On the square root of a positive B( )-valued function

Let (Ω, , μ) be a measure space, a separable Banach space, and ^* the space of all bounded conjugate linear functionals on . Let f be a weak^* summable positive B( ^*)-valued function defined on Ω. The existence of a separable Hilbert space , a weakly measurable B( )-valued function Q satisfying the relation Q^*(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺( ^*)-valued measures, the concepts of weak^*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

The isometries of LP(Ω, X)

Let (Ω, ∑, μ) be a finite measure space and X a separable Banach space. We characterize the linear isometries of $\text{L}{}^{\mathrm{p}}\text{(Ω, X)}$ onto itself for 1 ? p < ∞, p ≠ 2 under the condition that X is not the l^p-direct sum of two nonzero spaces (for the same p). It is shown that T is such an isometry if and only if (Tf)(·) = S(·)h(·)(Φ(f))(·), where Φ is a set isomorphism of ∑ onto itself, S is a strongly measurable operator-valued map such that S(t) is a.e. an isometry of X onto itself, and h is a scalar function which is related to Φ. It is further shown that for a big class of measure spaces (perhaps all nontrivial ones) the condition on X is also a necessary condition for the above conclusion to hold. In the case when X is a Hilbert space the injective isometries of $\text{L}{}^{\mathrm{p}}\text{(Ω, X)}$ are also characterized. They have the same form as above, except that Φ and S(t) are not necessarily onto.     

Stochastic processes with independent increments,taking values in a Hilbert space

Let $\text{H}$ be a real separable Hilbert space; let X(t), t?[0, 1], be a separable, stochastically continuous, $\text{H}$-valued stochastic process with independent increments. Then a decomposition of X(t) into a uniformly convergent sum of independent processes is found. In this decomposition one of the processes is Gaussian with continuous sample functions, and the remainder of the processes have sample functions whose discontinuities correspond to those of certain real-valued Poisson processes. The decomposition of X(t) leads to a Levy-Khintchine representation of the characteristic functional of X(t). In addition, the case when X(t) has finite variance is explored, and, as a consequence of the above decomposition, a Kolmogorov-type representation of the characteristic functional of X(t) is derived.     

Scaling limits of Gaussian vector fields

For Gaussian vector fields {X(t) ∈ Rⁿ:t ∈ R^d} we describe the covariance functions of all scaling limits Y(t) = $\text{L}$lim_α↓0 B^?1(α) X(αt) which can occur when B(α) is a d × d matrix function with B(α) → 0. These matrix covariance functions $\text{r(t, s) = EY(t) Y}{}^1\text{(s)}$ are found to be homogeneous in the sense that for some matrix L and each α > 0, $\text{(}{}^1\text{) r(αt, αs) = α}{}^{\mathrm{L1}}\text{r(t, s) α}{}^{\mathrm{L}}$. Processes with stationary increments satisfying (¹) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion.     

Minimax estimation of location parameters for certain spherically symmetric distributions

Families of minimax estimators are found for the location parameters of a p-variate distribution of the form

, where G(·) is a known c.d.f. on (0, ∞), p ≥ 3 and the loss is sum of squared errors. The estimators are of the form $\text{(1 ? ar(X′X)}\text{E}{}_0\text{(}\text{1}\text{X′X}\text{)X′X)X}$ where 0 ≤ a ≤ 2, r(X′X) is nondecreasing, and $\text{r(X′X)}\text{X′X}$ is nonincreasing. Generalized Bayes minimax estimators are found for certain G(·)'s.     

Inner functions in the unit ball of Cn

Let B be the open unit ball of $\text{C}$ⁿ, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have $\text{¦u¦ = 1}$ a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let $\text{¦X¦}$ be the norm closure, in L^∞ = L^∞(S), of X. Some results: set I is dense in the unit ball of H^∞(B) in the compact-open topology. On $\text{S,}\text{Q}\text{?}\text{Q}$ is weak¹-dense in $\text{L}{}^{\mathrm{\infty }}\text{, ¦}\text{Q}\text{?}\text{Q¦}$ does not contain $\text{H}{}^{\mathrm{\infty }}\text{, C(S) ?¦}\text{Q}\text{?}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ ¦}\text{H}\text{?}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ L}{}^{\mathrm{\infty }}$. (When $\text{n = 1, ¦Q¦ = H}{}^{\mathrm{\infty }}\text{and}\text{¦}\text{Q}\text{?}\text{Q¦ = L}{}^{\mathrm{\infty }}$.) Every unimodular $\text{?}\text{?}\text{L}{}^{\mathrm{\infty }}$ is a pointwise limit a.e. of products $\text{u}\text{v}\text{?}\text{, u}\text{?}\text{I, ν}\text{?}\text{I}$. The zeros of every $\text{? ? 0}$ in the ball algebra (but not of every H^∞-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if $\text{∥?∥}{}_{\mathrm{\infty }}\text{< 1}$. However, $\text{?}\text{u}$ cannot be bounded in B if u ? I is non-constant.     

