On asymptotic solutions of the renewal equation

Consider the renewal equation in the form (¹) $\text{u(t) = g(t) + ∝}{}_{\mathrm{o}}{}^{\mathrm{t}}\text{u(t ? τ) ?(τ) dτ}$, where $\text{?(t)}$ is a probability density on [0, ∞) and lim_{t → ∞}g(t) = g₀. Asymptotic solutions of (¹) are given in the case when f(t) has no expectation, i.e., $\text{∝}{}_0{}^{\mathrm{\infty }}\text{t?(t)dt = ∞}$. These results complement the classical theorem of Feller under the assumption that f(t) possesses finite expectation.     

Some asymptotic formulas on generalized divisor functions,II

Let A be an infinite sequence of positive integers a₁ < a₂ <… and put $\text{f}{}_{\mathrm{A}}\text{(x) =}\text{Σ}{}_{\mathrm{a\in A,\; a\le x}}\text{(}\text{1}\text{a}\text{)}$, $\text{D}{}_{\mathrm{A}}\text{(x) =}\text{max}{}_{\mathrm{1\le n\le x}}\text{Σ}{}_{\mathrm{<mtext>a\in A,</mtext><mtext>a</mtext><mtext>n</mtext>}}\text{1}$. In Part I, it was proved that $\text{lim}{}_{\mathrm{x\to +\infty }}\text{sup}\text{D}{}_{\mathrm{A}}\text{(x)}\text{f}{}_{\mathrm{A}}\text{(x)}\text{= +∞}$. In this paper, this theorem is sharpened by estimating D_A(x) in terms of f_A(x). It is shown that lim_x→+∞sup D_A(x) exp(?c₁(logf_A(x))²) = +∞ and that this assertion is not true if c₁ is replaced by a large constant c₂.     

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.     

Randomness implies order

Let τ: [0, 1] → [0, 1] possess a unique invariant density $\text{f}{}^1$. Then given any ? > 0, we can find a density function p such that $\text{∥ p ? f}{}^1\text{∥ < ?,}\text{and}\text{p}$ is the invariant density of the stochastic difference equation x_{n + 1} = τ(x_n) + W, where W is a random variable. It follows that for all starting points $\text{x}{}_0\text{? [0, 1],}\text{lim}{}_{\mathrm{n\to \infty }}\text{(1}\text{n)}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n\; ?\; 1}}\text{χ}{}_{\mathrm{B}}\text{(x}{}_{\mathrm{i}}\text{) = ∝}{}_{\mathrm{B}}\text{p(ξ) dξ}$.     

Some divided difference inequalities for n-convex functions

Using results from the theory of B-splines, various inequalities involving the nth order divided differences of a function f with convex nth derivative are proved; notably, $\text{f}{}^{\mathrm{(n)}}\text{(z)}\text{n! ? [x}{}_0\text{,…, x}{}_{\mathrm{n}}\text{]f}\text{?}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{(f}{}^{\mathrm{(n)}}\text{(x}{}_{\mathrm{i}}\text{)}\text{(n + 1)!)}$, where z is the center of mass $\text{(}\text{1}\text{(n + 1))}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n}}\text{x}{}_{\mathrm{i}}$.     

On the expansion of Cϱ1(V + I) as a sum of zonal polynomials

The coefficients a_τ?, sometimes called “generalized binomial coefficients” in the expansion $\text{C}{}_{\mathrm{?}}{}^1\text{(V +I) = Σ}{}_{\mathrm{\tau }}\text{a}{}_{\mathrm{?\tau }}\text{C}{}_{\mathrm{\tau }}{}^1\text{(V)}$, are computed explicitly when t = r + 1, where ? is a partition of r and τ a partition of t. A recursion formula permits the calculation of the general a_τ?. Several properties of a_τ? are proved. A connection between the a_τ? and other coefficients is established. The main tools used are Bingham's identity, results from the theory of invariant differential operators, and a lemma concerning zonal polynomials.     

