ
jℓ
, (5.8)
where the sum is taken over all distinct permutations (j1, . . . , jℓ) of ℓ integers drawn from the
set {1, . . . , d}. James [17] (see also Muirhead [30, Section 7.2]) proved that
Cκ(Σ) =
∑
λ≤κ
cκ,λMλ(Σ), (5.9)
where the constants cκ,λ are nonnegative and the summation is over all partitions λ such
that λ ≤ κ, i.e., λ is less than or equal to κ in the lexicographic ordering.
5.2 Inequalities for the partitional shifted factorials
and the zonal polynomials
Lemma 5.1. Let a1, . . . , ap be nonnegative integers such that a1 + · · ·+ ap ≥ 2, and define
αp = (2π)−(p−1)/4pp1/4p. (5.10)
Then
p∏
i=1
(pai)! ≤ α2p
p p
p(a1+···+ap)
[
(a1 + · · ·+ ap)!
]p
. (5.11)
Proof. Since the multinomial coefficient, (a1 + · · ·+ ap)/a1! · · · ap!, is a positive integer, then
p∏
i=1
ai! ≤ (a1 + · · ·+ ap)!. (5.12)
It is well known that the classical gamma function is increasing on the interval [2,∞); there-
fore, Γ
(
a+ (i− 1)p−1) ≤ Γ
(
a+ (p− 1)p−1) = Γ
(
a+ 1− p−1) for a ≥ 2 and all i = 1, . . . , p.
Applying Gauss’ multiplication formula for the gamma function, we obtain
Γ(pa) = (2π)−(p−1)/2ppa−(1/2)
p∏
i=1
Γ
(
a+ (i− 1)p−1)
≤ (2π)−(p−1)/2ppa−(1/2)[Γ(a+ 1− p−1)]p, (5.13)
where the latter inequality holds for a ≥ 2.
Setting a = a1 + · · ·+ ap + p−1 in (5.13), we obtain
(p(a1 + · · ·+ ap))! = Γ(p(a1 + · · ·+ ap) + 1)
≤ (2π)−(p−1)/2pp(a1+···+ap)+(1/2)[Γ(a1 + · · ·+ ap + 1)]p
= (2π)−(p−1)/2pp(a1+···+ap)+(1/2)[(a1 + · · ·+ ap)!]
p. (5.14)
10 A. Bagyan and D. Richards
Applying (5.14) to (5.12), we find that
p∏
i=1
(pai)! ≤
(
p(a1 + · · ·+ ap)
)
! ≤ (2π)−(p−1)/2pp(a1+···+ap)+(1/2)((a1 + · · ·+ ap)!
)p
≡ α2p
p p
p(a1+···+ap)[(a1 + · · ·+ ap)!]
p.
This completes the proof. ■
Lemma 5.2. Let d ≥ p and let κ = (κ1, . . . , κp) be a partition of length ℓ(κ) ≤ p and
weight |κ| ≥ 2. Then
(2p)−|κ|d|κ| ≤ (d/2)κ ≤ 2−|κ|(d+ |κ|)|κ|. (5.15)
Also, with γ1 =
(√
3 + 1
)
/2 ≃ 1.366025, we have
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
≤
[
αp(|κ|!)
1/2γ
|κ|
1 p|κ|/2d−|κ|/2]p. (5.16)
Proof. By (2.1), we have
(d/2)κ =
p∏
j=1
(
1
2
d− 1
2
(j − 1)
)
κj
=
p∏
j=1
κj∏
i=1
(
1
2
d− 1
2
(j − 1) + i− 1
)
. (5.17)
For all i ≥ 1, j ≤ p, and d ≥ p, we also have
1
2
d− 1
2
(j − 1) + i− 1 ≥ 1
2
d− 1
2
(p− 1) =
1
2
(
1− p− 1
d
)
d ≥ 1
2
(
1− p− 1
p
)
d = (2p)−1d.
Therefore, by (5.17),
(d/2)κ ≥
p∏
j=1
κj∏
i=1
[
(2p)−1d
]
= (2p)|κ|d−|κ|.
This establishes the lower bound in (5.15).
Next, we apply to the product in (5.17) the arithmetic-geometric mean inequality. Then we
obtain
(d/2)κ ≤
(
1
|κ|
p∑
j=1
κj∑
i=1
(
1
2
d− 1
2
(j − 1) + i− 1
))|κ|
=
(
1
2|κ|
(
d|κ| −
p∑
j=1
jκj +
p∑
j=1
κ2j
))|κ|
≤
(
1
2|κ|
(
d|κ|+
(
p∑
j=1
κj
)2))|κ|
= 2−|κ|(d+ |κ|)|κ|.