Homogeneous quantization and multiplicities of group representations

Let B be a compact manifold. A cone over B is a principal R⁺-bundle, X, with base B. Let (a, x) → ?_a(x) be the mapping associated with the action of a? R⁺ on X. X is called a symplectic cone if it possesses a symplectic form, ω, such that $\text{?}{}_{\mathrm{a}}{}^1\text{ω = aω}$. A compact Lie group, G, is said to act in a homogeneous fashion on X if it acts on X in such a way that both ω and the principal bundle structure are preserved. It is known that to such an action one can associate in a fairly canonical way a representation of G on a Hilbert space H. (See [3].) In this paper we propose a symplectic recipe for the multiplicities with which H decomposes into G-irreducibles and show that this recipe is correct “generically”.     

Central limit theorems for C(S)-valued random variables

Let C(S) be the space of real-valued continuous functions on a compact metric space S. Let {X_n, n ? 1} be a sequence of independent identically distributed C(S)-valued random variables with mean zero and sup_t?sE[X₁²(t)] = 1. We show that the measures induced by ($\text{X}{}_1\text{+ ··· + X}{}_{\mathrm{n}}\text{) n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ converge weakly to a Gaussian measure on C(S) under different conditions on X₁, one of which consolidates and extends results of Strassen and Dudley, Giné, and Dudley. Our method of proof is different from the methods employed by these authors.     

The inverse balayage problem for Markov chains,part II

This paper continues the study of the inverse balayage problem for Markov chains. Let X be a Markov chain with state space $\text{A ? B}{}_2$, let v be a probability measure on B₂ and let M(v) consist of probability measures μ on A whose X-balayage onto B₂ is v. The faces of the compact, convex set M(v) are characterized. For fixed μ?M(v) the set M(μ,v) of the measures ? of the form $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S) ? ·}}$, where S is a randomized stopping time, is analyzed in detail. In particular, its extreme points and edge are explicitly identified. A naturally defined reversed chain X, for which v is an inverse balayage of μ, is introduced and the relation between X and X^ is studied. The question of which $\text{? ? M(μ, v)}$ admit a natural stopping time $\text{S}{}_{\mathrm{?}}$ of X (not involving an independent randomization) such that $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S}{}_{\mathrm{?}}\text{) ? ·}}$, is shown to have rather different answers in discrete and continuous time. Illustrative examples are presented.     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

An ∞-dimensional inhomogeneous Langevin's equation

On a modified space Φ′ from the space $\text{J}$′ of tempered distributions, it is proven that a stochastic equation, $\text{X(t) = γ + W(t) + ∝}{}_0{}^{\mathrm{t}}\text{L}{}^1\text{(s) X(s) ds}$, has a unique solution, where W(t) is a Φ′-valued Brownian motion independent of a Φ′-valued Gaussian random variable γ and $\text{L}{}^1\text{(s)}$ is an integro-differential operator. As an application, a fluctuaton result (or central limit theorem) is shown for interacting diffusions.     

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).     

Minimax estimation of a multivariate normal mean under polynomial loss

Let X be an observation from a p-variate (p ≥ 3) normal random vector with unknown mean vector θ and known covariance matrix

. The problem of improving upon the usual estimator of θ, δ⁰(X) = X, is considered. An approach is developed which can lead to improved estimators, δ, for loss functions which are polynomials in the coordinates of (δ ? θ). As an example of this approach, the loss L(δ, θ) = |δ ? θ|⁴ is considered, and estimators are developed which are significantly better than δ⁰. When

is the identity matrix, these estimators are of the form $\text{δ(X) = (1 ? (}\text{b}\text{(d + |X|}{}^2\text{)}\text{))X}$.     

