A new property of reproducing kernels for classical orthogonal polynomials

We exhibit a second-order differential operator commuting with the reproducing kernel $\text{∑}{}_{\mathrm{n\; ?\; 0}}{}^{\mathrm{T}}\text{φ}{}_{\mathrm{n}}\text{(λ)}\text{φ}{}_{\mathrm{n}}\text{(μ)}\text{h}{}_{\mathrm{n}}$ each time that {φ_n(λ)} is one of the classical orthogonal polynomials: Jacobi, Laguerre, Hermite and Bessel. This is the analog of a known property in the study of time and band-limited signals.     

Minimum variance estimators for misclassification probabilities in discriminant analysis

Let α(n₁, n₂) be the probability of classifying an observation from population Π₁ into population Π₂ using Fisher's linear discriminant function based on samples of size n₁ and n₂. A standard estimator of α, denoted by T₁, is the proportion of observations in the first sample misclassified by the discriminant function. A modification of T₁, denoted by T₂, is obtained by eliminating the observation being classified from the calculation of the discriminant function. The UMVU estimators, $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$, of ET₁ = τ₁(n₁, n₂) and ET₂ = τ₂(n₁, n₂) = α(n₁ ? 1, n₂) are derived for the case when the populations have multivariate normal distributions with common dispersion matrix. It is shown that $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$ are nonincreasing functions of D², the Mahalanobis sample distance. This result is used to derive the sampling distributions and moments of $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$. It is also shown that α is a decreasing function of Δ² = (μ₁ ? μ₂)′Σ^?1(μ₁ ? μ₂). Hence, by truncating $\text{T}{}_1{}^1$ and $\text{T}{}_2{}^1$ (or any estimator) at the value of α for Σ = 0, new estimators are obtained which, for all samples, are as close or closer to α.     

Eigenvalues of p-summing and lp-type operators in Banach spaces

This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λ_n(T)) of a p-absolutely summing operator, p ? 2, satisfy

$\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{?π}{}_{\mathrm{p}}\text{(T).}$

This solves a problem of A. Pietsch. We give applications of this to integral operators in L_p-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π₂⁽ⁿ⁾, P = (p₁, …, p_n) and show that for T ∈ Π₂⁽ⁿ⁾, 2 = (2, …, 2) ∈ Nⁿ, the eigenvalues of T satisfy

$\text{∑}\text{i∈N}\text{|λ}{}_{\mathrm{i}}\text{(T)|}{}^{\mathrm{<mtext>2</mtext><mtext>n</mtext>}}{}^{\mathrm{<mtext>n</mtext><mtext>2</mtext>}}\text{?π}{}^{\mathrm{n}}{}_2\text{(T).}$

More generally, we show that for T ∈ Π_p⁽ⁿ⁾, P = (p₁, …, p_n), pi ? 2, the eigenvalues are absolutely p-summable,

$\text{1}\text{p}\text{=}\text{∑}\text{i=1}\text{n}\text{1}\text{p}{}_{\mathrm{i}}\text{and}\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}{}^{\mathrm{<mtext>1</mtext><mtext>p</mtext>}}\text{?C}{}_{\mathrm{p}}\text{π}{}^{\mathrm{n}}{}_{\mathrm{P}}\text{(T).}$

We also consider the distribution of eigenvalues of p-nuclear operators on L_r-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely

$\text{∑}\text{n∈N}\text{|λ}{}_{\mathrm{n}}\text{(T)|}{}^{\mathrm{p}}\text{? C}{}_{\mathrm{p}}\text{∑}\text{n∈N}\text{α}{}_{\mathrm{n}}\text{(T)}{}^{\mathrm{p}}$

, 0 < p < ∞, where α_n denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l₁ then X must be a Hilbert space.     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

On the maximum cardinality of a consistent set of arcs in a random tournament

For any tournament T on n vertices, let h(T) denote the maximum number of edges in the intersection of T with a transitive tournament on the same vertex set. Sharpening a previous result of Spencer, it is proved that, if T_n denotes the random tournament on n vertices, then, $\text{P}$$\text{(h(T}{}_{\mathrm{n}}\text{) ≤}\text{1}\text{2}\text{(}{}_2{}^{\mathrm{n}}\text{) + 1.73n}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}\text{) → 1}$ as n → ∞.     

