On martingale tail sums for the path length in random trees

For a martingale (X_n) converging almost surely to a random variable X, the sequence (X_n– X) is called martingale tail sum. Recently, Neininger (Random Structures Algorithms 46 (2015), 346–361) proved a central limit theorem for the martingale tail sum of Régnier's martingale for the path length in random binary search trees. Grübel and Kabluchko (in press) gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the law of the iterated logarithm for a family of trees containing binary search trees, recursive trees and plane‐oriented recursive trees. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 493–508, 2017     

The martingale hardy type inequality for Marcinkiewicz-Fejér means of two-dimensional conjugate Walsh-Fourier series

The main aim of this paper is to prove that for any 0 < p ≤ 2/3 there exists a martingale f ∈ H _p such that Marcinkiewicz-Fejér means of the two-dimensional conjugate Walsh-Fourier series of the martingale f is not uniformly bounded in the space L _p .     

Vector-valued weak martingale Hardy spaces and atomic decompositions

Some atomic decomposition theorems are proved in vector-valued weak martingale Hardy spaces w _p Σ_α(X), w _p Q _α(X) and wD _α(X). As applications of atomic decompositions, a sufficient condition for sublinear operators defined on some vector-valued weak martingale Hardy spaces to be bounded is given. In particular, some weak versions of martingale inequalities for the operators f*, S ^(p)(f) and σ^(p)(f) are obtained. This research was supported by the National Science Foundation of China (No. 10371093).     

A note on Thewalsh-Fejér means

The main aim of this paper is to prove that the maximal operator σ # is not bounded from the martingale Hardy space Hp (G) to the martingale Hardy space Hp (G) for 0 < p ≤ 1.     

Interpolation for weak Orlicz spaces with M Δ condition

An interpolation theorem for weak Orlicz spaces generalized by N-functions satisfying M _Δ condition is given. It is proved to be true for weak Orlicz martingale spaces by weak atomic decomposition of weak Hardy martingale spaces. And applying the interpolation theorem, we obtain some embedding relationships among weak Orlicz martingale spaces. This work was supported by the National Natural Science Foundation of China (Grant No. 10671147)     

Martingale transforms and Hardy spaces

Summary Burkholder's martingale transforms are especially useful in studying predictable martingale Hardy spaces. Characterizations of such spaces via martingale transforms are provided. In particular, it is shown that for 0<p<, a martingale inh ^p , defined by the conditioned square function, is the martingale transform of a bmo₂ martingale with a multiplier sequence whose maximal function is inL ^p .     

A new characterization of <Emphasis Type="Italic">p</Emphasis>-smoothable Banach spaces

In this paper an atomic decomposition theorem for Banach-space-valued weak Hardy regular martingale space w _p H _α ^S (X) is given. As an application, p-smoothable Banach spaces are characterized in terms of bounded sublinear operators defined on Banach-space-valued weak Hardy regular martingale space w _p H _α ^S (X).     

GIRSANOV TRANSFORMATION AND ITS APPLICATION TO THE THEORY OF ENLARGEMENT OF FILTRATIONS

Given a complete filtered probability space (Ω, (F _t ) _{t ∈ [0,1]}, F,P). If we enlarge the filtration by random variables satisfying condition A given in [Ja] and condition B ^X which is defined in this paper (or in [FI]), then the semimartingale property also preserves. Moreover, the invariance of martingale property for Poisson martingale under a simultaneous enlargement of filtration and change of equivalent probability measure can be obtained.     

The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes

For a wide class of local martingales (M _t ) there is a default function, which is not identically zero only when (M _t ) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability of functions of sup _s≤t M _s in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting and last exit times. Received: 6 August 1996 / Revised version: 27 July 1998     

Limit theorems for a class of diffusion processes

We consider a diffusion process {x(t)} on a compact Riemannian manifold with generator δ/2 + b. A current‐valued continuous stochastic process {X _t} in the sense of Itô [8] corresponds to {x(t)} by considering the stochastic line integral X _t(a) along {x(t)} for every smooth 1-form a. Furthermore {X _t} is decomposed into the martingale part and the bounded variation part as a current-valued continuous process. We show the central limit theorems for {X _t} and the martingale part of {X _t}. Occupation time laws for recurrent diffusions and homogenization problems of periodic diffusions are closely related to these theorems     

Martingales with given maxima and terminal distributions

Let μ be any probability measure onR with λ |x|dμ(x)<∞, and let μ^* denote its associated Hardy and Littlewood maximal p.m. It is shown that for any p.m.v for which μ<ν<μ^* in the usual stochastic order, there is a martingale (X _t)_0≦t≦1 for which sup_0≦t≦1 X _t andX ₁ have respective p.m. 'sv and μ. The proof uses induction and weak convergence arguments; in special cases, explicit martingale constructions are given. These results provide a converse to results of Dubins and Gilat [6]; applications are made to give sharp martingale and ‘prophet’ inequalities. Supported in part by NSF grants DMS-86-01153 and DMS-88-01818.     

