Martingale difference arrays and stochastic integrals

Lecture Notes in Mathematics, 1983

Lithuanian Mathematical Journal, 1990

In this note the following problem is considered. Suppose given a sequence of martingales (M n) and a sequence of predictable processes (w n) with nonincreasing trajectories. Under what conditions does one have weak convergence of the sequence of processes (X n) having the form X. (t) = w. (t) Mn (t), te[O, T], (1) in the Banach space D = D([0, T]) with the uniform norm? As a rule the trajectories of the processes w n do not belong to the space D. But the conditions given in the node (cf. Theorems 1 and 3) do not exclude the case w n = i and thus let one get convergence simply for a sequence of martingales. The assertions obtained are later applied to questions of the asymptotic behavior of processes connected with empirical ones.

Lithuanian Mathematical Journal, 1982

From Stochastic Calculus to Mathematical Finance, 2006

Consider a semimartingale of the form Yt = Y0 + t 0 asds + t 0 σs− dWs, where a is a locally bounded predictable process and σ (the "volatility") is an adapted right-continuous process with left limits and W is a Brownian motion. We define the realised bipower variation process V (Y ; r, s) n t = n

arXiv (Cornell University), 2004

Let (Ω, A, µ) be a Lebesgue space and T : Ω → Ω an ergodic measure preserving automorphism with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.

Bulletin of the Korean Mathematical Society

In Bae et al. [2], we have considered the uniform CLT for the martingale difference arrays under the uniformly integrable entropy. In this paper, we prove the same problem under the bracketing entropy condition. The proofs are based on Freedman inequality combined with a chaining argument that utilizes majorizing measures. The results of present paper generalize those for a sequence of stationary martingale differences. The results also generalize independent problems.

Probability Theory and Related Fields, 1980

2006

Notes on the Maxwell-Woodroofe martingale approach to central limit theorems for Markov chains.

2012

We derive Central Limit Theorems for the convergence of approximate quadratic variations, computed on the basis of regularly spaced observation times of the underlying process, toward the true quadratic variation. This problem was solved in the case of an Itô semimartingale having a non-vanishing continuous martingale part. Here we focus on the case where the continuous martingale part vanishes and find faster rates of convergence, as well as very different limiting processes. AMS 2000 subject classifications: 60F05, 60F17, 60G51, 60G07.

Lithuanian Mathematical Journal, 1985

I. Introduction. Basic Result Before formulating the basic result, we dwell on some necessary information connected with the concept of semimartingale. We shall assume that (~,~,P) is a complete probability space with a chosen nondecreasing family f={~,t~0} of o-algebras on it, satisfying the standard conditions, i.e., ~0 is completed by sets of measure zero from ~ and the family F is right continuous. We denote by (Dr0,+), r~[0,~)) the measure space of right continuous functions x:[0, ~) § R having left limits, with the Skorokhod topology ~z (cf. [I-3]). The notation (X, F) is used to denote an Fcompatible stochastic process X with trajectories from the space D[0,~]. In those cases when it is obvious or it is not necessary to indicate with which o-algebra the stochastic process X is compatible, we shall simply omit X, not indicating the o-algebra. We consider a function h:R § R such that h(x) = x for Ix[ ~ I/2, h(x) = 0 for Ix[ > I and lh(x) l < I. Then we can represent any semimartingale (X, F) uniquely in the form t r X,=X,+~+X~+ f f h~)(p-n)(ds, dx)+ f f (x-h~))p(ds, dx), t~o,