Note on multidimensional empirical processes for φ-mixing random vectors

In 1974, Sen proved weak convergence of the empirical processes (in the J₁-topology on D^p[0, 1]) for a stationary φ-mixing sequence of stochastic p( 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants {φ_n}, i.e., φ_n = O(n^−1−δ) for some δ > 0.     

Asymptotic properties of Cramér-Smirnov statistics—A new approach

Let Y₁,…, Y_n be independent identically distributed random variables with distribution function F(x, θ), θ = (θ′₁, θ′₂), where θ_i (i = 1, 2) is a vector of p_i components, p = p₁ + p₂ and for θI, an open interval in ^p, F(x, θ) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under H_n: θ₁ = θ₁₀ + n^−1/2γ, θ₂ unspecified, where γ is a given vector independent of n, is the distribution of a sum of weighted noncentral χ₁² variables whose weights are eigenvalues of a covariance function of a Gaussian process and noncentrality parameters are Fourier coefficients of the mean function of the Gaussian process. Further, the author exploits the special form of the covariance function by using perturbation theory to obtain the noncentrality parameters and the weights. The technique is applicable to other goodness-of-fit statistics such as U² [G. S. Watson, Biometrika 48 (1961), 109–114].     

Removable Singularities for Lu=Ψ(u) and Orlicz Capacities

Suppose L is a second order elliptic differential operator in ^d and let α>1. Baras and Pierre have proved in 1984 that Γ is removable for Lu=u^α if and only if its Bessel capacity Cap_2, α′(Γ)=0. We extend this result to a general equation Lu=Ψ(u) where Ψ(u) is an increasing convex function subject to Δ₂ and ₂ conditions. Namely, we prove that Γ is removable for Lu=Ψ(u) if and only if its Orlicz capacity is zero, that is, the integral ∫_B dx Ψ(∫_Γ |x−y|^2−d ν(dy)) is equal to 0 or ∞ for every measure ν concentrated on Γ, where B stands for any ball containing Γ.     

From Constructive Field Theory to Fractional Stochastic Calculus. (I) An introduction: Rough Path Theory and Perturbative Heuristics

Let B = (B ₁(t), . . . , B _d (t)) be a d-dimensional fractional Brownian motion with Hurst index α ≤ 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low H?lder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using “standard” tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a non-perturbative effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read for a second time the companion article (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (II) The rough path for \frac16 < a < \frac14{\frac{1}{6} < \alpha < \frac{1}{4}}: constructive proof of convergence, 2011, preprint) for the constructive proofs.     

Capture in linear functional differential games of pursuit

The problem of capture in a pursuit game which is described by a linear retarded functional differential equation is considered. The initial function belongs to the Sobolev space W₂⁽¹⁾. The target is either a subset of W₂⁽¹⁾ a point in W₂⁽¹⁾, a subset of the Euclidean space Eⁿ or a point of Eⁿ. There is capture if the initial function can be forced to the target by the pursuer no matter what the quarry does. The concept of capture therefore formalizes the concepts of controllability under unpredictable disturbances. This is proved to be equivalent to the controllability of an associated linear retarded functional differential equation. There is nothing in (2) (6) or (7) below which restricts the control sets to be of the same dimension as the phase space. Our results can be applied in (2) for example, if the constraint sets Q′, P′ are subsets of E^m and Eⁱ respectively with q(t) = C(t) q′(t), − p(t) = B(t) p′(t), q′(t) ε E^mp′(t) ε E^r and B(t) is an n × r′-matrices and C(t) an n × m-matrix.     

Strong Consistency of Bayes Estimates in Stochastic Regression Models

Under minimum assumptions on the stochastic regressors, strong consistency of Bayes estimates is established in stochastic regression models in two cases: (1) When the prior distribution is discrete, the p.d.f.fof i.i.d. random errors is assumed to have finite Fisher informationI=∫^∞_−∞(f′)²/f dx<∞; (2) for general priors, we assumefis strongly unimodal. The result can be considered as an application of a theorem of Doob to stochastic regression models.     

Interpretation of the Lavrentiev phenomenon by relaxation

We consider functionals of the calculus of variations of the form F(u)= ∝₀¹ f(x, u, u′) dx defined for u ε W^1,∞(0, 1), and we show that the relaxed functional with respect to weak W^1,1(0, 1) convergence can be written as

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, where the additional term L(u), called the Lavrentiev term, is explicitly identified in terms of F.     

Convergence of Certain Functionals of Integral Fractional Processes

In this paper we consider the asymptotic behavior of functionals of processes of the form ∫ ₀ ^t u _s dB _s ^H , where B ^H is a fractional Brownian motion with Hurst parameter H, and u is a process with finite q-variation, q<1/(1−H). We establish the stable convergence of the corresponding fluctuations.     

The set of values of b for which there exist block designs with b blocks

Let B denote the set of values of b for which there exists a block design with b blocks and for k3, let B_k denote the subset of B determined by the designs with block size k. We present some information about B and the sets B_k. In particular, we discuss, for certain integers h, the question as to whether there exist integers k and k^′ such that the equation b^′=b+h has infinitely many solutions b,b^′ satisfying bB_k and b^′B_k^′. The study is restricted to the case λ=1.     

