Bull. Korean Math. Soc. 4 (2004), No. 2, pp. 275 282 WEAK LAWS FOR WEIGHTED SUMS OF RANDOM VARIABLES Soo Hak Sung Abstract. Let {a ni, u n i, n } be an array of constants. Let {X ni, u n i, n } be {a ni }-uniformly integrable random variables. Weak laws for the weighted sums i=un a ni X ni are obtained.. Introduction Let {u n, n } and {, n } be two sequences of integers (not necessarily positive or finite), and let {, n } be a sequence of positive integers such that as n. Consider an array of constants {a ni, u n i, n } and an array of random variables {X ni, u n i, n } defined on a probability space (Ω, F, P ). When u n =, =, n, weak laws of large numbers for the array {X ni } have been established by several authors (see, Gut [2], Hong and Lee [3], Hong and Oh [4], and Sung [8]). An array {X ni, i, n } is said to be Cesàro uniformly integrable if () lim a sup E X ni I( X ni > a) = 0. This condition was introduced by Chandra []. Ordonez Cabrera [5] extended the notion of Cesàro uniform integrability to {a ni }-uniform integrability as follows. Received July 6, 2003. 2000 Mathematics Subject Classification: 60F05, 60F25. Key words and phrases: weak law of large numbers, weighted sums, r-mean convergence, convergence in probability, arrays, uniform integrability. This work was supported by Korea Research Foundation Grant (KRF-2002-05- CP0037).
276 Soo Hak Sung An array of random variables {X ni, u n i, n } is said to be {a ni }-uniformly integrable if (2) lim a sup a ni E X ni I( X ni > a) = 0. Note that {a ni }-uniform integrability reduces to Cesàro uniform integrability when a ni = kn for i and 0 elsewhere. Ordonez Cabrera [5] obtained weak laws for weighted sums of {a ni }-uniformly integrable random variables, where the weights satisfy lim n sup un i a ni = vn 0 and sup a ni r < for some 0 < r. In this paper, we prove the results of Ordonez Cabrera [5] under a weaker condition on the weights. 2. Main result Throughout this section, let {a ni, u n i, n } be an array of constants, and let {X ni, u n i, n } be an array of random variables. To prove the main result, we will need the following lemmas. Lemma. (Sung [9]). Suppose that {X ni, u n i, n } is an array of random variables satisfying the following conditions. (3) sup and (4) lim a sup E X ni r < E X ni r I( X ni r > a) = 0, where r > 0 and {, n } is a sequence of positive numbers such that as n. If β > r, then E X ni β I( X ni r ) = o(k β/r n ).
Weak Laws for Weighted Sums of Random Variables 277 Lemma 2. (Sung [9]). Suppose that {X ni, u n i, n } is an array of random variables satisfying (3) and (4) for some 0 < r < and {, n }. Then vn X ni kn /r 0 in L r and, hence, in probability as n. Lemma 3. Let r > 0. Let { X ni r, u n i, n } be { a ni r }- uniformly integrable random variables, where {a ni, u n i, n } is an array of constants satisfying lim n sup un i a ni = 0. Assume that (5) sup If β > r, then a ni r E X ni r <. a ni β E X ni β I( X ni r ) = o(), where = /sup un i a ni r. Proof. Take kn /r a ni X ni instead of X ni in Lemma. Then we have by (5) that sup E k /r n a ni X ni r <. Since a ni r for u n i, it follows that lim sup a lim a sup E k /r n a ni X ni r I( k /r n a ni X ni r > a) Thus we obtain by Lemma that k β/r n E k /r a ni r E X ni r I( X ni r > a) = 0. n a ni X ni β I( kn /r a ni X ni r ) = o(). So the result follows since / a ni r for u n i. Now, we state and prove our main result which generalizes some results in the literature in this area. See the corollaries and example following Theorem.
278 Soo Hak Sung Theorem. Let 0 < r. Let {a ni, u n i, n } and {X ni, u n i, n } be as in Lemma 3. When r =, we assume further that {X ni } is an array of (rowwise) pairwise independent random variables with EX ni = 0, i.e., for each fixed n, X n,un,, X n,vn are pairwise independent. Then a ni X ni 0 in L r and, hence, in probability as n. Proof. Take = / sup un i a ni r and kn /r a ni X ni instead of X ni in Lemma 2. When 0 < r <, the result follows from Lemma 2. We now prove the result when r =. Define X ni = X nii( X ni ) and X ni = X ni X ni for u n i and n. Since X n,u n,, X n,v n are pairwise independent random variables, we have by Lemma 3 with r = and β = 2 that E a ni (X ni EX ni) 2 a 2 nie X ni 2 0 as n. Also we obtain by the definition of {a ni }-uniform integrability that E a ni (X ni EX ni) 2 a ni E X ni 0 as n, since as n. Thus we have E E (E a ni X ni a ni (X ni EX ni) + E a ni (X ni EX ni) 2 ) /2 + E a ni (X ni EX ni) a ni (X ni EX ni) 0 as n.