Deficiencies of smooth manifolds

Being given a closed manifold Mⁿ, there are involutions (X²ⁿ, T) on closed manifolds of twice the dimension having fixed point set M. Kulkarni defined the deficiency of M for a class of involutions to be $\text{min}\text{(}\text{1}\text{2}\text{{}\text{dim}\text{H}{}^1\text{(X;Z}{}_{\mathrm{<mtext>2</mtext>}}\text{)?}\text{dim}\text{H}{}^1\text{(M;Z}{}_{\mathrm{<mtext>2</mtext>}}\text{)})}$ for all involutions (X, T) in the class. This paper exhibits manifolds for which the deficiency is positive for all involutions and studies the deficiencies for other classes.     

On a conjecture by D. Ž. D̄okovic

It is shown that if $\text{A?Ω}{}_{\mathrm{n}}\text{?{J}{}_{\mathrm{n}}\text{}}$ satisfies

$\text{nkσ}{}_{\mathrm{k}}\text{(A)?(n?k+1)}{}^2\text{σ}{}_{\mathrm{k?1}}\text{(A)}$

$\text{(k=1,2,…,n)}$

, where σ_k(A) denotes the sum of all kth order subpermanent of A, then Per[λJ_n+(1?λ)A] is strictly decreasing in the interval 0<λ<1.     

Construction of a recurrence relation for modified moments

In this paper we are constructing a recurrence relation of the form

$\text{∑}\text{i=0}\text{r}\text{ω}{}_{\mathrm{i}}\text{(k)m}{}_{\mathrm{k+i}}{}^{\mathrm{\{\lambda \}}}\text{[f] = ω(k)}$

for integrals (called modified moments)

$\text{m}{}_{\mathrm{k}}{}^{\mathrm{\{\lambda \}}}\text{[f]df=}\text{∫}\text{?1}\text{1}\text{f(x)C}{}_{\mathrm{k}}{}^{\mathrm{(\lambda )}}\text{(x)dx (k = 0,1,…)}$

in which C_k^(λ) is the k-th Gegenbauer polynomial of order $\text{λ(λ > ?}\text{1}\text{2}\text{)}$, and f is a function satisfying the differential equation

$\text{∑}\text{i=0}\text{n}\text{P}{}_{\mathrm{i}}\text{(x)f}{}^{\mathrm{(i)}}\text{(x) = p(x) (?1?x?1)}$

of order n, where p₀, p₁, …, p_n ? 0 are polynomials, and m_k^〈λ〉[p] is known for every k. We give three methods of construction of such a recurrence relation. The first of them (called Method I) is optimum in a certain sense.     

Exposants de Lyapounov pour opérateurs de Schrödinger discrètes quasi-périodiquesLyapounov exponents for discrete,quasi-periodic Schrödinger operators

We consider quasi-periodic Schrödinger operators H on $\text{Z}$ of the form H=H_λ,x,ω=λv(x+nω)δ_n,n′+Δ where v is a non-constant real analytic function on the d-torus $\text{T}{}^{\mathrm{d}}\text{(d?1)}$ and Δ denotes the discrete lattice Laplacian on $\text{Z}$. Denote by L_ω(E) the Lyapounov exponent, considered as function of the energy E and the rotation vector $\text{ω∈}\text{T}{}^{\mathrm{d}}$. It is shown that for |λ|>λ₀(v), there is the uniform minoration $\text{L}{}_{\mathrm{\omega }}\text{(E)>}\text{1}\text{2}\text{log}\text{|λ|}$ for all E and ω. For all λ and ω, L_ω(E) is a continuous function of E. Moreover, L_ω(E) is jointly continuous in (ω,E), at any point $\text{(ω}{}_0\text{,E}{}_0\text{)∈}\text{T}{}^{\mathrm{d}}\text{×}\text{R}$ such that k·ω₀≠0 for all $\text{k∈}\text{Z}{}^{\mathrm{d}}\text{?{0}}$. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 529–531.     

Concrete model theory for a class of operators

The operator $\text{L?(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∝}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∝}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ acting on H=∝₀^2π⊕L²(v_t), where m and v_t, 0 ? t ? 2π are measures on [0, 2π] with m smooth and e(s, t) = exp[?∝_t^s∝_Tdv_λ(θ) dm(λ)], satisfies $\text{rank}\text{(I ? LL}{}^1\text{) =}\text{rank}\text{(I ? L}{}^1\text{L) = 1}$. It is, therefore, unitarily equivalent to a scalar Sz.-Nagy-Foia? canonical model. The purpose of this paper is to determine the model explicitly and to give a formula for the unitary equivalence.     

