Martingale Convergence Research Articles

Weighted estimating equations for semiparametric transformation models with censored data from a case-cohort design

SUMMARY In a case-cohort design introduced by Prentice (1986), covariates are assembled only for a subcohort randomly selected from the entire cohort, and any additional cases outside the subcohort. Semiparametric transformation models are considered here for failure time data from the case-cohort design. Weighted estimating equations are proposed for esti mation of the regression parameters. The estimation procedure of survival probability at given covariate levels is also provided. Asymptotic properties are derived for the estimators using finite population sampling theory, U-statistics theory and martingale convergence results. The finite-sample properties of the proposed estimators, as well as the efficiency relative to the full cohort estimators, are assessed via simulation studies. A case-cohort dataset from the Atherosclerosis Risk in Communities study is used to illustrate the estimating procedure.

Local extinction versus local exponential growth for spatial branching processes

Let X be either the branching diffusion corresponding to the operator $Lu+\beta (u^2-u)$ on $D\subseteq $ $\mathbb{R}^{d}$ [where $\beta (x) \geq 0$ and $\beta\not\equiv 0$ is bounded from above] or the superprocess corresponding to the operator $Lu+\beta u -\alpha u^2$ on $D\subseteq $ $\mathbb{R}^{d}$ (with $\alpha>0$ and $\beta$ is bounded from above but no restriction on its sign). Let $\lambda _{c}$ denote the generalized principal eigenvalue for the operator $L+\beta $ on $D$. We prove the following dichotomy: either $\lambda _{c}\leq 0$ and X exhibits local extinction or $\lambda _{c}> 0$ and there is exponential growth of mass on compacts of D with rate $\lambda _{c}$. For superdiffusions, this completes the local extinction criterion obtained by Pinsky [Ann. Probab. 24 (1996) 237--267] and a recent result on the local growth of mass under a spectral assumption given by Engländer and Turaev [Ann. Probab. 30 (2002) 683--722]. The proofs in the above two papers are based on PDE techniques, however the proofs we offer are probabilistically conceptual. For the most part they are based on "spine'' decompositions or "immortal particle representations'' along with martingale convergence and the law of large numbers. Further they are generic in the sense that they work for both types of processes.

Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis

Recently Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503], using probabilistic methods alone, has given new proofs for the existence, asymptotics and uniqueness of travelling wave solutions to the K-P-P equation. Following in this vein we outline alternative probabilistic proofs. Specifically the techniques are confined to the study of additive and multiplicative martingales and spinal path decompositions along the lines of [B. Chauvin, A. Rouault, Probab. Theory Related Fields 80 (1988) 299], [R. Lyons, in: K.B. Athreya, P. Jagers (eds.), Classical and Modern Branching Processes, Vol. 84, Springer-Verlag, New York, 1997, pp. 217–222] and [R. Lyons et al., Ann. Probab. 23 (1995) 1125]. We also make use of a new decomposition where the spine is a conditioned process. Some new results concerning martingale convergence are obtained as a by-product of the analysis.

On Convergence of Series of Random Elements via Maximal Moment Relations with Applications to Martingale Convergence and to Convergence of Series with 𝑝-Orthogonal Summands. Correction

Abstract A result by Móricz, Su, and Taylor from Acta Math. Hungar. 65(1994), 1–16, was misstated in the authors' paper in Georgian Math. J. 8(2001), No. 2, 377–388, where due to this misstatement the invalid formulation and proof of a corollary is given. In this correction note, the needed result is correctly stated and a corrected version of the invalid corollary is proved.

Spatial autoregression model: strong consistency

Let ( α ̂ n, β ̂ n) denote the Gauss–Newton estimator of the parameter ( α, β) in the autoregression model Z ij = αZ i−1, j + βZ i, j−1 − αβZ i−1, j−1 + ε ij . It is shown in an earlier paper that when α=β=1, {n 3/2( α ̂ n−α, β ̂ n−β)} converges in distribution to a bivariate normal random vector. A two-parameter strong martingale convergence theorem is employed here to prove that n r( α ̂ n−α, β ̂ n−β)→ 0 ̄ almost surely when r< 3 2 .