This proves the upper bound in (5.15).
Further, it is known from [1, Lemma A.2] that, for all r ≥ 1, (d
1/2
/2)r
(d/2)r
≤ [(r − 1)!]1/2γr−1
1 d−r/2,
and since γ1 > 1, then we also have(
d1/2/2
)
r
(d/2)r
≤ [(r − 1)!]1/2γr−1
1 d−r/2 · γ1r
1/2 = (r!)1/2γr1d
−r/2.
Complete Asymptotic Expansions for the Normalizing Constants 11
Since the latter inequality is valid for r = 0, then we have shown that, for all r ≥ 0,(
d1/2/2
)
r
(d/2)r
≤ (r!)1/2γr1d
−r/2. (5.18)
Now we apply (5.18) to obtain
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
≤
p∏
i=1
[(pκi)!]
1/2γ
pκi
1 d−pκi/2 = γ
p|κ|
1 d−p|κ|/2
p∏
i=1
[(pκi)!]
1/2. (5.19)
By (5.11) in Lemma 5.1 with ai = κi, i = 1, . . . , p, we also have
p∏
i=1
(pκi)! ≤ α2p
p p
p|κ|(|κ|!)p,
and by substituting this bound into (5.19), we obtain (5.16). ■
For Σ ∈ Rd×d
sym, let σ1, . . . , σd denote the eigenvalues of Σ and define the diagonal matrix
Σ+ = diag(|σ1|, . . . , |σd|). The following result is a consequence of (5.9), the combinatorial
formula for the zonal polynomials.
Lemma 5.3. Let κ be a partition and Σ ∈ Rd×d
sym. Then |Cκ(Σ)| ≤ Cκ(Σ+).
Proof. By (5.9), the nonnegativity of the constants cκ,λ, and the triangle inequality, we have
|Cκ(Σ)| ≤
∣∣∣∣∑
λ≤κ
cκ,λMλ(Σ)
∣∣∣∣ ≤∑
λ≤κ
cκ,λ|Mλ(Σ)|.
By (5.8), for all partitions λ,
|Mλ(Σ)| =
∣∣∣∑σ
λ1
j1
· · ·σλℓ
jℓ
∣∣∣ ≤∑ |σj1 |
λ1 · · · |σjℓ |
λℓ =Mλ(Σ+).
Therefore,
|Cκ(Σ)| ≤
∑
λ≤κ
cκ,λMλ(Σ+) = Cκ(Σ+).
The proof now is complete. ■
The following result will be crucial for deriving bounds on the tails of the asymptotic series
for the normalizing constants of the matrix Bingham and matrix Langevin distributions.
Proposition 5.4. Let κ be a partition such that ℓ(κ) ≤ p, where p ≤ d, and let Σ ∈ Rd×d
sym. Then
|Cκ(Σ)| ≤
(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
Cκ(Id)∥Σ∥
|κ|. (5.20)
Proof. By Lemma 5.3, |Cκ(Σ)| ≤ Cκ(Σ+). Since ∥Σ∥ = ∥Σ+∥, then it suffices to assume,
without loss of generality, that Σ = Σ+, i.e., that Σ is positive semidefinite and diagonal, with
diagonal entries σ1, . . . , σd.
Since Σ is positive semidefinite, then so is every principal submatrix of H ′ΣH, for all
H ∈ Or(d). Therefore, detj
(
H ′ΣH
)
≥ 0 for all j = 1, . . . , d.
12 A. Bagyan and D. Richards
For ℓ(κ) ≤ p, it follows from (5.1) that
Cκ(Σ) = Cκ(Id)
∫
Or(d)
p∏
j=1
[detj
(
H ′ΣH
)
]κj−κj+1 dH. (5.21)
Denote by
(
H ′ΣH
)
ii
the ith diagonal entry of H ′ΣH, i = 1, . . . , d. By Hadamard’s inequality
for the principal minors of a positive semidefinite matrix [14, p. 505],
detj
(
H ′ΣH
)
≤
j∏
i=1
(
H ′ΣH
)
ii
for all j = 1, . . . , d; therefore,
p∏
j=1
[
detj
(
H ′ΣH
)]κj−κj+1 ≤
p∏
j=1
j∏
i=1
[(
H ′ΣH
)
ii
]κj−κj+1
=
p∏
i=1
p∏
j=i
[(
H ′ΣH
)
ii
]κj−κj+1 =
p∏
i=1
[(
H ′ΣH
)
ii
]κi .
