ON THE MARCINKIEWICZ ZYGMUND LAWS OF LARGE NUMBERS FOR NEGATIVELY DEPENDENT RANDOM FIELDS

Abstract

Abstract. In this paper we provide extensions of the MarcinkiewiczZygmund laws of large numbers for i.i.d random variables with mul-tidimensional indices to the case of negatively dependent random elds. 1. Introduction Let Z d+ , where dis a positive integer, denote the positive integer d-dimensional lattice points. The notation m n, where m = (m 1 ;m 2 ; ;m d ) and n = (n 1 ;n 2 ; ;n d ) in Z d+ , means that m i n i for all 1 id. The following multiindex version of the Marcinkiewicz Zygmundstrong laws of large numbers was given in Gut(1978).Theorem A Let 0 <r<2, and suppose that X, fX i ;i 2Z d+ gis a eld of i.i.d random variables with partial sums S n =P in X i ;i 2Z d+ :If EjXj r (log + jXj) d 1 <1and EX= 0 when 1 r<2, then(1:1)S n jnj 1r !0 a:s:asn !1:Conversely, if (1.1) holds, then EjXj r (log + jXj) d 1 <1and EX= 0when 1 r<2. Here jnj=Q di=1 n i and n !1means min 1id n i !1, that is, all coordinates tend to in nity. Also, throughout the paper,log + x= maxf1;logxg.Next, we turn to our attention to the negative dependence. Tworandom variables Xand Y are negative quadrant dependent(NQD) ifP(Xx;Y y) P(Xx)P(Y y) for all x;y2R. A nite family