Exact Rate Of Convergence Research Articles

Convergence Rate of Extremes for the General Error Distribution

Let {Xn, n ≥ 1} be an independent, identically distributed random sequence with each Xn having the general error distribution. In this paper we derive the exact uniform convergence rate of the distribution of the maximum to its extreme value limit.

Consistency of the kernel density estimator: a survey

Various consistency proofs for the kernel density estimator have been developed over the last few decades. Important milestones are the pointwise consistency and almost sure uniform convergence with a fixed bandwidth on the one hand and the rate of convergence with a fixed or even a variable bandwidth on the other hand. While considering global properties of the empirical distribution functions is sufficient for strong consistency, proofs of exact convergence rates use deeper information about the underlying empirical processes. A unifying character, however, is that earlier and more recent proofs use bounds on the probability that a sum of random variables deviates from its mean.

Computing the Minimum-Phase Filter Using the QL-Factorization

We investigate the QL-factorization of a time-invariant convolutive filtering matrix and show that this factorization not only provides the finite length equivalent to the minimum-phase filter, but also gives the associated all-pass filter. The convergence properties are analyzed and we derive the exact convergence rate and an upper bound for a simple single-input single-output system with filter length . Finally, this upper bound is used to derive an approximation of the convergence rate for systems of arbitrary length. Implementation-wise, the method has the advantage of being numerically stable and straight forward to extend to the multiple-input multiple-output case. Furthermore, due to the existence of fast QL-factorization methods, it is possible to compute the filters efficiently.

On Regularly Varying and History-Dependent Convergence Rates of Solutions of a Volterra Equation with Infinite Memory

We consider the rate of convergence to equilibrium of Volterra integrodifferential equations with infinite memory. We show that if the kernel of Volterra operator is regularly varying at infinity, and the initial history is regularly varying at minus infinity, then the rate of convergence to the equilibrium is regularly varying at infinity, and the exact pointwise rate of convergence can be determined in terms of the rate of decay of the kernel and the rate of growth of the initial history. The result is considered both for a linear Volterra integrodifferential equation as well as for the delay logistic equation from population biology.

Asymptotics for the Moment Convergence of U-Statistics in LIL

Let be a -statistic based on a symmetric kernel and i.i.d. samples . In this paper, the exact moment convergence rates in the law of the iterated logarithm and the law of the logarithm of are obtained, which extend previous results concerning partial sums.

Second Moment Convergence Rates for Uniform Empirical Processes

Let be a sequence of independent and identically distributed -distributed random variables. Define the uniform empirical process as , . In this paper, we get the exact convergence rates of weighted infinite series of .

Discretizing the fractional Lévy area

In this article, we give sharp bounds for the Euler discretization of the Lévy area associated to a d -dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean square convergence rate of the Euler scheme, depending on the Hurst parameter H ∈ ( 1 / 4 , 1 ) . For H < 3 / 4 the exact convergence rate is n − 2 H + 1 / 2 , where n denotes the number of the discretization subintervals, while for H = 3 / 4 it is n − 1 log ( n ) and for H > 3 / 4 the exact rate is n − 1 . Moreover, we also show that a trapezoidal scheme converges (at least) with the rate n − 2 H + 1 / 2 . Finally, we derive the asymptotic error distribution of the Euler scheme. For H ≤ 3 / 4 one obtains a Gaussian limit, while for H > 3 / 4 the limit distribution is of Rosenblatt type.

Moment convergence rates in the law of the logarithm for dependent sequences

Let {Xn; n ≥ 1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set Sn = Σk=1nXk, Mn = maxk≤n |Sk|, n ≥ 1. Suppose σ2 = EX12 + 2Σk=2∞ EX1Xk (0 < σ < ∞). In this paper, the exact convergence rates of a kind of weighted infinite series of E{Mn−σɛ√n log n}+ and E{|Sn| − σɛ√n log n}+ as ɛ ↘ 0 and E{σɛ√π2π/8logn − Mn}+ as ɛ ↗ ∞ are obtained.

Convergence studies on static and dynamic analysis of beams by using the U-transformation method and finite difference method

The static and dynamic analyses of simply supported beams are studied by using the U-transformation method and the finite difference method. When the beam is divided into the mesh of equal elements, the mesh may be treated as a periodic structure. After an equivalent cyclic periodic system is established, the difference governing equation for such an equivalent system can be uncoupled by applying the U-transformation. Therefore, a set of single-degree-of-freedom equations is formed. These equations can be used to obtain exact analytical solutions of the deflections, bending moments, buckling loads, natural frequencies and dynamic responses of the beam subjected to particular loads or excitations. When the number of elements approaches to infinity, the exact error expression and the exact convergence rates of the difference solutions are obtained. These exact results cannot be easily derived if other methods are used instead.

A wetting and drying treatment for the Runge–Kutta discontinuous Galerkin solution to the shallow water equations

This paper proposes a wetting and drying treatment for the piecewise linear Runge–Kutta discontinuous Galerkin approximation to the shallow water equations. The method takes a fixed mesh approach as opposed to mesh adaptation techniques and applies a post-processing operator to ensure the positivity of the mean water depth within each finite element. In addition, special treatments are applied in the numerical flux computation to prevent an instability due to excessive drying. The proposed wetting and drying treatment is verified through comparisons with exact solutions and convergence rates are examined. The obtained orders of convergence are close to or approximately equal to 1 for solutions with discontinuities and are improved for smooth solutions. The combination of the proposed wetting and drying treatment and a TVB slope limiter is also tested and is found to be applicable on condition that they are applied exclusively to an element at the same Runge–Kutta step.