Inserting this bound into (5.21) and then applying Hölder’s inequality, we obtain
Cκ(Σ) ≤ Cκ(Id)
∫
Or(d)
p∏
i=1
[(
H ′ΣH
)
ii
]κi dH
≤ Cκ(Id)
(
p∏
i=1
∫
Or(d)
[(
H ′ΣH
)
ii
]pκi dH
)1/p
. (5.22)
Denote by hi,j the (i, j)th entry of H, then hi = (h1,i, . . . , hd,i)
′ is the ith column of H,
i = 1, . . . , d. Since Σ = diag(σ1, . . . , σd), then for all i = 1, . . . , d,
(
H ′ΣH
)
ii
=
d∑
l=1
h2l,iσl = h′iΣhi.
Now regard H as a random matrix having the uniform distribution (Haar measure) on Or(d).
Since the Haar measure is orthogonally invariant, and therefore invariant under permutation of
the columns of H, it follows that the marginal distribution of hi does not depend on i. Denoting
equality in distribution by
L
=, we obtain(
H ′ΣH
)
ii
= h′iΣhi
L
= h′1Σh1 =
(
H ′ΣH
)
11
,
for all i = 1, . . . , d. Consequently,∫
Or(d)
[(
H ′ΣH
)
ii
]pκi dH =
∫
Or(d)
[(
H ′ΣH
)
11
]pκi dH =
C(pκi)
(Σ)
C(pκi)
(Id)
, (5.23)
the second equality following from (5.1). Applying (5.23) to (5.22), we obtain
Cκ(Σ) ≤ Cκ(Id)
(
p∏
i=1
C(pκi)
(Σ)
C(pκi)
(Id)
)1/p
. (5.24)
Complete Asymptotic Expansions for the Normalizing Constants 13
By [1, equation (A.10)],
C(pκi)
(Σ) ≤
(d1/2/2)pκi
(1/2)pκi
∥Σ∥pκi ;
also, by (5.4),
C(pκi)
(Id) =
(d/2)pκi
(1/2)pκi
.
Substituting the two latter results into (5.24), we obtain
Cκ(Σ) ≤ Cκ(Id)
(
p∏
i=1
(
d1/2/2
)
pκi
(1/2)pκi
∥Σ∥pκi ·
(1/2)pκi
(d/2)pκi
)1/p
= Cκ(Id)∥Σ∥
|κ|
(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
,
and this establishes (5.20). ■
Remark 5.5. Suppose we evaluate both sides of the inequality (5.20) at Σ = cId, c > 0.
Since Cκ(cId) = c|κ|Cκ(Id) and ∥cId∥
|κ| = c|κ|d|κ|/2, then the inequality reduces to
1 ≤
(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
d|κ|/2. (5.25)
For d→ ∞, we have
(d1/2/2)pκi
(d/2)pκi
∼ (d1/2/2)pκi
(d/2)pκi
= d−pκi/2,
therefore(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
d|κ|/2 ∼
(
p∏
i=1
d−pκi/2
)1/p
d|κ|/2 = 1.
Hence for scalar matrices Σ, both sides of the inequality (5.20) are asymptotically equal as
d→ ∞. This indicates that, in high dimensions, the inequality is close to asymptotically tight
if Σ is within a small neighborhood of any scalar matrix.
Remark 5.6. In the setting of general symmetric cones, another inequality for the zonal (spher-
ical) polynomials was given by Faraut and Korányi [9, Theorem XII.1.1] (see also the proof
of Lemma 6.5 in [11], where the same inequality can be derived for the matrix cones). In the
notation used here, the inequality in [9, loc. cit.] states that if κ = (κ1, . . . , κp) is a partition of
length p, Σ is a d× d positive definite matrix, and σ(1) ≥ · · · ≥ σ(d) are the ordered eigenvalues
of Σ, then
Cκ(Σ) ≤ Cκ(Id)
p∏
j=1
σ
κj
(j). (5.26)
There is also the issue of which of the inequalities (5.20) and (5.26) is sharper. We now
provide two examples to show that the answer depends on d and on Σ.
14 A. Bagyan and D. Richards
For the case in which Σ = cId where c does not depend on d, we find that (5.26) is sharper
than (5.20) since, by (5.25),
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
≥ d−p|κ|/2.
More generally, it can be seen that if Σ is “close” to a scalar matrix and d is small, then we can
expect (5.26) to be sharper than (5.20), however both bounds are asymptotic to d−p|κ|/2Cκ(Id)
for large d.
On the other hand, suppose that Σ is “far” from the scalar matrices cId in the sense that c
does not depend on d and ∥Id∥/∥Σ∥ → ∞ as d → ∞. As an example, for constants σ, ρ > 0
and r ∈ [0, 1), let the eigenvalues of Σ be
σj =
{
σdr/2, j = 1, . . . , p,
ρd−(d−p+1)/2, j = p+ 1, . . . , d.