Two-parameter Turan inequalities for ultraspherical and Laguerre polynomials

It is known that the classical orthogonal polynomials satisfy inequalities of the form U_n²(x) ? U_{n + 1}(x) U_{n ? 1}(x) > 0 when x lies in the spectral interval. These are called Turan inequalities. In this paper we will prove a generalized Turan inequality for ultraspherical and Laguerre polynomials. Specifically if P_n^λ(x) and L_n^α(x) are the ultraspherical and Laguerre polynomials and $\text{F}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x) =}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(x)}\text{P}{}_{\mathrm{n}}{}^{\mathrm{\lambda }}\text{(1)}\text{, G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) =}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)}\text{L}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(0)}\text{,}\text{then}\text{F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? F}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, ? 1 < x < 1, ?}\text{1}\text{2}\text{< α ? β ? α + 1}\text{and}\text{G}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? G}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) G}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > 0, x > 0, 0 < α ? β ? α + 1}$. We also prove the inequality $\text{(n + 1) F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n}}{}^{\mathrm{\beta }}\text{(x) ? nF}{}_{\mathrm{n\; +\; 1}}{}^{\mathrm{\alpha }}\text{(x) F}{}_{\mathrm{n\; ?\; 1}}{}^{\mathrm{\beta }}\text{(x) > A}{}_{\mathrm{n}}\text{[F}{}_{\mathrm{n}}{}^{\mathrm{\alpha }}\text{(x)]}{}^2\text{, ?1 < x < 1, ?}\text{1}\text{2}\text{< α ? β < α + 1,}\text{where}\text{A}{}_{\mathrm{n}}$ is a positive constant depending on α and β.     

A queueing model and a set of orthogonal polynomials

A single serving queueing model is studied where potential customers are discouraged at the rate λ_n = λqⁿ, 0 < q < 1, n is the queue length. The serving rate is μ_n = μ(1 ? qⁿ), n = 0, 1,…. The spectral function is computed and the corresponding set of orthogonal polynomials is studied in detail. The slightly more general model with $\text{λ}{}_{\mathrm{n}}\text{=}\text{λq}{}^{\mathrm{n}}\text{(1 + bq}{}^{\mathrm{n}}\text{)}\text{, μ}{}_{\mathrm{n}}\text{=}\text{μ(1 ? q}{}^{\mathrm{n}}\text{)}\text{(1 + bq}{}^{\mathrm{n}}\text{)}$ and the analogous orthogonal polynomials are also investigated. In both cases a method developed by Pollaczek is used which has been used very successfully to study new sets of orthogonal polynomials by Askey and Ismail.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.     

Stationary queues with one autonomous server

Given a stationary stochastic continuous demand of service σ(θ_tω) dt with ∫ σ(ω)P(dω) < 1, we construct real stationary point processes $\text{(T}{}_{\mathrm{n}}\text{, n ∈}\text{Z}\text{)[T}{}_{\mathrm{n}}\text{< T}{}_{\mathrm{n}}\text{+1, lim}{}_{\mathrm{\pm \infty }}\text{T}{}_{\mathrm{n}}\text{= ±∞]}$ such that

for a given constant D \2>0. These point processes correspond to a service discipline for which a single server services during the time intervals [T_n, T_n+1[ the demand of service accumulated during the proceding intervals [T_n?1, T_n[ and take a rest of fixed duration D.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$

     

Limit distributions for the maxima of stationary Gaussian processes

Let {X_n} be a stationary Gaussian sequence with E{X₀} = 0, {X²₀} = 1 and E{X₀X_n} = rⁿ_n Let c_n = (2ln n)^{$\text{built1}\text{2}$}, b_n = c_n? $\text{1}\text{2}$c^-1_n ln(4π ln n), and set M_n = max_{0 ?k?n}X_k. A classical result for independent normal random variables is that

$\text{P[c}{}_{\mathrm{n}}\text{(M}{}_{\mathrm{n}}\text{?b}{}_{\mathrm{n}}\text{)?x]→exp[-e}{}^{\mathrm{-x}}\text{] as n → ∞ for all x.}$

Berman has shown that (1) applies as well to dependent sequences provided r_nlnn = o(1). Suppose now that {r_n} is a convex correlation sequence satisfying r_n = o(1), (r_nlnn)^-1 is monotone for large n and o(1). Then