Quantization for Probability Measures in the Prokhorov Metric

For a probability distribution P on ${\bf R}^d$ and $n\in{\bf N}$ consider $e_n = \inf \pi (P,Q)$, where $\pi$ denotes the Prokhorov metric and the infimum is taken over all discrete probabilities Q with $|\mbox{supp}(Q) | \le n$. We study solutions Q of this minimization problem, stability properties, and consistency of empirical estimators. For some classes of distributions we determine the exact rate of convergence to zero of the nth quantization error $e_n$ as $n \rightarrow\infty$.

Milstein’s type schemes for fractional SDEs

Weighted power variations of fractional Brownian motion B are used to compute the exact rate of convergence of some approximating schemes associated to one-dimensional stochastic differential equations (SDEs) driven by B. The limit of the error between the exact solution and the considered scheme is computed explicitly.

Exact rates in log law for positively associated random variables

Let { X n ; n ⩾ 1 } be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set S n = ∑ k = 1 n X k , M n = max k ⩽ n | S k | , n ⩾ 1 . Suppose σ 2 = E X 1 2 + 2 ∑ k = 2 ∞ E X 1 X k . In this paper, we study the exact convergence rates of a kind of weighted infinite series of P { M n ⩾ ε σ n log n } , P { | S n | ⩾ ε σ n log n } and I { | S n | ⩾ ε σ n log n } as ε ↘ 0 , respectively.

Empirical multifractal moment measures of self-similar measure for q &lt; 0*

For q ≥ 0, Olsen [1] has attained the exact rate of convergence of the L q -spectrum of a self-similar measure and showed that the so-called empirical multifractal moment measures converges weakly to the normalized multifractal measures. Unfortunately, nothing is known for q < 0. Indeed, the problem of analysing the L q - spectrum for q < 0 is generally considered significantly more difficult since the L q -spectrum is extremely sensitive to small variations of μ for q < 0. In [2] we showed that self-similar measures satisfying the Open Set Condition (OSC) are Ahlfors regular and, using this fact, we obtained the exact rate of convergence of the L q -spectrum of a self-similar measure satisfying the OSC for q < 0. In this paper, we apply the results from [2] to show the empirical multifractal q’th moment measures of self-similar measures satisfying the OSC converges weakly to the normalized multifractal Hausdorff measures for q < 0.

Fourier–Padé Approximants for Nikishin Systems

We study type I Fourier–Padé approximation for certain systems of functions formed by the Cauchy transform of finite Borel measures supported on bounded intervals of the real line. This construction is similar to type I Hermite–Padé approximation. Instead of power series expansions of the functions in the system, we take their development in a series of orthogonal polynomials. We give the exact rate of convergence of the corresponding approximants. The answer is expressed in terms of the extremal solution of an associated vector-valued equilibrium problem for the logarithmic potential.

Exact rates in density support estimation

Let f be an unknown multivariate probability density with compact support S f . Given n independent observations X 1 , … , X n drawn from f , this paper is devoted to the study of the estimator S ˆ n of S f defined as unions of balls centered at the X i and of common radius r n . To measure the proximity between S ˆ n and S f , we employ a general criterion d g , based on some function g , which encompasses many statistical situations of interest. Under mild assumptions on the sequence ( r n ) and some analytic conditions on f and g , the exact rates of convergence of d g ( S ˆ n , S f ) are obtained using tools from Riemannian geometry. The conditions on the radius sequence are found to be sharp and consequences of the results are discussed from a statistical perspective.

Exact Convergence Rate for the Maximum of Standardized Gaussian Increments

We prove an almost sure limit theorem on the exact convergence rate of the maximum of standardized gaussian random walk increments. This gives a more precise version of Shao's theorem (Shao, Q.-M., 1995. On a conjecture of Révész. Proc. Amer. Math. Soc. 123, 575-582) in the gaussian case.

Convergence to self-similar solutions for a semilinear parabolic equation

We study the behavior of solutions of the Cauchy problem for a parabolic equation with power nonlinearity. Our concern is the rate of convergence of solutions to forward self-similar solutions. We determine the exact rate of convergence which turns out to depend on the spatial decay rate of initial data.

Exact convergence rates in the central limit theorem for a class of martingales

We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and, for stationary sequences, the rate n−1/2log n is reached.

The exact rate of convergence of the [formula omitted]-spectra of self-similar measures for [formula omitted

The L q -spectrum of a Borel measure is one of the key objects in multifractal analysis, and it is widely believed that L q -spectrum associated with a fractal measure encode important information about the underlying dynamics and geometry. The study of the L q -spectrum therefore plays a fundamental role in the understanding of dynamical systems or fractal measures. For q ⩾ 0 Olsen [L. Olsen, Empirical multifractal moment measures and moment scaling functions of self-similar multifractals, Math. Proc. Cambridge Philos. Soc. 133 (2002) 459–485] recently determined the exact rate of convergence of the L q -spectra of a self-similar measure satisfying the Open Set Condition (OSC). Unfortunately, nothing is known about the rate of convergence for q < 0 . Indeed, the problem of analysing L q -spectra for q < 0 is generally considered significantly more difficult since the L q -spectra are extremely sensitive to small variations in the distribution of μ for q < 0 . The purpose of this paper is to overcome these obstacles and to investigate the more difficult problem of determining the exact rate of convergence of the multifractal L q -spectra of a self-similar measure satisfying the OSC for q < 0 .