Branching one-dimensional periodic diffusion processes

Let X be a nonsingular conservative one-dimensional periodic diffusion process, λ₀ its principal eigenvalue and X a binary splitting branching diffusion process with nonbranching part X. For each α > λ₀ we have two positive martingales Wⁱ_t(α), i = 1, 2, of X attached to the two positive α-harmonic functions of X. The main purpose of this paper is to show that their limit random variables are positive for all α? (λ₀, α_i), where α_i are some constants greater than λ₀.     

The supercritical multitype age-dependent branching process

Let X(t) = (X₁(t),…, X_p(t)) be a p-dimensional supercritical age-dependent branching process. For an appropriate α > 0, necessary and sufficient conditions are found for X(t) e^?αt to converge to a nondegenerate random vector W. Several properties of W are also determined.     

Global solutions of first order linear systems of ordinary differential equations with distributional coefficients

With the help of our distributional product we define four types of new solutions for first order linear systems of ordinary differential equations with distributional coefficients. These solutions are defined within a convenient space of distributions and they are consistent with the classical ones. For example, it is shown that, in a certain sense, all the solutions of X₁′=(1+δ)X₁−X₂, X₂′=(2+δ′)X₁+4X₂+δ″ have the form X₁(t)=c₁(e^2t−2e^3t)−14e^3t−δ(t), X₂(t)=c₁(4e^3t−e^2t−δ(t))+28e^3t−18δ(t)+δ′(t), where c₁ is an arbitrary constant and δ is the Dirac measure concentrated at zero. In the spirit of our preceding papers (which concern ordinary and partial differential equations) and under certain conditions we also prove existence and uniqueness results for the Cauchy problem.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.     

On limit distributions of first crossing points of Gaussian sequences

Let {X_k, k?Z} be a stationary Gaussian sequence with EX₁ – 0, EX²_k = 1 and EX₀X_k = r_k. Define τ_x = inf{k: X_k >– βk} the first crossing point of the Gaussian sequence with the function – βt (β > 0). We consider limit distributions of τ_x as β→0, depending on the correlation function r_k. We generalize the results for crossing points τ_x = inf{k: X_k >β?(k)} with ?(– t)?t^γL(t) for t→∞, where γ > 0 and L(t) varies slowly.     

Automorphisms of real rational surfaces and weighted blow-up singularities

Let X be a singular real rational surface obtained from a smooth real rational surface by performing weighted blow-ups. Denote by Aut(X) the group of algebraic automorphisms of X into itself. Let n be a natural integer and let e = [e ₁, . . . , e _? ] be a partition of n. Denote by X ^e the set of ?-tuples (P ₁, . . . , P _? ) of disjoint nonsingular curvilinear subschemes of X of orders (e ₁, . . . , e _? ). We show that the group Aut(X) acts transitively on X ^e . This statement generalizes earlier work where the case of the trivial partition e = [1, . . . , 1] was treated under the supplementary condition that X is nonsingular. As an application we classify singular real rational surfaces obtained from nonsingular surfaces by performing weighted blow-ups.     

Asymptotic distributions of functions of the eigenvalues of sample covariance matrix and canonical correlation matrix in multivariate time series

Let S = (1/n) Σ_t=1ⁿ X(t) X(t)′, where X(1), …, X(n) are p × 1 random vectors with mean zero. When X(t) (t = 1, …, n) are independently and identically distributed (i.i.d.) as multivariate normal with mean vector 0 and covariance matrix Σ, many authors have investigated the asymptotic expansions for the distributions of various functions of the eigenvalues of S. In this paper, we will extend the above results to the case when {X(t)} is a Gaussian stationary process. Also we shall derive the asymptotic expansions for certain functions of the sample canonical correlations in multivariate time series. Applications of some of the results in signal processing are also discussed.     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

Estimation of the Gauss–Markov process from observation of its sign

Let X(t) be the ergodic Gauss–Markov process with mean zero and covariance function e^?|τ|. Let D(t) be +1, 0 or ?1 according as X(t) is positive, zero or negative. We determine the non-linear estimator of X(t₁) based solely on D(t), ?T ? t ? 0, that has minimal mean–squared error ε²(t₁, T). We present formulae for ε²(t₁, T) and compare it numerically for a range of values of t₁ and T with the best linear estimator of X(t₁) based on the same data.     