On integral-valued additive functions,II

A pair [₁,f₂] of integral-valued arithmetical functions is said to be uniformly distributed modulo [q₁, q₂], where q₁ and q₂ are integers > 1, if, for every pair [r₁, r₂] of integers, the set of those positive integers n which satisfy

$\text{f}{}_1\text{(n)≡(}\text{mod}\text{q}{}_1\text{)}\text{and}\text{f}{}_2\text{(n)≡r}{}_2\text{(}\text{mod}\text{q}{}_2$

has density $\text{1}\text{(q}{}_1\text{q}{}_2\text{)}$.We give necessary and sufficient conditions for a pair of integral-valued additive functions to be uniformly distributed modulo [q₁, q₂] in the case when (q₁, q₂) > 1.     

Generalization of a theorem by Hardy,Littlewood, and Pólya

Let τ be a non-expanding map which admits a σ-finite, absolutely continuous invariant measure μ. The measure $\text{μ¦}{}_{\mathrm{E}}$ is approximated as closely as desired by a measure μ_n invariant under the transformation τ_n satisfying $\text{inf}{}_{\mathrm{x}}\text{¦τ′}{}_{\mathrm{n}}\text{(x)¦ > 1}$. The orbit {τ_n^j(x)}_{j = 0}^∞ “exhibits” $\text{μ¦}{}_{\mathrm{E}}$ much more quickly than does the orbit {τ^j(x)}_{j = 0}^∞.     

A transformation theorem for one-dimensional F-expansions

If f is a monotone function subject to certain restrictions, then one can associate with any real number x between zero and one a sequence {a_n(x)} of integers such that

$\text{x=f(a}{}_1\text{(x) + f(a}{}_2\text{(x) +f(a}{}_3\text{(x) +…)))}$

. In this paper properties of the function F defined by

$\text{Fx=g(a}{}_1\text{(x) + g(a}{}_2\text{(x) +g(a}{}_3\text{(x) +…)))}$

, where g is any function satisfying the same restrictions as f, are discussed. Principally, F is found to be useful in finding stationary measures on the sequences {a_n(x)}.     

F-mixing and weak disjointness

A topological system (X,f) is $\text{F}$-transitive if for each pair of opene subsets U and V of X, $\text{N}{}_{\mathrm{f}}\text{(U,V)={n∈}\text{Z}{}_{\mathrm{+}}\text{:}\text{f}{}^{\mathrm{n}}\text{U∩V≠∅}∈}\text{F}$, where $\text{F}$ is a collection of subsets of $\text{Z}{}_{\mathrm{+}}$ which is hereditary upward. (X,f) is $\text{F}$-mixing if (X×X,f×f) is $\text{F}$-transitive. In this paper $\text{F}$-mixing systems are characterized in terms of the chaoticity of the systems. Moreover, weak disjointness is studied via family. We will give conditions such that a dual theorem of the Weiss–Akin–Glasner theorem holds. Examples with this dual theorem fails for some “good” families are obtained.     

The one-dimensional diffusion process and unitary representations of R1

Given a cocycle a(t) of a unitary group {U¹}, ?∞ < t < ∞, on a Hilbert space $\text{H}$, such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;^t₀U^sxds + β(t) for a unique x ? $\text{H}$, β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as $\text{σ}{}^2\text{({rmU}{}^{\mathrm{t}}\text{}, a(t)) =}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{)∥∝}{}^{\mathrm{t}}{}_0\text{U}{}^{\mathrm{s}}\text{x ds∥}{}^2$ if existing. For a stationary diffusion process on $\text{R}$¹, with Ω₁, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on $\text{Ω}{}_1$, based on the paths restricted to the time interval [0, 1]. It is shown that $\text{I(Ω)}$ is continuous and bounded on $\text{Ω}{}_1$. The shift τ_t, defines a unitary representation {U^t}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, $\text{I(Ω)}$ has a C^∞ spectral density function f;. It is then shown that σ²({U^t}, I) = f;(O).     

