tailieunhanh - Independent And Stationary Sequences Of Random Variables - Chapter 18

Chapter 18 THE CENTRAL LIMIT THEOREM FOR STATIONARY PROCESSES 1 . Statement of the problem This chapter contains the main objective of the second part of the book, the investigation of the limiting behaviour of the distributions of sums or integrals of the form | Chapter 18 THE CENTRAL LIMIT THEOREM FOR STATIONARY PROCESSES 1. Statement of the problem This chapter contains the main objective of the second part of the book the investigation of the limiting behaviour of the distributions of sums or integrals of the form T a . a T Bt1 E Xt AT Bi1 Xtdt AT t a J a as T- oo where Xt is a stationary process. If no assumptions except stationarity are made it is not generally possible to prove anything stronger than an ergodic theorem. Thus for instance we may take Xt X for all i AT 0 Bt T and obtain any distribution as the limiting distribution of . However in this example there is strong dependence between Xtl and Xt2 even for very large values of 111 12 . This shows that to obtain theorems of interest we must impose conditions of weak dependence between the past 901- and future 9ft of the process. We shall therefore study processes satisfying the strong or uniform mixing conditions and functionals of such processes. There is one other sort of trivial behaviour which must be excluded which arises when the sums Xt do not grow as T increases. Suppose for example that J is a sequence of independent identically distributed random variables then Xt t l- t defines a stationary process which is in any reasonable sense weakly dependent. But t l 316 CENTRAL LIMIT THEOREM FOR STATIONARY PROCESSES Chap. 18 so that converges in distribution in a trivial way taking BT 1. To exclude such behaviour we always require that lim Bt oo . With these restrictions it is possible to find all the possible limit distributions of . Theorem . Let Fn x be the distribution function of n J 1 where Xj is a strongly mixing stationary sequence with mixing coefficient a n and lim Bn oo . n co If Fn x converges weakly to a non-degenerate distribution function F x then F x is necessarily stable. If the latter distribution has exponent a then Bn n1 ah n where h n is slowly varying as n cc. Before proving the theorem we .

• Toán học
• biến cố ngẫu nghiên
• xác suất thống kê
• quản lí doanh nghiệp
• kế hoạch kinh doanh
• biến ngẫu nhiên độc lập
• Nghiên cứu khoa học
• Tập bài giảng
• Giải tích tổ hợp
• Biến cố ngẫu nhiên
• Đại lượng ngẫu nhiên