Two transformations of series that commute with compositional inversion

It is shown that the compositional inverse of either of two transformations of a given series can be determined from the compositional inverse of the series. Specifically, if t · f(t) and t · g(t) are compositional inverses, then so are t · f_k(t) and $\text{t · g}{}_{\mathrm{k}}{}^1\text{(t)}$, where f_k(t) is the kth Euler transformation of f(t) and $\text{g}{}_{\mathrm{k}}{}^1\text{(t) =}\text{g(t)}\text{(1 ? kt · g(t))}$.     

The maximal Fejér operator on the spacesH p (R×···×R)

It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH _p (R×···×R) toL _p (R ^d ) (1/2<p<∞) and is of weak type (H ₁ ^?i ,L ₁) (i=1,…,d), where the Hardy spaceH ₁ ^?i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H ₁ ^?i ?L(logL) ^d?1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH _p (R×···×R) whenever 1/2<p<∞. Thus, in casef ∈H _p (R×···×R) the Fejér means converge tof inH _p (R×···×R) norm. The same results are proved for the conjugate Fejér means, too.     

Locally adaptive fitting of semiparametric models to nonstationary time series

We fit a class of semiparametric models to a nonstationary process. This class is parametrized by a mean function μ(·) and a p-dimensional function θ(·)=(θ⁽¹⁾(·),…,θ^(p)(·))′ that parametrizes the time-varying spectral density f_θ(·)(λ). Whereas the mean function is estimated by a usual kernel estimator, each component of θ(·) is estimated by a nonlinear wavelet method. According to a truncated wavelet series expansion of θ⁽ⁱ⁾(·), we define empirical versions of the corresponding wavelet coefficients by minimizing an empirical version of the Kullback–Leibler distance. In the main smoothing step, we perform nonlinear thresholding on these coefficients, which finally provides a locally adaptive estimator of θ⁽ⁱ⁾(·). This method is fully automatic and adapts to different smoothness classes. It is shown that usual rates of convergence in Besov smoothness classes are attained up to a logarithmic factor.     

Rigidity and stability of the Leibniz and the chain rule

We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T ₁, T ₂,A: C ^k (?) → C(?) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ C ^k (?), namely, V (f · g) = (T ₁ f) · g + f · (T ₂ g) for k = 1, V (f · g) = (T ₁ f) · g + f · (T ₂ g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T ₁ f) ○ g · (T ₂ g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T ₁ f) · (Ag) + (Af) · (T ₂ g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T ₁ and T ₂ must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · Tg + B(f, g), and show under suitable conditions on B in the first case that B = 0 and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation.     

Stochastic gradient algorithm with random truncations

Let f : R^d × R^d → R be a Borel-measurable function which satisfies ∫_R^d′|f(θ, x) < ∞, ∨θ ϵ R^d, where q₀(·) is a probability measure on (R^d′, B^d′). The problem of minimization of the function f₀(θ) = ∫_R^d′(θ, x)q₀(d), θ ϵ R^d, is considered for the case when the probability measure q₀(·) is unknown, but a realization of a non-stationary random process {X_n}_n⩾1 whose single probability measures in a certain sense tend to q₀(·), is available. The random process {X_n}_n⩾1 is defined on a common probability space, R^′-valued, correlated and satisfies certain uniform mix conditions. The function f(·, ·) is completely known. A stochastic gradient algorithm with random truncations is used for the minimization of f₀(·), and its almost sure convergence is proved.     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

A generalization of bounded variation

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric system is established and the estimation of ‖S _n(·, f)-f(·)‖_{C [0,2π]} where S _n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

NONLINEAR WAVELET SMOOTHING OF ERROR DENSITY IN A SEMIPARAMETRIC REGRESSION MODEL

In this paper,a semlparametrie resresaion model in which errors are i. i. d random variables from an unknown density f( ) is considered. Based on Hall et al. (1995),a nonlinear wavelet estimation of f( ) without restrictions of continuity everywhere on f( ) is given,and the convergence rate of the estimators in L2 is obtained.     

Extreme points, support points and the Loewner variation in several complex variables

In this paper we consider extreme points and support points for compact subclasses of normalized biholomorphic mappings of the Euclidean unit ball Bn in Cn. We consider the class S0(Bn) of biholomorphic mappings on Bn which have parametric representation, i.e., they are the initial elements f (·, 0) of a Loewner chain f (z, t) = etz + ··· such that {e-tf (·, t)}t 0 is a normal family on Bn. We show that if f (·, 0) is an extreme point (respectively a support point) of S0(Bn), then e-tf (·, t) is an extreme point of S0(Bn) for t 0 (respectively a support point of S0(Bn) for t ∈[0, t0] and some t0 > 0). This is a generalization to the n-dimensional case of work due to Pell. Also, we prove analogous results for mappings which belong to S0(Bn) and which are bounded in the norm by a fixed constant. We relate the study of this class to reachable sets in control theory generalizing work of Roth. Finally we consider extreme points and support points for biholomorphic mappings of Bn generated by using extension operators that preserve Loewner chains.     