Then Σ is non-singular and
∥Σ∥ =
(
d∑
j=1
σ2j
)1/2
=
[
pσ2dr + ρ2(d− p)d−(d−p+1)]1/2 ∼ p1/2σdr/2,
as d → ∞, so Σ satisfies the hypotheses of Theorem 2.1. Further, Σ is far from the scalar
matrices cId since ∥Id∥/∥Σ∥ ∼ p−1/2σ−1d(1−r)/2 → ∞ as d→ ∞.
For this choice of Σ and for a partition κ of length p, the ratio of the right-hand sides of (5.20)
and (5.26) equals(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
∥Σ∥|κ|∏p
j=1 σ
κj
(j)
∼
(
d1/2/2
)|κ|
(d/2)|κ|
(
p1/2σdr/2
)|κ|(
σdr/2
)|κ| = d−|κ|/2p|κ|/2 → 0
as d → ∞. In such instances, the bound in (5.20) is sharper than the bound in (5.26) for
sufficiently large d, and it becomes increasingly sharper afterwards as d increases.
Next, we obtain a lower bound for the zonal polynomials. This bound tightens an inequality
stated in [9, Theorem XII.1.1 (ii)]. (Readers who wish to match the notation used in [9] with the
notation in the present article will find it helpful to review [6, Section 1.12], where the notation
used for general irreducible symmetric cones is made explicit in the case of each of the five classes
of cones.)
Proposition 5.7. Let κ = (κ1, . . . , κp) be a partition of length p, Σ be a d× d positive definite
matrix with ordered eigenvalues σ(1) ≥ · · · ≥ σ(d), and Nd = 1
2(d+ 2)(d− 1). Then
Cκ(Σ) ≥ (1 + |κ|)−NdCκ(Id)σ
κ1
(1) · · ·σ
κp
(p). (5.27)
Proof. By [9, p. 241, line 4],
Cκ(Σ)
Cκ(Id)
≥ T1
T2
σ
κ1
(1) · · ·σ
κp
(p), (5.28)
where T1 =
1
dκ
, with dκ being the dimension of a certain vector space of polynomials that are
homogeneous of degree k = |κ|, and
T2 =
∏
1≤i<j≤p
B
(
κi − κj +
1
2(j − i− 1) + 1, 12
)
B
(
1
2(j − i− 1) + 1, 12
) . (5.29)
Complete Asymptotic Expansions for the Normalizing Constants 15
In [9, p. 242, line 9], it is shown that
dκ ≤
(
Nd + k
Nd
)
=
(Nd + k)!
Nd!k!
=
Nd∏
j=1
j + k
j
≤ (1 + k)Nd
since (j + k)/j ≤ 1 + k, for all j = 1, . . . , Nd. Therefore,
T1 =
1
dκ
≥ (1 + |κ|)−Nd , (5.30)
As regards an upper bound for T2 we now prove, using an elementary probabilistic approach,
that
T2 ≤ 1. (5.31)
For b, c > 0, let V denote a random variable having a beta distribution with probability density
function
vb−1(1− v)c−1
B(b, c)
, 0 < v < 1.
Then for α ≥ 0,
B(α+ b, c)
B(b, c)
=
1
B(b, c)
∫ 1
0
vα+b−1(1− v)c−1dv = E
(
V α),
and it is trivial that E
(
V α) ≤ 1 since α ≥ 0 and 0 < V < 1, almost surely. Setting α = κi − κj ,
b = 1
2(j − i− 1) + 1, and c = 1
2 , we obtain
B
(
κi − κj +
1
2(j − i− 1) + 1, 12
)
B
(
1
2(j − i− 1) + 1, 12
) ≤ 1 (5.32)
for all 1 ≤ i < j ≤ p. Applying (5.32) to each term in (5.29), we obtain (5.31), and by
applying (5.30) and (5.31) to (5.28), then we obtain (5.27). ■
5.3 The proofs for Section 2
Proof of Theorem 2.1. We begin by applying to (2.4) the triangle inequality, the upper bound
from Proposition 5.4, and the inequality |Cκ(A)| ≤ Cκ(A+). Then we obtain
|Φd,p;m(A,Σ)| ≤
∞∑
k=m
1
k!
∑
|κ|=k
|Cκ(A)|
Cκ(Id)
(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
Cκ(Id)∥Σ∥
|κ|
≤
∞∑
k=m
∥Σ∥k
k!
∑
|κ|=k
Cκ(A+)
(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
. (5.33)
By (5.16), p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
1/p
≤ αp(|κ|!)
