Ukrainian Mathematical Journal, Vol. 61, No. 9, 2009 CONVERGENCE OF DIFFERENCE ADDITIVE FUNCTIONALS UNDER LOCAL CONDITIONS ON THEIR CHARACTERISTICS Yu. N. Kartashov 1 and A. M. Kulik 2 UDC 519.21 For additive functionals defined on a sequence of Markov chains that approximate a Markov process, we establish the convergence of functionals under the condition of local convergence of their characteristics (mathematical expectations). 1. Introduction In the present paper, we investigate the limit behavior of local-time-type functionals of Markov chains and processes. We consider functionals of the form  s;t n df D X kWst n;k <t F n;k  X n  t n;k  ;X n  t n;kC1  ;:::;X n  t n;kCL1   ;0  s<t; (1) where T n df Dft n;k g;n  1; is a sequence of partitions of R C ;X n ;n  1; is a sequence of processes that converge in a proper sense to a certain process X; and F n;k ;n  1; k  0; are nonnegative Borel functions. Since every functional  n possesses the property of additivity at the points of the corresponding partition T n ; we call these functionals difference additive. Examples of functionals of the form (1) are the number of hits of a certain set K n by values of the process X n at the points of the partition T n ; the number of sign changes for successive values of the process X n at the points of the partition T n ; etc. The main structural assumption is that the limit process X is a homogeneous Markov process, and each process X n possesses the Markov property at the points of the corresponding partition T n : Thus, X n ;n  1; can naturally be interpreted as a sequence of Markov chains with properly scaled time variable. This paper is a continuation of [1], where an approach to the investigation of the limit behavior of difference additive functionals was proposed. This approach is a generalization of the Dynkin approach to the investiga- tion of the limit behavior of W -functionals of a Markov process. The Dynkin sufficient condition establishes the convergence of W -functionals of a given Markov process under the condition of uniform convergence of their characteristics, i.e., mathematical expectations (see [2], Chap. 6). In [1], the notion of characteristic was general- ized to functionals of the form (1) and a result was proved that is, in a certain sense, an exact analog of the Dynkin theorem for difference additive functionals. The aim of the present paper is to weaken the assumptions of the main result of [1], in particular, the con- dition of the uniform convergence of characteristics and the continuity of the limit characteristic. To show that this weakening is not purely technical, we clarify the meaning of these conditions using the W -functionals of a multidimensional Brownian motion as an example. 1 Shevchenko Kiev National University, Kiev, Ukraine. 2 Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, Ukraine. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 9, pp. 1208–1224, September, 2009. Original article submitted February 9, 2009. 1428 0041–5995/09/6109–1428 c  2009 Springer Science+Business Media, Inc.