Differentiable and analytic families of continuous martingales in manifolds with connection

Summary.  We prove that the derivative of a differentiable family X _t (a) of continuous martingales in a manifold M is a martingale in the tangent space for the complete lift of the connection in M, provided that the derivative is bicontinuous in t and a. We consider a filtered probability space (Ω,(ℱ _t )_{0≤ t ≤1}, ℙ) such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a↦L(a) of random variable with values in M and such that L(0)≡x ₀∈M, there exists an analytic family a↦X(a) of continuous martingales such that X ₁(a)=L(a). For this, we investigate the convexity of the tangent spaces T ^{( n )} M, and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time T such that ℙ(T<1) is arbitrarily small. We use this construction of families of martingales in complex analytic manifolds to prove that every ℱ₁-measurable random variable with values in a compact convex set V with convex geometry in a manifold with a C ¹ connection is reachable by a V-valued martingale. Received: 14 March 1996/In revised form: 12 November 1996     

Characterization of submartingales of a new class (Σr)

In this work, we shall consider a new class (Σ^r) of local submartingales of the form X_t = M_t + A_t, where (M_t)_{t ? 0} is a càdlàg (right continuous with left limits) local martingale, (A_t)_{t ? 0} is a càdlàg increasing process, and the measure (dA) is carried by the set {t: X_{t ?} = 0}.

The aim of the present paper is to study the positive and negative parts of processes of this class and establish some martingale characterizations. The formula of relative martingales is derived in terms of last passage time. Finally, by using balayage formula, we calculate predictable compensator.     

Local times for a class of purely discontinuous martingales

Summary Suppose X _t is a purely discontinuous martingale. A sufficient condition for X _t to have a local time is given in terms of the local characteristics of X. An example is constructed to show that this condition is nearly optimal.     

The maximum maximum of a martingale constrained by an intermediate law

Let (M _t ) be any martingale with M ₀≡ 0, an intermediate law M ₁∼μ₁, and terminal law M ₂∼μ₂, and let Mˉ ₂≡ sup_{0≤ t ≤2} M _t . In this paper we prove that there exists an upper bound, with respect to stochastic ordering of probability measures, on the law of Mˉ ₂. We construct, using excursion theory, a martingale which attains this maximum. Finally we apply this result to the robust hedging of a lookback option. Received: 26 December 1998 / Revised version: 20 April 2000 /?Published online: 15 February 2001     

Martingales in Banach Lattices

In this article, we present a version of martingale theory in terms of Banach lattices. A sequence of contractive positive projections (E_n) on a Banach lattice F is said to be a filtration if E_nE_m = E_{n∧ m}. A sequence (x_n) in F is a martingale if E_nx_m = x_n whenever n ≤ m. Denote by M = M(F, (E_n)) the Banach space of all norm uniformly bounded martingales. It is shown that if F doesn’t contain a copy of c₀ or if every E_n is of finite rank then M is itself a Banach lattice. Convergence of martingales is investigated and a generalization of Doob Convergence Theorem is established. It is proved that under certain conditions one has isometric embeddings . Finally, it is shown that every martingale difference sequence is a monotone basic sequence. Mathematics Subject Classification (2000). 60G48, 46B42     

Moderate Deviations for Linear Processes Generated by Martingale-Like Random Variables

In this paper we study the moderate deviation principle for linear statistics of the type S _n =∑ _i∈Z c _n,i ξ _i , where c _n,i are real numbers, and the variables ξ _i are in turn stationary martingale differences or dependent sequences satisfying projective criteria. As an application, we obtain the moderate deviation principle and its functional form for sums of a class of linear processes with dependent innovations that might exhibit long memory. A new notion of equivalence of the coefficients allows us to study the difficult case where the variance of S _n behaves slower than n. The main tools are: a new type of martingale approximations and moment and maximal inequalities that are important in themselves.     

Orlicz-Hardy空间与BMO空间之间的鞅变换

以鞅变换为工具,刻画了Orlicz-Hardy鞅空间与BMO空间之间的相互关系.证明了如下结论:对任意上指标有限(等价于满足△_2-条件)的Young函数Φ,鞅f∈H_Φ{P_Φ,Q_Φ}的充分必要条件是,f是BMO∈{BMO_1,BMO_2}中某个鞅g的鞅变换.     

Restricted maximal operators of Fejér means of double Walsh-Fourier series

The main aim of this paper is to prove that there exists a martingale f ∈ H ₁^2/▭ such that the restricted maximal operators of Fejér means of twodimensional Walsh-Fourier series and conjugate Walsh-Fourier series does not belong to the space weak-L _1/2.     

Functional law of iterated logarithm for additive functionals of reversible Markov processes

1.IntroductionandMainResultsAssumethat(Xt),.T(T~NorAl)isaPolishspaceE-valuedMarkovprocess,definedon(fi,F,(R),(ot),(P-c)..E),withitssemigroupoftransitionkernels(Pt).Here(ot)isthesemigroupofshiftsonfisuchthatX.(otw)~X. t(w),Vs,tET;(R)isthenaturalfiltration.Throughoutthispaperweassumethat(Pt)issymmetricandergodicwithrespectto(w.r.t.forshort)aprobabilitymeasurepon(E,e)(eistheBorela--fieldofE),i.e.,.Symmetry:(Ptf,g)~(f,Pig):~isfptgdp,acET,if,gCL'(P);.ErgodicitytFOranyfEL'(P),ifPtf~f…