1/2γ
|κ|
1 p|κ|/2d−|κ|/2,
16 A. Bagyan and D. Richards
and by inserting this bound into (5.33), we obtain
Φd,p;m(A,Σ)| ≤
∞∑
k=m
∥Σ∥k
k!
∑
|κ|=k
Cκ(A+)αp(|κ|!)
1/2γ
|κ|
1 p|κ|/2d−|κ|/2
= αp
∞∑
k=m
γk1p
k/2d−k/2∥Σ∥k
(k!)1/2
∑
|κ|=k
Cκ(A+). (5.34)
By (5.3),
∑
|κ|=k Cκ(A+) = (trA+)
k, and, by assumption, ∥Σ∥ ≤ γ0d
r/2. Therefore, we
obtain from (5.34) the inequality
Φd,p;m(A,Σ)| ≤ αpRm
(
γ0γ1p
1/2(trA+)d
−(1−r)/2), (5.35)
where
Rm(t) =
∞∑
k=m
tk
(k!)1/2
, t > 0. (5.36)
By applying the ratio test, we find that the series (5.36) converges for all t, and therefore (5.35)
converges for all d. This establishes the bound (2.5).
To obtain a closed-form upper bound for Rm(t) we begin by applying Stirling’s well-known
inequality for the factorial function [15, Theorem 1]; viz., for all k ≥ 1,
k! ≥ (2π)1/2k(2k+1)/2e−k.
Inverting Stirling’s inequality and applying the result to (5.36), we obtain
Rm(t) ≤ (2π)−1/4
∞∑
k=m
k−(2k+1)/4ek/2tk. (5.37)
We now apply the telescoping method to obtain an upper bound for the right-hand side of (5.37).
For j ≥ m, let
aj = j−(2j+1)/4ej/2tj , (5.38)
then we obtain through straightforward algebraic manipulations
aj+1
aj
=
(
1 + j−1)−j/2 j1/4
(j + 1)3/4
e1/2t. (5.39)
Since the function j 7→
(
1 + j−1)−j/2
, j ≥ m, is decreasing, then(
1 + j−1)−j/2 ≤
(
1 +m−1)−m/2
, (5.40)
for all j ≥ m, and by applying the latter inequality to (5.39), we obtain
aj+1
aj
≤
(
1 +m−1)−m/2
e1/2t
j1/4
(j + 1)3/4
= c1/2m t
j1/4
(j + 1)3/4
,
where cm =
(
1 +m−1)−m
e. It follows that, for all k ≥ m,
ak = am
k−1∏
j=m
aj+1
aj
≤ am
k−1∏
j=m
c1/2m t
j1/4
(j + 1)3/4
= amc
(k−m)/2
m tk−m
k−1∏
j=m
j1/4
(j + 1)3/4
. (5.41)
Complete Asymptotic Expansions for the Normalizing Constants 17
Since
∏k−1
j=m j = (k − 1)!/(m− 1)!, then
k−1∏
j=m
j1/4
(j + 1)3/4
=
[(k − 1)!/(m− 1)!]1/4
[k!/m!]3/4
=
m3/4[(m− 1)!]1/2
k3/4[(k − 1)!]1/2
. (5.42)
On applying to (5.41) the identity (5.42) and the explicit expression for am, as given in (5.38),
we obtain
ak ≤ m−(2m+1)/4em/2tm · c(k−m)/2
m tk−mm
3/4[(m− 1)!]1/2
k3/4[(k − 1)!]1/2
= (e/mcm)m/2(m!)1/2
ck/2m tk
k3/4[(k − 1)!]1/2
. (5.43)
Therefore, by (5.37), (5.38), and (5.43),
Rm(t) ≤ (2π)−1/4(e/mcm)m/2(m!)1/2
∞∑
k=m
ck/2m tk
k3/4[(k − 1)!]1/2
≤ (2π)−1/4(e/mcm)m/2(m!)1/2
( ∞∑
k=m
1
k3/2
)1/2( ∞∑
k=m
ckmt
2k
(k − 1)!
)1/2
, (5.44)
where the latter inequality follows by the Cauchy–Schwarz inequality.
To obtain an upper bound for the first sum in (5.44), we observe from graphical considerations
using Riemann sums that, for m ≥ 2,
∞∑
k=m
1
k3/2
≤
∫ ∞
m
dt
(t− 1)3/2
=
2
(m− 1)1/2
≤
(
8
m
)1/2
. (5.45)
As for the second sum in (5.44), we apply the inequality 1
(k+m−1)! ≤
1
k!(m−1)! to obtain
∞∑
k=m
t2k
(k − 1)!
=
∞∑
k=0
t2(k+m)
(k +m− 1)!
