Convergence of linear functions of Pfeifer records

The probabilistic behaviour of partial sums of upper and lower records has been studied in the literature. In this article, we take a broader view and study partial sums of record like sequences. We show that such sequences converge in distribution to normal and lognormal distribution. In particular our results apply to Pfeifer records. We also show the strong convergence of partial sums of lower Pfeifer records under suitable assumptions.     

On limit laws for sums of Pfeifer records

The problem of determining limiting distributions for sums of records has been studied by several authors who have considered a variety of assumptions sufficient to ensure that sums of records properly normalized will converge to a non-degenerate distribution. As a parallel to these endeavors, it is of interest to establish conditions under which the sum of Pfeifer records, properly normalized, converges. Pfeifer records are defined under the assumption that initial observations are i.i.d. with common survival function and following the (n−1)-th record value the observations are assumed to have survival function ,n=1,2,.... The study of the asymptotic behavior of sums of Pfeifer records constitutes a natural generalization of work on sums of classical records. The present paper introduces conditions under which the limit distribution of sums of Pfeifer records is non-degenerate.      

关于记录值序列部分和的渐近正态性

证明了 Burr分布和 Frechét分布的记录值序列的部分和是渐进对数正态的 ,从而解决了文[2 ]中的一个悬而未决的问题     

Partial Sum Process for Records

Suppose the upper records from a sequence of i.i.d. random variables is in the domain of attraction of a normal distribution. Consider the D(0,1]-valued process {Z_n(·)} constructed by usual interpolation of the partial sums of the records. We prove that under some mild conditions, {Z_n} converges to a limiting Gaussian process in D(0,1]. As a consequence, the partial sums of records is asymptotically normal. AMS 2000 Subject Classification Primary—60F17 Secondary—60G70     

A remark on LSL for weighted sums of i.i.d random elements

In the paper, the upper bound and lower bound of the law of the single logarithm (LSL) are established under the condition that the sequence of the normalized weighted sums of random elements is bounded in probability. The main result improves the upper bound in [Sung, S.H., 2009. A law of the single logarithm for weighted sums of i.i.d. random elements. Statist. Probab. Lett., 79, 1351–1357] and hence extends the result in [Chen, P., Gan, S., 2007. Limiting behavior of weighted sums of i.i.d. random variables. Statist. Probab. Lett., 77, 1589–1599].     

Laws of the iterated logarithm for partial sum processes indexed by functions

We establish a bounded and a compact law of the iterated logarithm for partial sum processes indexed by classes of functions. We assume a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables.     

Invariance principles for non-isotropic long memory random fields

We prove that when a random field with bounded spectral density satisfies a Donsker type theorem, its dilated and properly normalised spectral field admits a weak limit. We apply this result to establish the convergence of partial sums for random fields obtained by filtering a white noise. In particular, we prove the convergence of partial sums for strongly-dependent fields whose memory does not satisfy the regularity conditions previously met in the literature.     

Bounds for Asian basket options

In this paper we propose pricing bounds for European-style discrete arithmetic Asian basket options in a Black and Scholes framework. We start from methods used for basket options and Asian options. First, we use the general approach for deriving upper and lower bounds for stop-loss premia of sums of non-independent random variables as in Kaas et al. [Upper and lower bounds for sums of random variables, Insurance Math. Econom. 27 (2000) 151–168] or Dhaene et al. [The concept of comonotonicity in actuarial science and finance: theory, Insurance Math. Econom. 31(1) (2002) 3–33]. We generalize the methods in Deelstra et al. [Pricing of arithmetic basket options by conditioning, Insurance Math. Econom. 34 (2004) 55–57] and Vanmaele et al. [Bounds for the price of discrete sampled arithmetic Asian options, J. Comput. Appl. Math. 185(1) (2006) 51–90]. Afterwards we show how to derive an analytical closed-form expression for a lower bound in the non-comonotonic case. Finally, we derive upper bounds for Asian basket options by applying techniques as in Thompson [Fast narrow bounds on the value of Asian options, Working Paper, University of Cambridge, 1999] and Lord [Partially exact and bounded approximations for arithmetic Asian options, J. Comput. Finance 10 (2) (2006) 1–52]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.     

Approximation of Cauchy-Type Integrals

We investigate approximations of analytic functions determined by Cauchy-type integrals in Jordan domains of the complex plane. We develop, modify, and complete (in a certain sense) our earlier results. Special attention is given to the investigation of approximation of functions analytic in a disk by Taylor sums. In particular, we obtain asymptotic equalities for upper bounds of the deviations of Taylor sums on the classes of -integrals of functions analytic in the unit disk and continuous in its closure. These equalities are a generalization of the known Stechkin's results on the approximation of functions analytic in the unit disk and having bounded rth derivatives (here, r is a natural number).On the basis of the results obtained for a disk, we establish pointwise estimates for the deviations of partial Faber sums on the classes of -integrals of functions analytic in domains with rectifiable Jordan boundaries. We show that, for a closed domain, these estimates are exact in order and exact in the sense of constants with leading terms if and only if this domain is a Faber domain.     