The Vector QD Algorithm for Smooth Functions (f, f′)

We deal with the functionz(f(z), f′(z)) wheref(z)=∑_i0 a_izⁱ, (a_i ) with lim_i→∞ a_i+1×a_i−1/(a_i)²=q. We investigate the convergence of the vector QD algorithm. We give the asymptotic behaviour of the generalized Hankel determinants. A convergence result on the vector orthogonal polynomials is proved.     

Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter

We define a stochastic integral with respect to fractional Brownian motion B^H with Hurst parameter that extends the divergence integral from Malliavin calculus. For this extended divergence integral we prove a Fubini theorem and establish versions of the formulas of Itô and Tanaka that hold for all . Then we use the extended divergence integral to show that for every and all , the Russo–Vallois symmetric integral exists and is equal to , where G^′=g, while for , does not exist.     

On the Rate of Convergence of Two Bernstein–Bézier Type Operators for Bounded Variation Functions

In this paper we study the rate of convergence of two Bernstein–Bézier type operatorsB^(α)_nandL^(α)_nfor bounded variation functions. By means of construction of suitable functions and the method of Bojanic and Vuillemier (J. Approx. Theory31(1981), 67–79), using some results of probability theory, we obtain asymptotically optimal estimations ofB^(α)_nandL^(α)_nfor bounded variation functions at points of continuity and points of discontinuity.     

Laguerre Expansions of Gel′fand-Shilov Spaces

The subspaces G_α, G^β, and G^β_α (α, β ≥ 0)of Schwartz′ space S⁺ in (0, + ∞) are associated with the Hankel transform in the same way as the Gel′fand-Shilov spaces S_α, S^β, and S^β_α are associated with the Fourier transform. Indeed, if we consider the Hankel transform H_γ (γ < −1) defined by _γ(ƒ)(t) = ∫^∞₀ (xt)^−γ/2x^γJ_γ([formula]) ƒ(x) dx then _γ is an isomorphism from G_α, G^β, and G^β_α onto G^α, G_β, and G^α_β respectively. So. the spaces G^α_α are invariant for _γ. In this paper, we characterize the spaces G^α_α (α > 1) in terms of their Fourier-Laguerre coefficients. Also, we characterize the range of the Fourier-Laplace operator _D defined by _D(ƒ)(w) = ∫^∞₀ ƒ(t) e^{−(1/2)((1 + w)/(1 − w))t} for w D = {w : |w| ≤ 1} when it acts on the space G^α_α.     

Stochastic differential equations with fractal noise

Stochastic differential equations in ?ⁿ with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W^β₂ for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in W^α ₂ for 0 < α < 1. Linear equations are considered as a special case. This approach leads to fast computer algorithms basing on Picard's iteration method. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case

Under general growth assumptions, that include some cases of linear growth, we prove existence of Lipschitzian solutions to the problem of minimizing ∫_a^bL(x(s),x′(s)) ds with the boundary conditions x(a)=A, x(b)=B.     

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index {{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}

Let B?=?(B ₁(t), . . . ,B _d (t)) be a d-dimensional fractional Brownian motion with Hurst index ???<?1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low H?lder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using ??standard?? tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates and call for an extension of Gaussian tools such as, for instance, the Malliavin calculus. After a first introductory paper (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics, 2011), this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as Lévy area. A summary in French may be found in Unterberger (Mode d??emploi de la théorie constructive des champs bosoniques, avec une application aux chemins rugueux, 2011).     

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.     

Asymptotic distributions of regression and autoregression coefficients with martingale difference disturbances

In this paper a form of the Lindeberg condition appropriate for martingale differences is used to obtain asymptotic normality of statistics for regression and autoregression. The regression model is y_t = Bz_t + v_t. The unobserved error sequence {v_t} is a sequence of martingale differences with conditional covariance matrices {Σ_t} and satisfying sup_{t=1,…, n} {v′_tv_tI(v′_tv_t>a) |z_t, v_t−1, z_t−1, …} 0 as a → ∞. The sample covariance of the independent variables z₁, …, z_n, is assumed to have a probability limit M, constant and nonsingular; max_t=1,…,nz′_tz_t/n 0. If (1/n)Σ_t=1ⁿΣ_t Σ, constant, then √nvec( _n−B) N(0,M⁻¹Σ) and _n Σ. The autoregression model is x_t = Bx_{t − 1} + v_t with the maximum absolute value of the characteristic roots of B less than one, the above conditions on {v_t}, and (1/n)Σ_t=max(r,s)+1(Σ_tv_t−1−rv′_t−1−s) δ_rs(ΣΣ), where δ_rs is the Kronecker delta. Then √nvec( _n−B) N(0,Γ⁻¹Σ), where Γ = Σ_{s = 0}^∞B^sΣ(B′)^s.     

An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise

We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion W. For this algorithm we derive an error bound in terms of its number of evaluations of one-dimensional components of W. The rate of convergence depends on the spatial dimension of the heat equation and on the decay of the eigenfunctions of the covariance of W. According to known lower bounds, our algorithm is optimal, up to a constant, and this optimality cannot be achieved by uniform time discretizations. AMS subject classification (2000)  60H15, 60H35, 65C30     

Stochastic integration on partially ordered sets

The stochastic integral is introduced with respect to a stochastic process X = (X_s)_sεV, where V is any general partially ordered set satisfying some mild regularity conditions. As important examples the stochastic integral is constructed with respect to a class of Gaussian processes having similarities to the Brownian motion on the real line, and also with respect to L²-martingales under an assumption of conditional independence on the underlying σ-fields.