Random Harmonic Series

Here the positive and negative terms partly cancel, allowing the series to converge. To a probabilist, this alternating series suggests choosing plus and minus signs at random, by tossing a fair coin. Formally, let (ej) ∞ j=1 be independent random variables with common distribution P (ej = 1) = P (ej = −1) = 1/2. Then, Kolmogorov’s three series theorem [1, Theorem 22.8] or the martingale convergence theorem [1, Theorem 35.4] shows that the sequence ∑n j=1 ej/j converges almost surely. In this note, we investigate the distribution of the sum X := ∑∞ j=1 ej/j.

On‐line almost‐sure parameter estimation for partially observed discrete‐time linear systems with known noise characteristics

AbstractIn this paper we discuss parameter estimators for fully and partially observed discrete‐time linear stochastic systems (in state‐space form) with known noise characteristics. We propose finite‐dimensional parameter estimators that are based on estimates of summed functions of the state, rather than of the states themselves. We limit our investigation to estimation of the state transition matrix and the observation matrix. We establish almost‐sure convergence results for our proposed parameter estimators using standard martingale convergence results, the Kronecker lemma and an ordinary differential equation approach. We also provide simulation studies which illustrate the performance of these estimators. Copyright © 2002 John Wiley &amp; Sons, Ltd.

Nonlinear martingale theory for processes with values in metric spaces of nonpositive curvature

We develop a nonlinear martingale theory for time discrete processes $(Y_n)_{n\in \NN_0}$. These processes are defined on any probability space $(\O,\F,\F_n,\P)_{n\in\NN_0}$ and have values in a metric space (N,d) of nonpositive curvature (in the sense of A. D. Alexandrov). The defining martingale property for such processes is \[ \E(Y_{n+1}|\F_n)=Y_n, \qquad \P\mbox{-a.s.,} \] where the conditional expectation on the left-hand side is defined as the minimizer of the functional \[ Z\mapsto\E d^2(Z,Y_{n+1}) \] within the space of $\F_n$-measurable maps $Z\dvtx \O\to N$. We give equivalent characterization of N-valued martingales (using merely the usual linear conditional expectations) and derive fundamental properties of these martingales, for example, a martingale convergence theorem. Finally, we exploit the relation with harmonic maps. It turns out that a map $f\dvtx M\to N$ is harmonic w.r.t. a given Markov kernel p on M if and only if it maps Markov chains $(X_n)_{n\in\NN}$ (with transition kernel p) on M onto martingales $(f(X_n))_{n\in\NN}$ with values in $N$. The nonlinear heat flow $f\dvtx \N_0\times M\to N$ of a given initial map $f(0,\cdot)\dvtx M\to N$ at time n is obtained as the filtered expectation, \[ f(n,x) := \E_x [ f(X_n) |\!|\!| (\F_k)_{k\ge 0}] \] of the random map $f(X_n)$. Similarly, the unique solution to the Dirichlet problem for a given map $g\dvtx M\to N$ and a subset $D\subset M$ is obtained as \[ f(x) := \E_x [ g(X_{\tau(D)})|\!|\!| (\F_k)_{k\ge 0}]. \] In both cases, a crucial role is played by the notion of expectation $\E_x [\cdot |\!|\!| (\F_k)_{k\ge 0}]$ which will be discussed in detail. Moreover, we prove Jensen's inequality for expectations and expectations and we prove (weak and strong) laws of large numbers for sequences of i.i.d. random variables with values in N. Our theory is an extension of the classical linear martingale theory and of the nonlinear theory of martingales with values in manifolds as developed, for example, in Emery and Kendall. The goal is to extend the previous framework towards processes with values in metric spaces. This will lead to a stochastic approach to the theory of (generalized) harmonic maps with values in such singular spaces as developed by Jost and Korevaar and Schoen.

Continuous Ocone Martingales as Weak Limits of Rescaled Martingales

Consider a martingale $M$ with bounded jumps and two sequences $a_n, b_n \to \infty$. We show that if the rescaled martingales $$ M^n_t =\frac{1}{\sqrt{a_n}}M_{b_n t}$$ converge weakly, then the limit is necessarily a continous Ocone martingale. Necessary and sufficient conditions for the weak convergence of the rescaled martingales are also given.

On the conditional expectation on probability MV-Algebras with product

In this paper the conditional expectation is defined and a variant of the martingale convergence theorem on probability MV-algebras with product is proved.