Arithmetic properties of certain functions in several variables

If T is an n × n matrix with nonnegative integral entries, we define a transformation T: Cⁿ → Cⁿ by w = Tz where

$\text{W}{}_1\text{=}\text{∏}\text{j=1}\text{n}\text{z}{}_{\mathrm{j}}{}^{\mathrm{t}}\text{ij}\text{(1?i?n).}$

We consider functions f(z) of n complex variables which satisfy a functional equation giving f(Tz) as a rational function of ¹f(z) and we obtain conditions under which such a function f(z) takes transcendental values at algebraic points.     

Integral inequalities for generalized concave or convex functions

Let ψ be convex with respect to ?, B a convex body in Rⁿ and f a positive concave function on B. A well-known result by Berwald states that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{ψ(f(x)) dx ? n ∝}{}_0{}^1\text{ψ(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$ (1) if ξ is chosen such that $\text{1}\text{¦B¦}\text{∝}{}_{\mathrm{B}}\text{?(f(x)) dx = n ∝}{}_0{}^1\text{?(ξt)(1 ? t)}{}^{\mathrm{n\; ?\; 1)}}\text{dt}$.The main purpose in this paper is to characterize those functions f : B → R₊ such that (1) holds.     

Counterexample—An inadmissible estimator which is generalized bayes for a prior with “light” tails

Previous work on the problem of estimating a univariate normal mean under squared error loss suggests that an estimator should be admissible if and only if it is generalized Bayes for a prior measure, F, whose tail is “light” in the sense that $\text{∫}{}_1{}^{\mathrm{\infty }}\text{f}{}^{\mathrm{1?1}}\text{(x) dx = ∞ =}\text{∫}{}_{\mathrm{?\infty }}{}^{\mathrm{?1}}\text{f}{}^{\mathrm{1?1}}\text{(x) dx}$, where $\text{f}{}^1$ denotes the convolution of F with the normal density. (There is also a precise multivariate analog for this condition.) We provide a counterexample which shows that this suggestion is false unless some further regularity conditions are imposed on F.     

Elementary proof of the transformation formula for Lambert series involving generalized Dedekind sums

An elementary proof is given of the author's transformation formula for the Lambert series $\text{G}{}_{\mathrm{p}}\text{(x) =}\text{Σ}{}_{\mathrm{n?1}}{}^{\mathrm{<mtext>\infty </mtext>}}\text{n}{}^{\mathrm{?p}}\text{x}{}^{\mathrm{n}}\text{(1?x}{}^{\mathrm{n}}\text{)}$ relating G_p(e^2πiτ) to G_p(e^2πiAτ), where p > 1 is an odd integer and $\text{Aτ =}\text{(aτ + b)}\text{(cτ + d)}$ is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function $\text{log}\text{η(τ) =}\text{πiτ}\text{12}\text{? G}{}_1\text{(e}{}^{\mathrm{2\pi i\tau }}\text{)}$, and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions.     

On matrix functions which commute with their derivative

We improve several results published from 1950 up to 1982 on matrix functions commuting with their derivative, and establish two results of general interest. The first one gives a condition for a finite-dimensional vector subspace E(t) of a normed space not to depend on t, when t varies in a normed space. The second one asserts that if A is a matrix function, defined on a set ?, of the form A(t)= U diag(B₁(t),…,B_p(t)) U^-1, t ∈ ?, and if each matrix function B_k has the polynomial form

$\text{B}{}_{\mathrm{k}}\text{(t)=}\text{∑}\text{i=0}\text{α}{}_{\mathrm{k}}\text{f}{}_{\mathrm{ki}}\text{(t)C}{}_{\mathrm{k}}{}^{\mathrm{i}}\text{, t∈ ?, k∈{1,…,p}}$

then A itself has the polynomial form

$\text{A(t)=}\text{∑}\text{i=0}\text{d?1}\text{f}{}_{\mathrm{i}}\text{(t)C}{}^{\mathrm{i}}\text{,t∈?}$

, where

$\text{d=}\text{∑}\text{k=1}\text{p}\text{d}{}_{\mathrm{k}}$

, d_k being the degree of the minimal polynomial of the matrix C_k, for every k ∈ {1,…,p}.     

