A UNIFORM ESTIMATE FOR THE RATE OF CONVERGENCE IN THE MULTIDIMENSIONAL CENTRAL LIMIT THEOREM FOR HOMOGENEOUS MARKOV CHAINS

In this paper a uniform estimate is obtained for the remainder term in the central limit theorem (CLT) for a sequence of random vectors forming a homogeneous Markov chain with arbitrary set of states. The result makes it possible to estimate the rate of convergence in the CLT without assuming the finiteness of the absolute third moment of the transition probabilities. Some consequences are also proved


INTRODUCTION.
Let Xl, z2, be a homogeneous Markov process with an arbitrary set of states X, defined by the transition probability function P(, A), ( E X, A E x (a-algebra of subsets of X), and the initial distribution: P(Zl A)= 7r(A), A x.
Suppose that P(., .)satisfies the following condition ofuniform ergodicity: sup IP(5, A)-P(r/, A)Ip < 1. ,7 X,A 9x It is well known that (see [13]) if(1.1) is fulfilled, then a stationary distribution P(A), A E x exists.Let f()-(fl() ,fk()), where f,(.), (i 1, k) are real x-measurable functions defined on X.Without loss of generality, in the next we shall suppose that x .f,OT)P(drl) O,   1, k   Suppose that the matrix h ]]a,3]l, where (7.p[f.(Zi)f3(i) q-p[f.(,)f,(.+1)]q- r=l r=l 4/2 M GHARIB is positive definite Let, Bf(x,.),P,(A) P(S E A), S,-v ,.-1 A,(A) P,(A)-(A) A B where the matrix B is such that BB A -, (.) is the standard nodal measure in k, and B k is the class of all Borel sets in In many aicles (see for example, [2], [5], [6], [7], and [8]) ufo estimates (in paicul, the Be-Essen bounds) are obtained for the qutity" sup [A,(A)I, AB under the condition that the transition probability nction P(.,.) has a fit absolute moment of order not less than tee.(Here B is the class of all convex measurable sets in ).
In the present work a unifo estimate is obtained for (1.2) thout assung the fiteness of: ms sup

X dx
TMs estimate eends the results peMMng to the case of the estence of absolute moments for the trsition probabilities of order not less than the tMrd for a sequence of rdom vtors that e either independent d identicMly distfibut (P(.,.)= P(.)= n(.)), or lied in a homogeneous Mkov chMn (, for example, ], [19], d [20]).
FORMULATION OF THE RESULTS THEOREM 1.The inequality sup I(A)I _< C(,p) Lo + Lo + L,n + + p (2.1) AB holds tree.Wher here and in what follows, C(.) denote a positive constant, not the same at different places, which depends only on the arguments in the parentheses As it is shown in [1 ], [11], and 15] the expression L. characterizes the cosponding result in th case where suation of independent identically distributed random vectors (p(., .)=p(.)= n(.)) is considered.The expression L characterizes, in some sense, the dependence stmctur of the random (L, 0 for independent identically distributed random vectors).
Through the proof of Theorem 1, we shall need estimates for the derivatives of the characteristic function , (.).Let us fix a direction 0 and let te be the projection onto 0. The proof of this lemma is easily obtained by repeating, word for word, the arguments of the proof of Lemma 2 in [8] with the use of the general spectral theory of linear operators acting in the Banach space of all bounded functions with the uniform norm Through the proof of part b, Lemma 3 of Chapter 6 in 16] is used Let ,(A) =,(A) P(A) AEB k where P(.) is a distribution function in k.Let r] (1) be such a vector that P[(A) P(r/(1) E A) and where Xx X1 (k) depends only on k.
For any T > 0, put rl (T) T-in, P(r(A) P(rl (T) a) where denotes the composition operation.
PROOF OF THEOREM 2. The proof of Theorem 2 goes along the lines of the proof of Theorem 1, where the condition 7r(.)P(.) is taken into account.
Also, for L., one can show, using Holder's inequality, that Finally, from (4.12) and (4.14), we get the assertion of Theorem 3.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: CONVERGENCE IN THE MULTIDIMENSIONAL CENTRAL LIMIT THEOREM/443