WLLN for arrays of nonnegative random variables

Transcription

1 WLLN for arrays of nonnegative random variables Stefan Ankirchner Thomas Kruse Mikhail Urusov November 8, 26 We provide a weak law of large numbers for arrays of nonnegative and pairwise negatively associated in each row) random variables under a rather weak domination condition. Introduction 2 MSC : 6F5. Keywords : array of random variables, domination condition, negative association, negative quadrant dependence, weak law of large numbers. We start with quoting Gut [3]: There are two essential ways in which one might generalize the weak law of large numbers WLLN) for i.i.d. random variables; to relax the i.-assumption or the i.d.-assumption or both). The former amounts to introducing martingale assumptions, mixing assumptions, etc. The latter amounts to introducing conditions on how unequal the summands may be. Typical requirements are domination assumptions. Chandra [] introduced a rather weak domination condition, called Cesàro uniform integrability which is weaker than just uniform integrability, see Condition A) in Section 2), and proved a WLLN with L -convergence for sequences of Cesàro uniformly integrable pairwise independent random variables. Weakening the assumption of pairwise independence Landers and Rogge [6] proved a WLLN with L -convergence for sequences of Cesàro uniformly integrable pairwise nonpositively correlated random variables. The latter is possible only at the cost of assuming that the random variables are nonnegative. Moreover, Example 4 in [6] shows that the latter statement does not hold true if one omits the nonnegativity assumption. If we do not assume that the random variables are nonnegative, it is even possible to construct, for each p, 2), identically distributed pairwise uncorrelated random variables X n ) n N with E X p <, not fulfilling the WLLN see Theorem in [7]). On the contrary, if X n ) n N are pairwise independent instead of uncorrelated) identically distributed random variables with E X <, then even SLLN holds due to Etemadi [2] and there is no need to assume that the random variables are nonnegative). The aforementioned results in [] and in [6] can be easily extended to arrays of random variables X ni, i, n N) with as n, and the WLLNs have the form i= X ni a ni ), n, ) where the convergence holds in L, and a ni are suitable deterministic numbers in fact, a ni = EX ni in the settings of [] and [6]). There is another line of research on WLLNs, which is right from the outset written for arrays of random variables, where it is possible to weaken the assumption of pairwise independence in each row) without imposing the nonnegativity assumption. Here one obtains WLLNs in form ) but with suitable random variables a ni, that is, here we speak about a different class of results. More specifically, Gut [3] establishes L -convergence in ) under Cesàro uniform integrability, which is the same domination condition as in [] and [6], but with a ni = E[X ni X nj, j i ]. In comparison with the extension to arrays of) [6] there is no need to assume nonnegativity in [3], while, as we already mentioned, nonnegativity is essential in [6]. That is, if we want to have a WLLN in form ) with deterministic a ni, this comes at a price in the case of [6] the price is the assumptions of nonnegativity and pairwise nonpositive covariance in each row). More recent work in this line of research was devoted to further weakening the domination assumptions. For instance, Hong and Oh [4] introduced a weaker than Cesàro uniform integrability domination condition namely, Condition B) in Section 2) and proved a WLLN in form ). In contrast to [3], the convergence in ) is provided only in probability due to the weak domination condition) and with slightly different a ni : a ni = E [ X ni { Xni } X nj, j i ] without Cesàro uniform integrability E[X ni X nj, j i ] can Stefan Ankirchner, Institute for Mathematics, University of Jena, Ernst-Abbe-Platz 2, 7745 Jena, Germany. s.ankirchner@unijena.de, Phone: +49 ) ; Thomas Kruse, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 4527 Essen, Germany. thomas.kruse@uni-due.de, Phone: +49 ) ; Mikhail Urusov, Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 4527 Essen, Germany. mikhail.urusov@uni-due.de, Phone: +49 )