On the order of the error function of the (k,r)-integers

Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = a^kb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Q_k,r(x) denotes the number of (k, r)-integers ≤ x, then $\text{Q}{}_{\mathrm{k,r}}\text{(x) =}\text{xζ(k)}\text{ζ(r)}\text{+ Δ}{}_{\mathrm{k,r}}\text{(x)}$, where $\text{Δ}{}_{\mathrm{k,r}}\text{(x) = O(x}{}^{\mathrm{<mtext>1</mtext><mtext>r</mtext>}}\text{exp}\text{[?B}\text{log}{}^{\mathrm{<mtext>3</mtext><mtext>5</mtext>}}\text{x (}\text{log log}\text{x)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>5</mtext>}}\text{])}$, B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δ_k,r(x) and prove that

$\text{1}\text{x}\text{∫}\text{1}\text{α}\text{δ}{}_{\mathrm{k,r}}\text{(t)dt=0(x}\text{1}\text{k}\text{ω(x))}\text{or}\text{0(x}{}^{\mathrm{3/(4r+1)}}\text{ω(x))}$

, according as $\text{k ≤}\text{(4r + 1)}\text{3}$ or $\text{k >}\text{(4r + 1)}\text{3}$, where ω(x) = exp [B log x(log log x)^?1].     

The range of the derivative of a differentiable bumpL'image de la dérivée d'une fonction bosse différentiable

We study the range of the derivative of a Frechet differentiable bump. X is an infinite dimensional separable C^p-smooth Banach space. We first prove that any connected open subset of $\text{X}{}^1$ containing 0 is the range of the derivative of a C^p-bump. Next, analytic subsets of $\text{X}{}^1$ which satisfy a natural linkage condition are the range of the derivative of a C¹-bump. We find analogues of these results in finite dimensions. We finally show that $\text{f′(}\text{R}{}^2\text{)}$ is the closure of its interior, if f is a C²-bump on $\text{R}{}^2$. To cite this article: T. Gaspari, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 189–194.     

Enumeration of the square permutations in Sn

Let f(n) denote the number of square permutations in the symmetric group S_n. This paper proves a conjecture that f(2k + 1) = (2k + 1)f(2k) and provides efficient procedures for the computation of f(n). The behavior of f(n) as n → ∞ is investigated and an asymptotic result obtained which shows that f(n) ～ 2Knⁿe^?n, where $\text{K = Π}{}_1{}^{\mathrm{\infty }}\text{cosh}\text{(}\text{1}\text{2k}\text{)}$.     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.     

Algebraic braided model of the affine line and difference calculus on a topological spaceModèle algébrique tressé de la droite affine et calcul aux différences sur un espace topologique

In a previous Note [1], we suggested a quantum model of the unit interval [0,1], using convergent power series, parametrized by a variable q (a remarkable example is the quantum exponential, defined by Euler). In the present Note, we suggest a simpler model based on functions $\text{f=f(x):}\text{Z}\text{→k}$ (with an arbitrary commutative ring k) which are constant when x?+∞ or x??∞ and their “differentials” considered as functions x?f(x+1)?f(x) (difference calculus). Thanks to this new “differential calculus over the integers”, we can associate to any simplicial set or topological space X a braided differential graded algebra $\text{D}{}^1\text{(X)}$ which is similar in spirit to the algebra $\text{W}{}^1\text{(X)}$ introduced in [1]. We notice that the p-homotopy type of X can be read from the braiding of $\text{D}{}^1\text{(X)}$. In particular, if $\text{k=}\text{Z}$, we recover in a purely algebraic way the integral cohomology, Steenrod operations, homotopy groups from this braiding. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 121–126.     

Zeta matrices of elliptic curves

Let $\text{O =}\text{lim}{}_{\mathrm{n}}\text{Z/p}{}^{\mathrm{n}}\text{Z}$, let $\text{A}\text{= O[g}{}_2\text{, g}{}_3\text{]}{}_{\mathrm{\Delta }}$, where g₂ and g₃ are coefficients of the elliptic curve: Y² = 4X³ ? g₂X ? g₃ over a finite field and Δ = g₂³ ? 27g₃² and let $\text{B}\text{=}\text{A}\text{[X, Y]}\text{(Y}{}^2\text{? 4X}{}^3\text{+ g}{}_2\text{X + g}{}_3\text{)}$. Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free $\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q}$-module $\text{H}{}^1\text{(X,}\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$. Main results are; Theorem 1.1: X²dY and YdX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{, Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$; Theorem 1.2: YdX, X²dY, Y^?1dX, Y^?2dX and XY^?2dX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{? (Y = 0), Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$, where $\text{X}$ is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.