On the metric type of measurable functions and convergence in distribution

In the present paper, sequences of real measurable functions defined on a measure space ([0, 1], µ), where µ is the Lebesgue measure, are studied. It is proved that for every sequence f_n that converges to f in distribution, there exists a sequence of automorphisms S_n of ([0, 1], µ) such that f_n(S_n(t)) converges to f(t) in measure. Connection with some known results is also discussed.     

Hyperquasipolynomials for the Theta-Function

Let g be a linear combination with quasipolynomial coefficients of shifts of the Jacobi theta function and its derivatives in the argument. All entire functions f: ? → ? satisfying f(x+y)g(x?y) = α₁(x)β₁(y)+· · ·+α_r(x)β_r(y) for some r ∈ ? and α_j, β_j: ? → ? are described.     

Backward stochastic differential equations and applications to optimal control

We study the existence and uniqueness of the following kind of backward stochastic differential equation, $$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$ under local Lipschitz condition, where (Ω, ?,P, W(·), ?_t) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ?_t-adapted process, andX is ?_t-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.     

Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t?T} be observable for each x in some compact set X. Assume that Y(x, t) = θ₀f₀(x)(t) + … + θ_kf_k(x)(t) + N(t) where f_i are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with $\text{R}$ the closed convex hull of the set {(φ, f(x))_H : 6φ 6_H ≤ 1, x?X}, where (·, ·)_H is the inner product on H. It is shown that if X is convex and f_i are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c′θ can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.     

On equivalence of simple closed curves in flat surfaces

By explicit constructions,we give direct proofs of the following results: for any distinct homotopy classes of simple closed curves α and β in a closed surface of genus g 1,there exist a hyperbolic structure X and a holomorphic quadratic differential q on X such that lX(α) = lX(β),extX(α) = extX(β) and lq(α) = lq(β),where lX(·),extX(·) and lq(·) are the hyperbolic length,the extremal length and the quadratic differential length respectively.These imply that there are no equivalent simple closed curves in hyperbolic surfaces or in flat surfaces.     

Markov chain Monte Carlo Using an Approximation

This article presents a method for generating samples from an unnormalized posterior distribution f(·) using Markov chain Monte Carlo (MCMC) in which the evaluation of f(·) is very difficult or computationally demanding. Commonly, a less computationally demanding, perhaps local, approximation to f(·) is available, say f^**_x(·). An algorithm is proposed to generate an MCMC that uses such an approximation to calculate acceptance probabilities at each step of a modified Metropolis–Hastings algorithm. Once a proposal is accepted using the approximation, f(·) is calculated with full precision ensuring convergence to the desired distribution. We give sufficient conditions for the algorithm to converge to f(·) and give both theoretical and practical justifications for its usage. Typical applications are in inverse problems using physical data models where computing time is dominated by complex model simulation. We outline Bayesian inference and computing for inverse problems. A stylized example is given of recovering resistor values in a network from electrical measurements made at the boundary. Although this inverse problem has appeared in studies of underground reservoirs, it has primarily been chosen for pedagogical value because model simulation has precisely the same computational structure as a finite element method solution of the complete electrode model used in conductivity imaging, or “electrical impedance tomography.” This example shows a dramatic decrease in CPU time, compared to a standard Metropolis–Hastings algorithm.     

Ruzsa’s constant on additive functions

A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20.     

Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

Noncommutative analogs of probabilistic notions and results

Considering a random variable as a multiplication operator by a measurable function, a natural generalization consists in allowing noncommuting and unbounded operators defined on a common invariant domain with cyclic vector φ. By multiplication and addition, these operators generate a 1-algebra which in turn can be considered as a representation π of an abstract 1-(tensor) algebra. Moments are replaced by m(a₁ ··· a_n) = 〈φ, π(a₁) ··· π(a_n) φ〉. In analogy to the classical case the notions of cumulants, addition of independent random variables, and infinite divisibility are introduced, as well as Gaussianness as a generalization of normal random variables. Previous results are briefly reviewed, including a characterization of infinite divisibility. Among the new results are noncommutative analogs of versions of the central limit theorem and of Cramér's theorem. All results have direct applications to representations of Lie algebras, to quantum field theory, and to statistical mechanics.