Jongsig Bae

Uniform CLT for Markov chains and its invariance principle: A martingale approach

The convergence of stochastic processes indexed by parameters which are elements of a metric space is investigated in the context of an invariance principle of the uniform central limit theorem (UCLT) for stationary Markov chains. We assume the integrability condition on metric entropy with bracketing. An eventual uniform equicontinuity result is developed which essentially gives the invariance principle of the UCLT. We translate the problem into that of a martingale difference sequence as in Gordin and Lifsic.(7) Then we use the chaining argument with stratification adapted from that of Ossiander.(11) The results of this paper generalize those of Levental(10) and Ossiander.(11)

THE UNIFORM CLT FOR MARTINGALE DIFFERENCE ARRAYS UNDER THE UNIFORMLY INTEGRABLE ENTROPY

In this paper we consider the uniform central limit theorem for a martingale-diere nce array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-diere nce arrays of process indexed by a class of measurable functions by a method as Ziegler (19) did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-diere nce and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler (19) and other results of independent problems. The results also generalizes those of Bae and Choi (3) to martingale-diere nce array of a function-indexed stochastic process. Finally, an application to classes of functions changing with n is given.

THE UNIFORM LAW OF LARGE NUMBERS FOR THE KAPLAN-MEIER INTEGRAL PROCESS

Let Un{f) = / fd(Fn — F) be the function-indexed Kaplan-Meier integral process constructed from the random censorship model. We study a uniform version of the law of large numbers of Glivenko-Cantelli type for {Un} under the bracketing entropy condition. The main result is that the almost sure convergence and convergence in the mean of the process Un holds uniformly in T. In proving the result we shall employ the bracketing method which is used in the proof of the uniform law of large numbers for the complete data of the independent and identically distributed model. In this paper, we obtain a uniform version of the law of large numbers of GlivenkoCantelli type for the function-indexed Kaplan-Meier integral process based on the incomplete data of the random censorship model. In obtaining the uniform law of large numbers, we observe the role played by bracketing of the indexed class of functions of the process and slightly modify the underlying metric. Then we employ the idea of DeHardt [1] of the bracketing method to the Kaplan-Meier integral process. The uniform law of large numbers of the present paper extends the one dimensional law of large numbers for random censoring that was established by Stute and Wang [4] and the DeHardts uniform law of large numbers for independent and identically distributed random variables [1]. Among others our results are stated not only for almost sure convergence but also for convergence in the mean. The results may be used in nonparametric statistical inference in verifying uniform consistency. See Van de Geer [6] for applications. We begin by introducing the integral version of the usual empirical process based

The uniform central limit theorem for the Kaplan-Meier integral process

In this paper, we develop the uniform central limit theorem for the function indexed Kaplan-Meier integral process based on the incomplete data of the random censorship model. The main goal is to investigate a tightness for the process under the metric entropy with £ 2 bracketing condition for the indexed class of functions of the process and mild assumptions due to censoring effects as in Stute [7]. In the achieving the goal, because of the lack of exponential inequalities for the random censorship model, we do not attempt to use the usual chaining argument that depends on the use of exponential inequalities on the tail probabilities. Instead we investigate the uniform order of convergence of the remainder terms in the representation of Stute [7] on the Kaplan-Meier integral and use the result of Ossiander [5] on the complete data of independent and identically distributed random variables. The uniform central limit theorem of the present paper extends the one dimensional central limit theorem under random censoring that was established by Stute in 1995 [7] and the Ossianders uniform central limit theorem for independent and identically distributed random variables that appeared in 1987 ([5]). Among others a sequential integral process and an invariance principle of the Kaplan-Meier integral will be produced as a corollary of the main result. The results may be used in nonparametric statistical inference. We begin with introducing the integral version of the usual empirical process based on the complete data of independent and identically distributed random variables.

THE UNIFORM CONSISTENCY OF THE SAMPLE KERNEL QUANTILE PROCESS

We obtain a kernel quantile process based on the kernel quantile estimator and prove the uniform consistency of the kernel quantile process by developing that of the usual sample quantile process. We apply our result to the classical kernel type processes.

THE EMPIRICAL LIL FOR THE KAPLAN-MEIER INTEGRAL PROCESS

We prove an empirical LIL for the Kaplan-Meier inte- gral process constructed from the random censorship model under bracketing entropy and mild assumptions due to censoring eects. The main method in deriving the empirical LIL is to use a weak convergence result of the sequential Kaplan-Meier integral process whose proofs appear in Bae and Kim (2). Using the result of weak convergence, we translate the problem of the Kaplan Meier integral process into that of a Gaussian process. Finally we derive the re- sult using an empirical LIL for the Gaussian process of Pisier (6) via a method adapted from Ossiander (5). The result of this paper extends the empirical LIL for IID random variables to that of a random censorship model.

THE SEQUENTIAL UNIFORM LAW OF LARGE NUMBERS

Let be the sequential empirical process based on the independent and identically distributed random variables. We prove that convergence problems of to zero boil down to those of . We employ Ottavianis inequality and the complete convergence to establish, under bracketing entropy with the second moment, the almost sure convergence of to zero.

THE UNIFORM LOCAL ASYMPTOTIC NORMALITY: AN EMPIRICAL PROCESS THEORY APPROACH

We investigate a uniform local asymptotic normality for likelihood ratio processes based on an independent and identi- cally distributed local asymptotic problem. Our tool is an empirical process theory.