2 fail to be well-defined). For further results along this line of research, see Sung [], Sung, Hu and Volodin [2], Sung, Lisawadi and Volodin [3] and references therein. The aim of our paper is to weaken the domination assumption in the first mentioned line of research. Namely, we consider arrays of nonnegative random variables as in [6]) and prove a WLLN in form ) with deterministic a ni under a new domination condition, which is weaker than Cesàro uniform integrability. This comes at the price that we have only convergence in probability in ) convergence in L is impossible because our domination condition does not imply integrability of the random variables in the array) and that we need a stronger dependence condition in each row the random variables are pairwise negatively associated). We will see that this result is new even in the special case when the random variables in each row are pairwise independent. In Section 2 we introduce the setting and discuss the relationships between several domination conditions known from the literature and new ones). In Section 3 we formulate and discuss our main result Theorem 3.) and present an example Example 3.3), where the previously known domination conditions are not satisfied, but the WLLN holds true due to Theorem 3.. Section 4 contains the proof of Theorem Setting and domination conditions We consider an array of nonnegative random variables X ni, i, n N), as n, on a probability space Ω, F, P ). We denote X ni = X ni {Xni }, i, n N. Set S n = i= X ni, n N, and S n = i= X ni, n N. For all n N let f n : [, ] R + be the functions defined by the formula f n z) = i= P [X ni > z]. We now introduce several domination conditions and discuss the relationships between them. Notice that we do not write absolute values in the conditions below because all random variables X ni are nonnegative. Condition A) Cesàro uniform integrability): sup n N i= E X ni {Xni>y}), y. 2) It is easy to see that Cesàro uniform integrability of X ni, i, n N) is a weakening of the ordinary) uniform integrability of X ni, i, n N). Condition B): sup n N i= yp X ni > y), y. 3) Condition B) is strictly weaker than Condition A) notice that 3) does not necessarily imply integrability of all random variables X ni, while 2) does imply that). Condition C): Applying the substitution y = z we see that 4) is equivalent to zf n z) dz, n. 4) i= yp X ni > y) dy, n. 5) Comparing 3) and 5) we see that Condition C) is a weakening of Condition B). Condition D) consists of 6) and 7): i= P [X ni > z] f n ), n, 6) P [X ni u]du dz, n. 7) We recall in connection with 6) that f n ) = i= P [X ni > ] and further remark that neither of conditions 6) and 7) implies the other one: take X ni = + for all n and i to see that 7) does not imply 6); consider X ni to be uniformly distributed on [, ] for all n and i to see that 6) does not imply 7). Lemma 2.. Condition D) is strictly weaker than Condition C). Proof. The example of X ni = for all n and i shows that Condition D) does not imply Condition C). We now prove that Condition C) implies Condition D). 2

3 Since for all n N the functions f n are nonincreasing, it holds that f n ) = 2 zf n ) dz 2 zf n z) dz, that is, 4) implies 6). Finally, applying the estimate P [X ni u]du z in 7) we see that 7) also follows from 4). Conditions A) and B) were already used in the literature as domination assumptions for WLLNs as for Condition A) see e.g. [], [3] and [6]; concerning Condition B) see [4]). Conditions C) and D) are new. We need Condition D) for our main result, while Condition C) is an easier-to-check sufficient condition, which is given in terms of the functions f n only. 3 Main result Theorem 3.. Let X ni, i, n N), as n, be an array of nonnegative random variables. We assume Condition D) and that either n N, the random variables X n,..., X nkn are pairwise independent 8) or both the following conditions 9) and ) hold true: n N, i < j, y, z R +, P [X ni > y, X nj > z] P [X ni > y]p [X nj > z], f n ) f n z)dz, n. ) Then the sequence kn i= Xni E [ X ni {Xni }]) converges to zero in probability. As already mentioned in the introduction, typical assumptions in many WLLNs are domination assumptions and dependence assumptions. In Theorem 3. the domination assumption is Condition D), while the assumption either 8) or 9)&) is the dependence assumption, which we now discuss in more detail. Remark 3.2. i) Here are two further equivalent forms of condition 9): 9) n N, the random variables X n,..., X nkn n N, the random variables X n,..., X nkn are pairwise negatively quadrant dependent NQD); are pairwise negatively associated NA). It is worth noting that the word pairwise in both items of the preceding list is essential. See Section 2 in [8] for the definition of NQD and for the equivalence with 9). See Section 2 in [5] for the definition of NA and for the fact that, for a pair of random variables, NQD is equivalent to NA. ii) Pairwise NA as dependence assumptions in WLLNs also appeared in Theorem of [9] and in Theorem 3.2 of [3]. The main difference with our Theorem 3. is in the domination assumptions. The domination assumptions in [9] and in [3] are also weakenings of Cesàro uniform integrability, but they are qualitatively different from our Conditions C) and D). In particular, the domination assumptions in [9] and [3] imply that all random variables X ni are integrable, while our domination assumptions do not imply that. iii) It is obvious that 8) implies 9). However, we cannot simply assume 9) because, in general, we need a technical assumption ) together with 9), while in the case when the random variables in the array are pairwise independent in each row, ) is no longer necessary to assume. iv) The sequence of nonnegative nonincreasing functions f n, n N, given by the formula f n z) = log knz+, kn z [, ], provides an example, where 4), that is, Condition C), is satisfied, while ) is violated. In particular, we conclude that Condition D) does not imply ). Example 3.3. We take β ), 3 and α 2 β )., 3 By ξ we denote a random variable with the density α )x α {x } ) and write η d ζ to indicate that the random variable ζ stochastically dominates the random variable η. For all n N we consider pairwise NQD random variables X n,..., X nn that satisfy Cn β ξ d X ni d Mn β ξ, i =,..., n, 2) 3