≤ t2m
(m− 1)!
∞∑
k=0
t2k
k!
=
t2m
(m− 1)!
exp
(
t2
)
. (5.46)
Applying (5.45) and (5.46) to (5.44), we obtain
Rm(t) ≤ (2π)−1/4(e/mcm)m/2(m!)1/2
(
8
m
)1/4
(
cmmt
2m
(m− 1)!
exp
(
cmt
2))1/2
= (4e/π)1/4(e/m)(m/2)−(1/4)tm exp
(
cmt
2/2
)
, (5.47)
and by inserting this bound at (5.35), we obtain (2.6).
Finally, it follows from (2.5) that Φd,p;m(A,Σ) = O
(
d−(1−r)m/2) as d→ ∞. ■
Comments on Remark 2.3. Denote by bm the bound in (5.47). By direct calculation, we
obtain
bm+1
bm
= t
( m
m+ 1
)(m/2)−(1/4)( e
m+ 1
)1/2
exp
(
(cm+1 − cm)t2/2
)
. (5.48)
By (5.40), cm+1 − cm < 0; hence the latter three terms in (5.48) each are strictly less than 1, so
we obtain bm+1/bm < t. Therefore, if t ≤ 1, then bm+1 < bm, i.e., the sequence {bm,m ≥ 2}, is
strictly decreasing; and by applying this result to (2.6), we obtain the statement in the first part
18 A. Bagyan and D. Richards
of Remark 2.3. We have also determined from simple numerical calculations that bm remains
strictly decreasing for t ≤ 1.5.
For larger values of t, the sequence bm is unimodal. This can be proved by treating m
(temporarily) as a continuous variable, calculating the logarithmic derivative of (5.48) with
respect to m, and verifying that there exists an m0 such that the derivative is positive for
all m ≤ m0 and negative for m > m0. By graphing the bound (5.47) as a function of m, we
have observed that it appears to be unimodal for all t ≥ 1.6.
Proof of Theorem 2.4. Since A, Σ are positive definite, then Cκ(A) > 0 and Cκ(Σ) > 0 for
all partitions κ. By Proposition 5.7,
Φd,p;m(A,Σ) =
∞∑
k=m
1
k!
∑
|κ|=k
Cκ(A)Cκ(Σ)
Cκ(Id)
≥
∞∑
k=m
1
k!
∑
|κ|=k
Cκ(A)(1 + |κ|)−Ndσ
κ1
(1) · · ·σ
κp
(p)
=
∞∑
k=m
(1 + k)−Nd
k!
∑
|κ|=k
Cκ(A)σ
κ1
(1) · · ·σ
κp
(p).
Since σ(1) ≥ · · · ≥ σ(d) > 0, then σ
κ1
(1) · · ·σ
κp
(p) ≥ σ
|κ|
(p) for any partition κ of length p. Therefore,∑
|κ|=k
Cκ(A)σ
κ1
(1) · · ·σ
κp
(p) ≥ σk(p)
∑
|κ|=k
Cκ(A) = σk(p)(trA)
k,
so we obtain
Φd,p;m(A,Σ) ≥
∞∑
k=m
(1 + k)−Nd(σ(p) trA)
k
k!
.
We now apply the telescoping method to bound the latter series. Let τ ≡ σ(p) tr(A) and,
for j ≥ m, define
aj =
(1 + j)−Ndτ j
j!
.
Then
aj+1
aj
=
(1 + j
2 + j
)Nd
τ
1 + j
.
It is straightforward that, for j ≥ m, 1+j
2+j ≥ 1+m
2+m , and therefore
aj+1
aj
≥
(1 +m
2 +m
)Nd
τ
1 + j
.
Hence for all k ≥ m,
ak = am
k−1∏
j=m
aj+1
aj
≥ am
k−1∏
j=m
[(
1 +m
2 +m
)Nd τ
1 + j
]
= am
(1 +m
2 +m
)(k−m)Ndm!τk−m
k!
.
Substituting for am and then simplifying, we obtain
ak ≥ (2 +m)mNd(1 +m)−(1+m)Nd
(
1 +m
2 +m
)kNd τk
k!
,
and therefore
Φd,p;m(A,Σ) ≥
∞∑
k=m
ak ≥ (2 +m)mNd(1 +m)−(1+m)Nd
∞∑
k=m
(
1 +m
2 +m
)kNd τk
k!
.
The proof now is complete. ■
Complete Asymptotic Expansions for the Normalizing Constants 19
5.4 The proofs for Section 3
Proof of Theorem 3.1. Since B′B is positive semidefinite, then all its eigenvalues are non-
negative; hence, by (5.9), Cκ
(
1
4B
′B
)
≥ 0. Also, it follows from (2.1) that (d/2)κ > 0. Therefore,
it follows from (3.1) that Ψd,p;m(B) ≥ 0.