Weak Laws for Weighted Sums of Random Variables 279 Corollary. Let 0 < r <. Let { X ni r, u n i, n } be { a ni r }-uniformly integrable random variables, where {a ni, u n i, n } is an array of constants satisfying lim n sup un i a ni = 0 and for some constant C > 0 (6) Then a ni r < C for all n. a ni X ni 0 in L r and, hence, in probability as n. Proof. From Theorem, it is enough to show that (5) holds. Since { X ni r } is { a ni r }-uniformly integrable, there exists a > 0 such that Then sup a ni r E X ni r I( X ni r > a). E X ni r = E X ni r I( X ni r a) + E X ni r I( X ni r > a) which implies by (6) that sup a + E X ni r I( X ni r > a), a ni r E X ni r a sup a ni r + sup a ni r E X ni r I( X ni r > a) i=u n a C +. Hence (5) is satisfied. The above corollary has been proved by Ordonez Cabrera [5]. Rohatgi [7] established a weaker result (convergence in probability) under the stronger condition that {X n, n } is a sequence of independent random variables which is uniformly bounded by a random variable X with E X r <. Wang and Rao [0] extended Rohatgi s result to L r - convergence under the uniform integrability (without independent condition).
280 Soo Hak Sung Corollary 2. Let {X ni, u n i, n } be {a ni }-uniformly integrable (rowwise) pairwise independent random variables with EX ni = 0 for u n i and n, where {a ni, u n i, n } is an array of constants satisfying lim n sup un i a ni = 0 and for some constant C > 0 (7) Then a ni < C for all n. a ni X ni 0 in L and, hence, in probability as n. Proof. By Theorem, it is enough to show that (5) holds when r =. The proof of the rest is similar to that of Corollary and is omitted. The above corollary has been proved by Ordonez Cabrera [5]. Pruitt [6] established a weaker result (convergence in probability) under the stronger condition that {X n, n } is a sequence of independent and identically distributed random variables with EX n = 0 for n. Rohatgi [7] extended Pruitt s result to a sequence of independent random variables which is uniformly bounded by a random variable X with E X <. Wang and Rao [0] extended Rohatgi s result to L - convergence for uniformly integrable pairwise independent random variables. The following example shows that the conditions of Theorem are weaker than the conditions of Corollary 2. Example. Let {X n, n } be a sequence of pairwise independent random variables such that X n = ±/ log n with probability /2 if n is not a perfect cube, and X n = ±n /3 with probability /2 if n is a perfect cube (i.e., n = j 3 for some positive integer j). Define an array of constants {a ni, i, n } as follows. { log n/n if i n, a ni = 0 if i > n. Since a ni = log n, we can not apply this example to Corollary 2. Observe that E X i = / log i = O(n/ log n) i j 3,i n i j 3,i n
Weak Laws for Weighted Sums of Random Variables 28 and i=j 3,i n E X i = i=j 3,i n i /3 = j 3 n j = j 0(j 0 + ) 2 n/3 (n /3 + ), 2 where j 0 = max{j : j 3 n}. Thus we have If a >, then a ni E X i = log n n n E X i log n n (O( n log n ) + n/3 (n /3 + ) ) = O(). 2 Hence, for a >, we have sup a ni E X i I( X i > a) = log n n log n a ni E X i I( X i > a) = sup n>a 3 n i=j 3,a 3 <i n E X i. i=j 3,a 3 <i n E X i n /3 (n /3 + ) log n sup 0 n>a 3 2n as a. Therefore the conditions of Theorem with r = are satisfied. By Theorem, we obtain a ni X i 0 in L and, hence, in probability as n. References [] T. K. Chandra, Uniform integrability in the Cesàro sense and the weak law of large numbers, Sankhya, Ser. A 5 (989), no. 3, 307 37. [2] A. Gut, The weak law of large numbers for arrays, Statist. Probab. Lett. 4 (992), no., 49 52.
282 Soo Hak Sung [3] D. H. Hong and S. Lee, A general weak law of large numbers for arrays, Bull. Inst. Math. Acad. Sinica 24 (996), no. 3, 205 209. [4] D. H. Hong and K. S. Oh, On the weak law of large numbers for arrays, Statist. Probab. Lett. 22 (995), no., 55 57. [5] M. Ordonez Cabrera, Convergence of weighted sums of random variables and uniform integrability concerning the weights, Collect. Math. 45 (994), no. 2, 2 32. [6] W. E. Pruitt, Summability of independent random variables, J. Math. Mech. 5 (966), 769 776. [7] V. K. Rohatgi, Convergence of weighted sums of independent random variables, Proc. Cambridge Philos. Soc. 69 (97), 305 307. [8] S. H. Sung, Weak law of large numbers for arrays, Statist. Probab. Lett. 38 (998), no. 2, 0 05. [9], Weak law of large numbers for arrays of random variables, Statist. Probab. Lett. 42 (999), no. 3, 293 298. [0] X. C. Wang and M. B. Rao, A note on convergence of weighted sums of random variables, Internat. J. Math. Math. Sci. 8 (985), no. 4, 805 82. Department of Applied Mathematics, Pai Chai University, Taejon 302-735, Korea E-mail: sungsh@mail.pcu.ac.kr