Fourier transform methods for pathwise covariance estimation in the presence of jumps

We provide a new non-parametric Fourier procedure to estimate the trajectory of the instantaneous covariance process (from discrete observations of a multidimensional price process) in the presence of jumps extending the seminal work of Malliavin and Mancino (2002, 2009). Our approach relies on a modification of (classical) jump-robust estimators of integrated realized covariance to estimate the Fourier coefficients of the covariance trajectory. Using Fourier–Féjer inversion we reconstruct the path of the instantaneous covariance. We prove consistency and a central limit theorem (CLT) and in particular that the asymptotic estimator variance is smaller by a factor 2/3 in comparison to classical local estimators.     

Almost sure Cesàro and Euler summability of sequences of dependent random variables

For a sequence of real random variables C _α-summability is shown under conditions on the variances of weighted sums, comprehending and sharpening strong laws of large numbers (SLLN) of Rademacher-Menchoff and Cramér-Leadbetter, respectively. Further an analogue of Kolmogorov’s criterion for the SLNN is established for E _α-summability under moment and multiplicativity conditions of 4th order, which allows one to weaken Chow’s independence assumption for identically distributed square integrable random variables. The simple tool is a composition of Cesàro-type and of Euler summability methods, respectively. Received: 12 June 2006, Revised: 14 May 2007     

On the Marcinkiewicz–Zygmund Strong Law for U-Statistics

Sen (when 0<p<1) and Giné and Zin (1<p<2) have given sufficient conditions for a Marcinkiewicz–Zygmund type SLLN to hold for U-statistics. As demonstrated by the latter authors, their requirement is not necessary when U is completely degenerate or 0<p<1. Here, for p in (1, 2), a simplified proof of sufficiency is obtained which weakens their integrability condition and in the nondegenerate case also extends the permissible values of p. This, in turn, leads to improved conditions for Hoeffding's CLT (likewise for the LIL) in the non-degenerate case. Moreover, nonnecessity is shown to hold in the SLLN when U is degenerate but not completely so.     

High frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus

We prove a multivariate CLT for skewness and kurtosis of the wavelets coefficients of a stationary field on the torus. The results are in the framework of the fixed-domain asymptotics, i.e. we refer to observations of a single field which is sampled at higher and higher frequencies. We consider also studentized statistics for the case of an unknown correlation structure. The results are motivated by the analysis of high-frequency financial data or cosmological data sets, with a particular interest towards testing for Gaussianity and isotropy.     

Prokhorov Blocks and Strong Law of Large Numbers Under Rearrangements

We find conditions on a sequence of random variables to satisfy the strong law of large numbers (SLLN) under a rearrangement. It turns out that these conditions are necessary and sufficient for the permutational SLLN (PSLLN). By PSLLN we mean that the SLLN holds under almost all simple permutations within blocks the lengths of which grow exponentially (Prokhorov blocks). In the case of orthogonal random variables it is shown that Kolmogorov's condition, that is known not to be sufficient for SLLN, is actually sufficient for PSLLN. It is also shown that PSLLN holds for sequences that are strictly stationary with finite first moments. In the case of weakly stationary sequences a Gaposhkin result implies that SLLN and PSLLN are equivalent. Finally we consider the case of general norming and generalization of the Nikishin theorem. The methods of proof uses on the one hand the idea of Prokhorov blocks and Garsia's construction of product measure on the space of simple permutations, and on the other hand, a maximal inequality for permutations.     

END序列强大数律研究

本文研究了END序列的均值强大数律,获得了均值强大数律的充分条件. 作为应用,我们可以立即得到Marcinkiewicz均值和logarithmic均值强大数律.最后我们开展了有关大数定律的数据模拟工     

Implications of contrarian and one-sided strategies for the fair-coin game

We derive some results on contrarian and one-sided strategies of Skeptic for the fair-coin game in the framework of the game-theoretic probability of Shafer and Vovk [G. Shafer and V. Vovk. Probability and Finance — It’s Only a Game!, Wiley, New York, 2001]. In particular, as regards the rate of convergence of the strong law of large numbers (SLLN), we prove that Skeptic can force that the convergence has to be slower than or equal to O(n^−1/2)$O\left(n^{-1/2}\right)$. This is achieved by a very simple contrarian strategy of Skeptic. This type of result, bounding the rate of convergence from below, contrasts with more standard results of bounding the rate of SLLN from above by using momentum strategies. We also derive a corresponding one-sided result.     

Game-theoretic versions of strong law of large numbers for unbounded variables

We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (Shafer, G. and Vovk, V. 2001, Probability and Finance: It's Only a Game! (New York: Wiley)). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game-theoretic versions of SLLN largely correspond to standard measure-theoretic results. However game-theoretic proofs are different from measure-theoretic ones in the explicit consideration of various hedges. In measure-theoretic proofs existence of moments is assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.     

The strong law of large numbers for weighted averages under dependence assumptions

Strong laws of large numbers (SLLN) for weighted averages are proved under various dependence assumptions when the variables are not necessarily independent or identically distributed. The results considerably extend the existing results. Weighted versions of the Marcinkiewicz-Zygmund SLLN are also formulated and proved under a similar set up. It seems that such results are not known even for independent and identically distributed random variables.     