Time-Varying Fractionally Integrated Processes with Finite or Infinite Variance and Nonstationary Long Memory

Recently, Philippe et al. (C.R. Acad. Sci. Paris. Ser. I 342, 269–274, 2006; Theory Probab. Appl., 2007, to appear) introduced a new class of time-varying fractionally integrated filters A(d)x _t =∑ _j=0 ^∞ a _j (t)x _t?j , B(d)x _t =∑ _j=0 ^∞ b _j (t)x _t?j depending on arbitrary given sequence d=(d _t ,t∈?) of real numbers, such that A(d)^?1=B(?d), B(d)^?1=A(?d) and such that when d _t ≡d is a constant, A(d)=B(d)=(1?L) ^d is the usual fractional differencing operator. Philippe et al. studied partial sums limits of (nonstationary) filtered white noise processes X _t =B(d)ε _t and Y _t =A(d)ε _t in the case when (1) d is almost periodic having a mean value $\bar{d}\in (0,1/2)$ , or (2) d admits limits d _±=lim? _t→±∞ d _t ∈(0,1/2) at t=±∞. The present paper extends the above mentioned results of Philippe et al. into two directions. Firstly, we consider the class of time-varying processes with infinite variance, by assuming that ε _t ,t∈? are iid rv’s in the domain of attraction of α-stable law (1<α≤2). Secondly, we combine the classes (1) and (2) of sequences d=(d _t ,t∈?) into a single class of sequences d=(d _t ,t∈?) admitting possibly different Cesaro limits $\bar{d}_{\pm}\in(0,1-(1/\alpha))$ at ±∞. We show that partial sums of X _t and Y _t converge to some α-stable self-similar processes depending on the asymptotic parameters $\bar{d}_{\pm}$ and having asymptotically stationary or asymptotically vanishing increments.     

Arithmétique d'une famille de corps cubiques

In this Note, we study the family of polynomials: P(X)=X³?nX²?n, with n=3^sp₁…p_t, where s=0 or 1 and where the p_i, for 1?i?t, are distinct prime numbers and all different from 3, and (4n²+27)/9^s is squarefree. For this family, we determine the arithmetic invariants of the number field $\text{K=}\text{Q}\text{(α),}$ where α is the only real root of the polynomial P(X), and we find the following results: $\text{O}{}_{\mathrm{K}}\text{=}\text{Z}\text{[α]}$ is the ring of integers of K, d_K=?n²(4n²+27) is the discriminant of K; ε=α²+1 is the fundamental unit of O_K and R_K=Log(α²+1) is the regulator of K. To cite this article: O. Lahlou, M. El Hassani Charkani, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Novikov homology,jump loci and Massey products

Let X be a finite CW complex, and ρ: π ₁(X) → GL(l, ?) a representation. Any cohomology class α ∈ H ¹(X, ?) gives rise to a deformation γ _t of ρ defined by γ _t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ _gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case when X is a Kähler manifold and ρ is semi-simple. If α ∈ H ¹(X, ?) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ _gen. We investigate the dependence of these numbers on α and prove that they are constant in the complement to a finite number of proper vector subspaces in H ¹(X, ?).     

