ABSTRACT
If H:, is clloscn so that X, is not iiltlcper~derit I ~ l t "weakly" tlcy)c~lltlcr~t of Xi , , h E 11; we ciui ~dt~irriat,ely i)ound the litst tcmi of (1.3.1) by an appropriate quant,ity. For cxarnplc, consitlcr it scquc~ncc of ilonrrcgat~ive r.v.s X I , X2, ..., dY,, titltirig values in (0, 1; 2, ..., 7. - l } and assrrmc: t h t thc sequence (X,,} is (1,-irrixing, i.e. t h e exist ~ , , , . r r ~ = 1 , 2 , .. S I I C ~ that lIP(Ar?B) - liD(l'l)IP(fl) <- (L, , , for every A E a ( X I ,..., X , ) ; H Co(Xi 1 n , ; Xaf7,,+lr ...), 1: > l , 7 n -> l ; iwd U,,, 4 O (see c.g. Billiugsley (198G), p. 37.5). T l m ~
A final case where the last sunmland in (4.3.1) can he easily controlled (upper boiinded) arises when the r.v.s X I , X:!, ..., X , exhibit certain types of positive or negative dependence. Thus, when we deal with integer valued r.v.s and X,, xb: b E BA, Q = 1 , 2 , ..., n, are associated or negatively associated then one could exploit Theorem 1 in I3outsika.s and Koutras (2000a) to deduce the upper bound
It is worth rnentioning that when the X, are binary r.v.s, then the upper bound in (4.3.2) is of the form (bl + b2 + b,:)/2 where bl and h2 coincide to the quantities involved in the Chen-Stein upper bound, Arratia et al. (1989, 1990).