Generalized Law of the Iterated Logarithm and Its Convergence Rate

Abstract

This article considers the partial sums from a sequence of independent and identically distributed random variables. It is well-known that the Hartman-Wintner law of the iterated logarithm holds if and only if the second moment exists. This article studies the generalized law of the iterated logarithm for the partial sums when they are normalized by a sequence of constants that are regularly varying with index 1/2. As a result, two equivalent conditions for the law are obtained.     

B值独立随机元重对数律收敛速度的一般形式

本文讨论了Ｂ值独立同分布（ｉｉｄ）随机元重对数律收敛速度的一般形式,使得Ｄａｖｉｓ＾「１」及Ｇｕｔ＾「２,３」中的一些结果成为特款,同时减弱了Ｄａｖｉｓ结果中的矩条件,并且得到了Ｂ值ｉｉｄ随机元满足有界重对数律的一个充分性条件。作为应用,我们给出了随机足标和的相应结果。     

Convergence results for a normalized triangular array of symmetric random variables

For a triangular array of symmetric random variables (without any integrability condition) we replace the classical assumption of row-wise independence by that of row-wise joint symmetry. Under this weaker assumption we prove some results concerning the convergence in distribution of a suitable sequence of randomly normalized sums to the standard normal distribution. Then we exhibit a class of row-wise independent triangular arrays for which the ordinary sums fail to converge in distribution, while our results enable us to affirm the convergence in distribution of the normalized sums.     

Isoperimetric control of the Steklov spectrum

We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.     

Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables

We derive exponential bounds for the tail of the distribution of normalized sums of triangular arrays of random variables, not necessarily independent, under the law of ordinary logarithm.

Furthermore, we provide estimates for partial sums of triangular arrays of independent random variables belonging to suitable grand Lebesgue spaces and having heavy-tailed distributions.

     

Convergence of multiple fourier series for functions of bounded variation

For functions of bounded variation in the sense of Hardy, we consider the pointwise convergence of the partial sums of Fourier series over a given sequence of bounded sets in the space of harmonics. We obtain sufficient conditions for convergence; necessary and sufficient conditions are obtained for the case in which these sets are convex with respect to each coordinate direction. The Pringsheim convergence of Fourier series in this problem was established by Hardy. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 583–595, April, 1997. Translated by S. A. Telyakovskii and V. N. Temlyakov     

On a problem of littlewood

The theorem on the tending to zero of coefficients of a trigonometric series is proved when theL ¹-norms of partial sums of this series are bounded. It is shown that the analog of Helson's theorem does not hold for orthogonal series with respect to the bounded orthonormal system. Two facts are given that are similar to Weis' theorem on the existence of a trigonometric series which is not a Fourier series and whoseL ¹-norms of partial sums are bounded.     

Joint limit distributions of exceedances point processes and partial sums of gaussian vector sequence

In this paper, we study the joint limit distributions of point processes of exceedances and partial sums of multivariate Gaussian sequences and show that the point processes and partial sums are asymptotically independent under some mild conditions. As a result, for a sequence of standardized stationary Gaussian vectors, we obtain that the point process of exceedances formed by the sequence (centered at the sample mean) converges in distribution to a Poisson process and it is asymptotically independent of the partial sums. The asymptotic joint limit distributions of order statistics and partial sums are also investigated under different conditions.     

On algebrability of nonabsolutely convergent series

We prove that the set of all complex series which are nonabsolutely convergent is c-algebrable. We establish a similar result for the set of all divergent complex series with bounded partial sums.     

B值随机元和的完全收敛性

在B值随机元随机有界于一非负随机变量的情况下．讨论了B值独立随机元序列非随机足标和的完全收敛性，作为应用，得到了随机足标和完全收敛性的相应结果，将[5]中的一些结果推广到B值独立随机无情形，同时使[3]（d=1），[4]，[6]的相应结果成为特例．     

The Asymptotic Distributions of Sums of Records

LetX ₁,X ₂... be a sequence of i.i.d. random variables and let be the associated (upper) record sequence. Resnick (1973) identified the class of all possible limit distributions for . Here we focus on sums of records, . We describe three cases in which T _n can be normalized to have a non-trivial limiting distribution. The problem of identifying all possible limit laws for normalized sums of records remains open.