4 with some universal constants C < M in, ). Elementary calculations using the second stochastic dominance relation in 2) show that the array X ni ) n N, i n satisfies Condition C) and ). Therefore, Theorem 3. applies, and we conclude that n n i= X ni E[X ni {Xni n}]) converges to zero in probability. Notice that E[X ni ] <, while E[Xni 2 ] = in this example. Moreover, another elementary calculation that again uses only the second dominance relation in 2) yields n E [ X ni {Xni>n}], n, n hence we also have n i= n X ni E[X ni ]) P, n. i= Finally, observe that, due to the first dominance relation in 2), the array X ni ) n N, i n does not satisfy Condition B). Indeed, for every y > we have sup n N n n i= yp X ni > y) sup n N n n yp Cn β ξ > y) = y, which converges to infinity as y. Notice that the probabilities on the right-hand side of the last inequality are equal to for large n.) Remark 3.4. In order to illustrate that the assumptions in the example above allow for various dependence structures, we now show a possible specific construction of non-trivial pairwise NQD random variables X n,..., X nn satisfying 2). For all n N, let {Y nk : k N} {U nk : k N} be a family of 4-tuplewise independent random variables such that U nk have the density ) and P Y nk = ) = P Y nk = 2) = 2. We define i= X n,2k = n β Y nk U n,2k, X n,2k = n β Y nk U n,2k. Theorem in [8] implies that, for each n N, the random variables X nk, k N, are pairwise NQD, while 2) is clear from the construction. It is obvious that one can produce more sophisticated constructions on this way. Finally, we emphasise that the requirement for a sequence of random variables to be 4-tuplewise independent cf. above) is, in fact, much weaker than independence, and it allows for quite interesting dependence structures e.g. see [] for an example of a sequence of bounded N-tuplewise independent and identically distributed random variables that do not satisfy the central limit theorem). 4 Proof We first notice that, in the notation of Section 2, we have i= Xni E [ X ni {Xni }]) = Sn E[ S n ] ). Now Theorem 3. is a direct consequence of the following Lemmas 4. and 4.2. Lemma 4.. Under 6) it holds that P [S n S n ] as n. Proof. We have P [S n S n ] i= P [X ni > ] = f n ) as n. Lemma 4.2. Under the assumptions of Theorem 3. the sequence Sn E[S n ] ) converges to zero in L 2 as n. Proof. For all n N it holds that [ E Sn E[S n ] ) ) ] 2 = kn 2 VarS n ) = kn 2 VarX ni ) + 2 CovX ni, X nj ). i= i<j For all i, n N we have E[X 2 kn ni] = 2 yp [X ni > y]dy 2 yp [X ni > y]dy = 2kn 2 zp [X ni > z]dz. 3) 4