Asymptotic expansions in the integral and local limit theorems in banach spaces with applications to ω-statistics

LetB be a real separable Banach space and letX,X ₁,X ₂,...∈B denote a sequence of independent identically distributed random variables taking values inB. DenoteS _n =n ^?1/2(X ₁+...X _n ). Let π:B→R be a polynomial. We consider (truncated) Edgeworth expansions and other asymptotic expansions for the distribution function of the r.v. π(S _n ) with uniform and nonuniform bounds for the remainder terms. Expansions for the density of π(S _n ) and its higher order derivatives are derived as well. As an application of the general results we get expansions in the integral and local limit theorems for ω-statistics $$\omega _n^p (q)\mathop { = n^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} }\limits^\Delta \smallint _{(0,1)} \{ F_n (x) - x\} ^p q(x)dx$$ and investigate smoothness properties of their distribution functions. Herep≥2 is an even number,q: [0, 1]→[0, ∞] is a measurable weight function, andF _n denotes the empirical distribution function. Roughly speaking, we show that in order to get an asymptotic expansion with remainder termO(n ^?α), α<p/2, for the distribution function of the ω-statistic, it is sufficient thatq is nontrivial, i.e., mes{t∈(0, 1):q(t)≠0}>0. Expansions of arbitrary length are available provided the weight functionq is absolutely continuous and positive on an nonempty subinterval of (0, 1). Similar results hold for the density of the distribution function and its derivatives providedq satisfies certain very mild smoothness condition and is bounded away from zero. The last condition is essential since the distribution function of the ω-statistic has no density whenq is vanishing on an nonempty subinterval of (0, 1).     

Asymptotic behavior of the convex hull of a stationary Gaussian process?

Let X = {X(t), t ?? T} be a stationary centered Gaussian process with values in ? ^d , where the parameter set T equals ? or ?+. Let ?? _t = Cov(X ₀ ,X _t ) be the covariance function of X, and (??,?, P) be the underlying probability space. We consider the asymptotic behavior of convex hulls W _t = conv{X _u , u ?? T ?? [0, t]} as t ?? +?? and show that under the condition ??t ?? 0, t????, the rescaled convex hull (2 ln t) ^?1/2 W _t converges almost surely (in the sense of Hausdorff distance) to an ellipsoid ? associated to the covariance matrix ?? ₀. The asymptotic behavior of the mathematical expectations E f(W _t ), where f is a homogeneous function, is also studied. These results complement and generalize in some sense the results of Davydov [Y. Davydov, On convex hull of Gaussian samples, Lith. Math. J., 51(2): 171?C179, 2011].     

On the observation closest to the origin

Out of n i.i.d. random vectors in $\text{R}$^d let X¹_n be the one closest to the origin. We show that X¹_n has a nondegenerate limit distribution if and only if the common probability distribution satisfies a condition of multidimensional regular variation. The result is then applied to a problem of density estimation.     

Temps de sortie d’un intervalle borné pour l’intégrale du mouvement brownien

Let (B_t)_t≥0 be a standard Brownian motion starting at y, X_t = x+ ∫₀^tB_s, ds, x ∈ (a, b). Let us set T_ab = inf{t > 0 : X_t ∉ (a,b)}. In this paper, we compute the moments of the random variable B_Ta,b, and deduce the probability law of B_Ta,b. We show how to obtain the expectation E_(x,y)(T_ab^mB_Tabⁿ). We also explicitly determine the probabilities P_(x,y){X_Tab = a} and P_(x,y) { X_Tab = b}.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

Spaces Λα(X) and interpolation

For two pairs of rearrangement invariant spaces α = [(X₁, Y₁), (X₂, Y₂)] we give necessary and sufficient conditions for pairs (X, Y) to be weak intermediate for σ, i.e., each operator which is of weak types (X_i, Y_i), i = 1, 2, also maps X boundedly to Y. Spaces Λ_α(X) are introduced and are shown to have many of the properties that characterize Lorentz L^pq spaces. Necessary and sufficient conditions in terms of a simple function F(s, t) are given in order that (Λ_α(X), Λ_α(Y)) be weak intermediate for σ. Other properties of the function F(s, t) yield sufficient conditions and necessary conditions for interpolation theorems.