Martingale Convergence and the Functional Equation in the Multi-Type Branching Random Walk

A generalization of Biggins Martingale Convergence Theorem is proved for the multitype branching random walk The proof appeals to modern techniques involving the construction of sizebiased measures on the space of marked trees generated by the branching process As a simple conse quence we obtain existence and uniqueness of solutions within a specied class to a system of functional equations

On Convergence of Series of Random Elements via Maximal Moment Relations with Applications to Martingale Convergence and to Convergence of Series with p-Orthogonal Summands

Abstract The rate of convergence for an almost surely convergent series of Banach space valued random elements is studied in this paper. As special cases of the main result, known results are obtained for a sequence of independent random elements in a Rademacher type p Banach space, and new results are obtained for a martingale difference sequence of random elements in a martingale type p Banach space and for a p-orthogonal sequence of random elements in a Rademacher type p Banach space. The current work generalizes, simplifies, and unifies some of the recent results of Nam and Rosalsky [Teor. Īmovīr. ta Mat. Statist. 52: 120–131, 1995] and Rosalsky and Rosenblatt [Bull. Inst. Math. Acad. Sinica 11: 185–208, 1983, Nonlinear Anal. 30: 4237–4248, 1997].

Projective limits of complex measures and martingale convergence

In this paper we prove, for signed or complex Radon measures on completely regular spaces, the analogue of Prokhorov's criterion on the existence of the projective limit of a compatible system of measures. Because of loss of mass under projections this cannot be reduced to the case of positive measures. Countable projective limits are, as in the case of positive measures, particularly simple, the sole condition now being the boundedness of the total variations. It is shown, with the help of the martingale convergence theorem, that the densities of these complex measures with respect to their variations, converge in an appropriate sense. This work is part of an extended project on the mathematical theory of path integrals.

Spreading-invariant sequences and processes on bounded index sets

We say that a random sequence is spreadable if all subsequences of equal length have the same distribution. For infinite sequences the notion is equivalent to exchangeability but for finite sequences it is more general. The present paper is devoted to a systematic study of finite spreadable sequences and of processes on [0, 1] with spreadable increments. In particular, we show how many basic results in the exchangeable case—notably the predictable sampling theorem, the Wald-type identities, and various martingale and weak convergence results—admit extensions to a spreadable setting. We also identify some additional conditions that ensure the exchangeability of a spreadable sequence or process.

The Markov approximation of the sequences of N-valued random variables and a class of small deviation theorems

Let {Xn,n⩾0} be a sequence of random variables on the probability space (Ω,F,P) taking values in the alphabet S={1,2,…,N}, and Q be another probability measure on F, under which {Xn,n⩾0} is a Markov chain. Let h(P|Q) be the sample divergence rate of P with respect to Q related to {Xn}. In this paper the Markov approximation of {Xn,n⩾0} under P is discussed by using the notion of h(P|Q), and a class of small deviation theorems for the averages of the bivariate functions of {Xn,n⩾0} are obtained. In the proof an analytic technique in the study of a.e. convergence together with the martingale convergence theorem is applied.

On the probability theory on MV algebras

The paper contains a review of some results concerning probability theory on MV algebras (laws of large numbers, central limit theorem, martingale convergence theorem). Also some algebraic and methodical aspects are discussed.

A note on the conditional probability on product MV algebras

A conditional probability (state) of an element from a product MV algebra with respect to a product MV valued observable is formulated and a version of the martingale convergence theorem is proved.

Martingale convergence and the stopped branching random walk

We discuss the construction of stopping lines in the branching random walk and thus the existence of a class of supermartingales indexed by sequences of stopping lines. Applying a method of Lyons (1997) and Lyons, Pemantle and Peres (1995) concerning size biased branching trees, we establish a relationship between stopping lines and certain stopping times. Consequently we develop conditions under which these supermartingales are also martingales. Further we prove a generalization of Biggins' Martingale Convergence Theorem, Biggins (1977a) within this context.

On the conditional expectation of observables in MV algebras of fuzzy sets

A quantum mechanical model is considered based on an MV algebra of fuzzy sets. The conditional expectation is studied of an observable with respect to another one. The main result is a variant of the martingale convergence theorem.

Convergence of set-valued and fuzzy-valued martingales

The purpose of this paper is to prove the convergence theorems of set-valued and fuzzy-valued martingales in Kuratowski-Mosco sense without assuming that their values are compact or of compact level sets.