On the qth convergence exponent of entire functions

In this paper we obtain a growth relation for entire functions of qth order with respect to the distribution of its zeros. We also derive certain relations between the qth convergence exponents of two or more entire functions. The most striking result of the paper is: If f(z) has at least one zero, then

$\text{lim sup}\text{r→∞}\text{log n}\text{(r)}\text{log}{}^{\mathrm{[q+1]}}{}_{\mathrm{r}}\text{=?(q)}$

, where n(r) is the number of zeros of f(z) in $\text{¦}\text{z}\text{¦ ? r}$ and

$\text{?(q)=g.l.b.}\text{α:α>0 and}\text{∑}\text{∞}\text{n=1}\text{(log}{}^{\mathrm{[q]}}\text{r}{}_{\mathrm{n}}\text{)}{}^{\mathrm{-\alpha }}\text{<∞}$

.     

Zur theorie der Ringklassenkörper über imaginär-quadratischen Zahlkörpern

Let Σ be an imaginary quadratic number field, and $\text{Ω}{}_{\mathrm{f}}$ the ring class field extension of Σ for a natural number f as conductor. For the investigations of class number and unit group of the subfields of $\text{Ω}{}_{\mathrm{f}}$ by means of the methods of complex multiplication one often uses the quotients $\text{Δ(}\text{a}\text{)}\text{Δ(}\text{b}\text{)}$ of the singular values of the discriminant Δ occurring in the theory of modular functions. As is well known, they are contained in $\text{Ω}{}_{\mathrm{f}}$. But it is not known whether those quotients generate $\text{Ω}{}_{\mathrm{f}}$ over Σ. Corollary 1 of this paper solves this problem. Moreover Theorems 2, 3 exhibit more general methods of generating the subfields by means of relative norms.     

Subextension of plurisubharmonic functions with bounded Monge–Ampère mass

Let $\text{Ω?}\text{C}{}^{\mathrm{n}}$ be a hyperconvex domain. Denote by $\text{E}{}_0\text{(Ω)}$ the class of negative plurisubharmonic functions ? on $\text{Ω}$ with boundary values 0 and finite Monge–Ampère mass on $\text{Ω.}$ Then denote by $\text{F}\text{(Ω)}$ the class of negative plurisubharmonic functions ? on $\text{Ω}$ for which there exists a decreasing sequence (?)_j of plurisubharmonic functions in $\text{E}{}_0\text{(Ω)}$ converging to ? such that $\text{sup}{}_{\mathrm{j}}\text{∫}{}_{\mathrm{\Omega }}\text{(dd}{}^{\mathrm{c}}\text{?}{}_{\mathrm{j}}\text{)}{}^{\mathrm{n}}\text{+∞.}$It is known that the complex Monge–Ampère operator is well defined on the class $\text{F}\text{(Ω)}$ and that for a function $\text{?∈}\text{F}\text{(Ω)}$ the associated positive Borel measure is of bounded mass on $\text{Ω.}$ A function from the class $\text{F}\text{(Ω)}$ is called a plurisubharmonic function with bounded Monge–Ampère mass on $\text{Ω.}$We prove that if $\text{Ω}$ and $\text{Ω}$ are hyperconvex domains with $\text{Ω?}\text{Ω}\text{?}\text{C}{}^{\mathrm{n}}$ and $\text{?∈}\text{F}\text{(Ω),}$ there exists a plurisubharmonic function $\text{?}\text{?}\text{∈}\text{F}\text{(}\text{Ω}\text{)}$ such that $\text{?}\text{?}\text{??}$ on $\text{Ω}$ and $\text{∫}{}_{\mathrm{<mtext>\Omega </mtext>}}\text{(dd}{}^{\mathrm{c}}\text{?}\text{?}\text{)}{}^{\mathrm{n}}\text{?∫}{}_{\mathrm{\Omega }}\text{(dd}{}^{\mathrm{c}}\text{?)}{}^{\mathrm{n}}\text{.}$ Such a function is called a subextension of ? to $\text{Ω}\text{.}$From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on $\text{Ω.}$To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).