By [14, Section 5.6], ∥B′B∥ ≤ ∥B∥2. Denoting 1
4B
′B by Λ, it follows from the assumption
on B that ∥Λ∥ = 1
4∥B
′B∥ ≤ 1
4∥B∥2 ≤ γ0d
r/2. Since Λ is positive semidefinite, then by Proposi-
tion 5.4,
Cκ(Λ) ≤
(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
Cκ(Ip)∥Λ∥
|κ|.
Applying this bound to (3.1), we find that
Ψd,p;m(B) ≤
∞∑
k=m
∥Λ∥k
k!
∑
|κ|=k
Cκ(Ip)
(d/2)κ
(
p∏
i=1
(
d1/2/2
)
pκi
(d/2)pκi
)1/p
.
Next we apply the lower bound in (5.15) and the bound in (5.16) to obtain
Ψd,p;m(B) ≤
∞∑
k=m
∥Λ∥k
k!
∑
|κ|=k
Cκ(Ip) · (2p)
|κ|d−|κ| · αpγ
|κ|
1 d−|κ|/2p|κ|/2(|κ|!)1/2
= αp
∞∑
k=m
(
2γ1p
3/2d−3/2∥Λ∥
)k
(k!)1/2
∑
|κ|=k
Cκ(Ip).
By (5.3) and the bound ∥Λ∥ ≤ γ0d
r/2, we deduce that
Ψd,p;m(B) ≤ αp
∞∑
k=m
(
2γ1p
3/2d−3/2 · γ0d
r/2)k
(k!)1/2
pk = αp
∞∑
k=m
(
2γ0γ1p
5/2d−(3−r)/2)k
(k!)1/2
.
Noting that the latter series is identical with Rm
(
2γ0γ1p
5/2d−(3−r)/2), where Rm(t) is given
in (5.36), we apply the inequality (5.47) to obtain (3.3).
Finally, it follows from (3.2) that Ψd,p;m(B) = O
(
d−(3−r)m/2) as d→ ∞. ■
Proof of Theorem 3.2. Since B is of full rank, then B′B is positive definite. Letting β(1) ≥
· · · ≥ β(p) denote the ordered eigenvalues of 1
4B
′B, then by Proposition 5.7,
Ψd,p;m(B) =
∞∑
k=m
1
k!
∑
|κ|=k
Cκ
(
1
4B
′B
)
(d/2)κ
≥
∞∑
k=m
1
k!
∑
|κ|=k
(1 + |κ|)−NpCκ(Ip)
(d/2)κ
β
κ1
(1) · · ·β
κp
(p)
≥
∞∑
k=m
(1 + k)−Npβk(p)
k!
∑
|κ|=k
Cκ(Ip)
(d/2)κ
β
κ1
(1) · · ·β
κp
(p),
since the ordering of the β(j) implies that β
κ1
(1) · · ·β
κp
(p) ≥ β
|κ|
(p).
By the upper bound in (5.15), we have
1
(d/2)κ
≥ 2|κ|(d+ |κ|)−|κ|;
20 A. Bagyan and D. Richards
therefore
Ψd,p;m(B) ≥
∞∑
k=m
(1 + k)−Npβk(p)
k!
∑
|κ|=k
2|κ|(d+ |κ|)−|κ|Cκ(Ip)
=
∞∑
k=m
(1 + k)−Np2k(d+ k)−kβk(p)
k!
∑
|κ|=k
Cκ(Ip)
=
∞∑
k=m
(1 + k)−Np(d+ k)−k
k!
(2pβ(p))
k.
Since each term in the latter series is positive, then we obtain the single-term lower bound,
Ψd,p;m(B) ≥ (1 +m)−Np(d+m)−m
m!
(2pβ(p))
m,
and this bound also converges to 0 as d→ ∞ or m→ ∞.
We apply now the telescoping method to obtain a sharper lower bound for Ψd,p;m(B).