The class of tenable zero-balanced Pólya urns with an initially dominant subset of colors

In Kholfi and Mahmoud (2011) the class of tenable irreducible nondegenerate zero-balanced Pólya urn schemes is introduced and its asymptotic behavior in various phases is studied. In the absence of an initially dominant subset of colors, the counts of balls of all the colors satisfy multivariate central limit theorems. It is reported there that the case of an initially dominant subset of colors poses challenges requiring finer asymptotic analysis. In the present investigation we follow up on this. Indeed, we characterize noncritical cases with an initially dominant subset of colors in which not all ball counts satisfy one multivariate central limit theorem, but rather a subset of the ball counts satisfies a singular multivariate central limit theorem. The rest of the cases are critical, in which all the ball counts satisfy a multivariate central limit theorem, but under a different scaling. However, for these critical cases the Gaussian phases are delayed considerably.     

Regeneration for chains with infinite memory

Summary A regeneration structure is established for chains with infinite memory. The memory is required to decay only along a single recurrent path. When there are many recurrent paths (e.g. under conservativity) the construction yields a decomposition into regenerative recurrent classes.Research supported by NSF Grant DMS 89-01464     

Competitive Lotka-Volterra population dynamics with jumps

This paper considers competitive Lotka-Volterra population dynamics with jumps. The contributions of this paper are as follows. (a) We show that a stochastic differential equation (SDE) with jumps associated with the model has a unique global positive solution; (b) we discuss the uniform boundedness of the pth moment with p>0 and reveal the sample Lyapunov exponents; (c) using a variation-of-constants formula for a class of SDEs with jumps, we provide an explicit solution for one-dimensional competitive Lotka-Volterra population dynamics with jumps, and investigate the sample Lyapunov exponent for each component and the extinction of our n-dimensional model.     

Komlós Theorem for Unbounded Random Sets

We present the Komlós theorem for multivalued functions whose values are closed (possibly unbounded) convex subsets of a separable Banach space. Komlós theorem can be seen as a generalization of the SLLN for it deals with a sequence of integrable multivalued functions that do not have to be identically distributed nor independent. The Artstein–Hart SLLN for random sets with values in Euclidean spaces is derived from the main result. Finally, since the main theorem concerns multifunctions whose values are allowed to be unbounded, we can restate it in terms of normal integrands (random lower semicontinuous functions).     

Some upper and lower bounds on the coupon collector problem

The classical coupon collector problem is closely related to probabilistic-packet-marking (PPM) schemes for IP traceback problem in the Internet. In this paper, we study the classical coupon collector problem, and derive some upper and lower bounds of the complementary cumulative distribution function (ccdf) of the number of objects (coupons) that one has to check in order to detect a set of different objects. The derived bounds require much less computation than the exact formula. We numerically find that the proposed bounds are very close to the actual ccdf when detecting probabilities are set to the values common to the PPM schemes.     

The linearization of the central limit operator in free probability theory

We interpret the Central Limit Theorem as a fixed point theorem for a certain operator, and consider the problem of linearizing this operator. In classical as well as in free probability theory [VDN92], we consider two methods giving such a linearization, and interpret the result as a weak form of the CLT. In the classical case the analysis involves dilation operators; in the free case more general composition operators appear. Received: 3 December 1997     

On the Gaussian approximation of vector-valued multiple integrals

By combining the findings of two recent, seminal papers by Nualart, Peccati and Tudor, we get that the convergence in law of any sequence of vector-valued multiple integrals F_n towards a centered Gaussian random vector N, with given covariance matrix C, is reduced to just the convergence of: (i) the fourth cumulant of each component of F_n to zero; (ii) the covariance matrix of F_n to C. The aim of this paper is to understand more deeply this somewhat surprising phenomenon. To reach this goal, we offer two results of a different nature. The first one is an explicit bound for d(F,N) in terms of the fourth cumulants of the components of F, when F is a R^d-valued random vector whose components are multiple integrals of possibly different orders, N is the Gaussian counterpart of F (that is, a Gaussian centered vector sharing the same covariance with F) and d stands for the Wasserstein distance. The second one is a new expression for the cumulants of F as above, from which it is easy to derive yet another proof of the previously quoted result by Nualart, Peccati and Tudor.     

A rough path over multidimensional fractional Brownian motion with arbitrary Hurst index by Fourier normal ordering

Fourier normal ordering (Unterberger, 2009) [34] is a new algorithm to construct explicit rough paths over arbitrary Hölder-continuous multidimensional paths. We apply in this article the Fourier normal ordering algorithm to the construction of an explicit rough path over multi-dimensional fractional Brownian motion B$B$ with arbitrary Hurst index α$\alpha$ (in particular, for α≤1/4$\alpha \le 1/4$, which was till now an open problem) by regularizing the iterated integrals of the analytic approximation of B$B$ defined in Unterberger (2009) [32]. The regularization procedure is applied to ‘Fourier normal ordered’ iterated integrals obtained by permuting the order of integration so that innermost integrals have highest Fourier modes. The algebraic properties of this rough path are best understood using two Hopf algebras: the Hopf algebra of decorated rooted trees (Connes and Kreimer, 1998) [6] for the multiplicative or Chen property, and the shuffle algebra for the geometric or shuffle property. The rough path lives in Gaussian chaos of integer orders and is shown to have finite moments.     

A limit theorem for trees of alleles in branching processes with rare neutral mutations

We are interested in the genealogical structure of alleles for a Bienaymé–Galton–Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Ji?ina process) in discrete time. Itô’s excursion theory and the Lévy–Itô decomposition of subordinators provide fundamental insights for the results.