$\text{P[r}{}_{\mathrm{n}}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{(M}{}_{\mathrm{n}}\text{? (1?r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{b}{}_{\mathrm{n}}\text{)?x] → Ф(x)}$

for all x, where Ф is the normal distribution function. While the normal can thus be viewed as a second natural limit distribution for {M_n}, there are others. In particular, the limit distribution is given below when r_n is (sufficiently close to) γ/ln n. We further exhibit a collection of limit distributions which can arise when r_n decays to zero in a nonsmooth manner. Continuous parameter Gaussian processes are also considered. A modified version of (1) has been given by Pickands for some continuous processes which possess sufficient asymptotic independence properties. Under a weaker form of asymptotic independence, we obtain a version of (2).     

Decomposability of symmetric multilinear functions

Let V denote a finite dimensional vector space over a field K of characteristic 0, let T_n(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let S_n(V) denote the subspace of T_n(V) whose members are the fully symmetric members of T_n(V). If $\text{L}$_n denotes the symmetric group on {1,2,…,n} then we define the projection $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{: T}{}_{\mathrm{n}}\text{(V) → S}{}_{\mathrm{n}}\text{(V)}$ by the formula , where P_σ : T_n(V) → T_n(V) is defined so that P_σ(A)(y₁,y₂,…,y_n = A(y_σ(1),y_σ(2),…,y_σ(n)) for each A?T_n(V) and y_i?V, 1 ? i ? n. If $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, then x₁?x₂? … ?x_n denotes the member of T_n(V) such that $\text{(x}{}_1\text{?x}{}_2\text{· ? ? ?x}{}_{\mathrm{n}}\text{)(y}{}_1\text{,y}{}_2\text{,…,y}{}_{\mathrm{n}}\text{) = П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{x}{}_{\mathrm{i}}\text{(y}{}_{\mathrm{i}}\text{)}$ for each y₁ ,₂,…,y_n in V, and x₁·x₂… x_n denotes $\text{P}{}_{\mathrm{<mtext>L</mtext>}}\text{(x}{}_1\text{?x}{}_2\text{? … ?x}{}_{\mathrm{n}}\text{)}$. If B? S_n(V) and there exists $\text{x}{}_{\mathrm{i}}\text{? V}{}^1\text{, 1 ? i ? n}$, such that B = x₁·x₂…x_n, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of S_n(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.     

The asymptotic equivalence of Bayes and maximum likelihood estimation

Let (X, $\text{A}$) be a measurable space, Θ ? $\text{R}$ an open interval and P_Ω ∥ $\text{A}$, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let $\text{Ω}{}_{\mathrm{n}}$ be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let $\text{R}{}_{\mathrm{<mtext>n,</mtext><mtext>x</mtext>}}$ be the posterior distribution for the sample size n given $\text{x}\text{? X}{}^{\mathrm{n}}$. $\text{L:}\text{Θ}\text{× Θ →}\text{R}$ denotes a loss function fulfilling certain regularity conditions and T_n denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists c_K ≥ 0 such that

This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.     

On the enumeration of self-complementary m-placed relations

A formula for the number a_m(2n) of self-complementary m-placed relations is given. Then we obtain from this formula asymptotic results, e.g.$\text{a}{}_{\mathrm{m}}\text{(2n) ～ (2(}{}^{\mathrm{2n}}\text{)}{}^{\mathrm{<mtext>m</mtext><mtext>2?n</mtext>}}\text{)/n}$     

A maximization problem and its application to canonical correlation

Let Σ be an n × n positive definite matrix with eigenvalues λ₁ ≥ λ₂ ≥ … ≥ λ_n > 0 and let M = {x, y | x?Rⁿ, y?Rⁿ, x ≠ 0, y ≠ 0, x′y = 0}. Then for x, y in M, we have that $\text{x′Σy}\text{(x′Σxy′Σy)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$ and the inequality is sharp. If

$\text{∑=}\text{∑}{}_{11}\text{∑}{}_{12}\text{∑}{}_{21}\text{∑}{}_{22}$

is a partitioning of Σ, let θ₁ be the largest canonical correlation coefficient. The above result yields $\text{θ}{}_1\text{≤}\text{(λ}{}_1\text{? λ}{}_{\mathrm{n}}\text{)}\text{(λ}{}_1\text{+ λ}{}_{\mathrm{n}}\text{)}$.