Let τ ≡ 2pβ(p) and, for j ≥ m, define aj = (1 + j)−Np(d+ j)−jτ j/j!. Then
aj+1
aj
=
(
1 + j
2 + j
)Np
(
d+ j
d+ 1 + j
)j+1 τ
(j + 1)(d+ 1 + j)
≥
(
1 +m
2 +m
)Np
e−1 τ
(j + 1)(d+ 1 + j)
since 1+j
2+j ≥ 1+m
2+m ,
( d+j
d+1+j
)j+1 ≥ e−1 for all j ≥ m and all d ≥ 1. Therefore, for all k ≥ m,
ak = am
k−1∏
j=m
aj+1
aj
≥ am
k−1∏
j=m
[(
1 +m
2 +m
)Np e−1τ
(j + 1)(d+ 1 + j)
]
= am
µk−m
(m+ 1)k−m(d+ 1 +m)k−m
,
where µ =
(
1+m
2+m
)Npe−1τ . Therefore,
Ψd,p;m(B) ≥
∞∑
k=m
ak ≥ am
∞∑
k=m
µk−m
(m+ 1)k−m(d+ 1 +m)k−m
= am
∞∑
k=0
µk
(m+ 1)k(d+ 1 +m)k
= am1F2(1;m+ 1, d+ 1 +m;µ).
This establishes (3.4). ■
6 Concluding remarks
In this article, we presented an approach, to calculating the normalizing constants of the matrix
Bingham and matrix Fisher distributions, that is alternative to numerical simulation methods.
Although Monte Carlo computations may be adequate for approximating the normalizing con-
stants Φd,p and Ψd,p, such numerical simulations can only yield estimates of the errors in the com-
puted values of the normalizing constants and cannot provide provable bounds on those errors.
Complete Asymptotic Expansions for the Normalizing Constants 21
The approach given in the present article provides, under stated hypotheses, guaranteed upper
bounds for the remainder terms, and therefore is able to calculate the normalizing constants to
any desired level of accuracy. Further, the methods developed here provides explicit rates of
convergence as d → ∞ or m → ∞, and also deduces in some instances lower bounds on the
accuracy of each remainder term.
We note that the methods used in this article are applicable to a general class of distributions,
on the Stiefel and Grassmann manifolds, as described by Chikuse [4, pp. 35–36]. Further, the
approach given here applies to the complex matrix Bingham and complex matrix Langevin
distributions; this can be done by making the necessary changes from the real zonal polynomials
to the complex zonal polynomials and from the generalized hypergeometric functions of real
symmetric matrix argument to the generalized hypergeometric functions of Hermitian matrix
argument [16]. In particular, we note that the complex analog of (5.1), the crucial integral
representation for the zonal polynomials, is provided in [11, Theorem 4.8].
Finally, we remark that all the results in this article extend, mutatis mutandis, to arbitrary
symmetric cones. A basic principle observed from [6, 11] is that methods that are successful
for the cone of real positive definite matrices will remain successful for any symmetric cone.
Indeed, the results that we have established can be extended to derive rates of convergence of
the spherical polynomial series expansions of all generalized hypergeometric functions defined
on the symmetric cones.
Acknowledgements
We are grateful to referees and an editor who read our manuscript with great thoroughness and
provided us with incisive and perceptive comments.
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1 Introduction
2 Complete asymptotics for the normalizing constant of the high-dimensional matrix Bingham distribution
3 Complete asymptotics for the normalizing constant of the high-dimensional matrix Langevin distribution
4 Implications for applications
5 Proofs
5.1 Properties of the zonal polynomials
5.2 Inequalities for the partitional shifted factorials and the zonal polynomials
5.3 The proofs for Section 2
5.4 The proofs for Section 3
6 Concluding remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-212652 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T06:55:41Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bagyan, Armine Richards, Donald 2026-02-09T09:34:00Z 2024 Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions. Armine Bagyan and Donald Richards. SIGMA 20 (2024), 094, 22 pages 1815-0659 2020 Mathematics Subject Classification: 60E05; 62H11; 62E20; 62R30 arXiv:2402.08663 https://nasplib.isofts.kiev.ua/handle/123456789/212652 https://doi.org/10.3842/SIGMA.2024.094 For positive integers and p such that ≥ , let ℝᵈˣᵖ denote the set of × real matrices, ₚ be the identity matrix of order , and d,ₚ = { ∈ ℝᵈˣᵖ ∣ ′ = ₚ} be the Stiefel manifold in ℝᵈˣᵖ. Complete asymptotic expansions as → ∞ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on d,ₚ. The accuracy of each truncated expansion is strictly increasing in ; also, for sufficiently large , the accuracy is strictly increasing in , the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both → ∞ and → ∞. Values of and arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as or increases. These results extend recently obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions. We are grateful to the referees and the editor who read our manuscript with great thoroughness and provided us with incisive and perceptive comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions Article published earlier |
| spellingShingle | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions Bagyan, Armine Richards, Donald |
| title | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_full | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_fullStr | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_full_unstemmed | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_short | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_sort | complete asymptotic expansions for the normalizing constants of high-dimensional matrix bingham and matrix langevin distributions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212652 |
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