5 Moreover, we have for all i, n N that 2 kn EX ni = P [X ni > y]dy) = P [X ni > y]dy = P [X ni > y] P [X ni > ]dy = P [X ni > y]dy P [X ni > ] = = 2 P [X ni > z]dz P [X ni > ] P [X ni > z]dz 2knP 2 [X ni > ] P [X ni > z] This implies for all i, n N that VarX ni ) = E[X 2 ni] EX ni 2 zp [X ni > z]dz 2 + 2P [X ni > ] = 2kn 2 P [X ni > z] P [X ni > z]dz P [X ni > u]du dz 2P [X ni > ] P [X ni > z]dz P [X ni > z] P [X ni u]du dz + P [X ni > ] P [X ni > u]du dz P [X ni > z]dz. ) P [X ni > z]dz. 4) 5) Hence, we obtain for all n N that Var[X ni ] 2 i= i= 2 = i= i= P [X ni > z] P [X ni > ] P [X ni > z] P [X ni > z] i= P [X ni u]du dz P [X ni > z]dz P [X ni u]du dz + 2 i= P [X ni u]du dz + 2f n ). P [X ni > ] 6) Then it follows from Condition D) that, as n, i= Var[X ni ]. Under 8), the covariances in 3) vanish, and the proof is completed. We proceed to work under 9) and ). For all i < j, n N, we have by 9) E[X ni X nj ] = P [X ni > y, X nj > z]dydz P [X ni > y]p [X nj > z]dydz. P [X ni > y, X nj > z]dydz 5

6 This implies for all i < j, n N, that This yields for all n N CovX ni, X nj ) = E[X ni X nj ] E[X ni ]E[X nj ] Then ) implies that = = P [X nj > ] i<j CovX ni, X nj ) P [X ni > y]p [X nj > z] P [X ni > y]p [X nj > z]dydz P [X ni > y]p [X nj > z] P [X ni > y] P [X ni > ])P [X nj > z] P [X nj > ])dydz P [X ni > y]p [X nj > ] + P [X nj > z]p [X ni > ]dydz lim sup n i,j= = P [X ni > y]dy + P [X ni > ] P [X nj > y]dy. = f n ) P [X nj > ] j= kn P [X nj > ] f n z)dz. k i= n i<j CovX ni, X nj ) =. Since the expression in 3) is nonnegative, this completes the proof. References P [X ni > y]dy P [X ni > y]dy [] T. K. Chandra. Uniform integrability in the Cesàro sense and the weak law of large numbers. Sankhyā Ser. A, 53):39 37, 989. [2] N. Etemadi. An elementary proof of the strong law of large numbers. Z. Wahrsch. Verw. Gebiete, 55):9 22, 98. [3] A. Gut. The weak law of large numbers for arrays. Statist. Probab. Lett., 4):49 52, 992. [4] D. H. Hong and K. S. Oh. On the weak law of large numbers for arrays. Statist. Probab. Lett., 22):55 57, 995. [5] K. Joag-Dev and F. Proschan. Negative association of random variables, with applications. Ann. Statist., ): , 983. [6] D. Landers and L. Rogge. Laws of large numbers for uncorrelated Cesàro uniformly integrable random variables. Sankhyā Ser. A, 593):3 3, 997. [7] D. Landers and L. Rogge. Identically distributed uncorrelated random variables not fulfilling the WLLN. Bull. Korean Math. Soc., 383):65 6, 2. [8] E. L. Lehmann. Some concepts of dependence. Ann. Math. Statist., 37:37 53, 966. [9] M. Ordóñez Cabrera and A. I. Volodin. Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability. J. Math. Anal. Appl., 352): , 25. [] A. R. Pruss. A bounded N-tuplewise independent and identically distributed counterexample to the CLT. Probab. Theory Related Fields, 3): , 998. [] S. H. Sung. Weak law of large numbers for arrays. Statist. Probab. Lett., 382): 5, 998. [2] S. H. Sung, T. C. Hu, and A. Volodin. On the weak laws for arrays of random variables. Statist. Probab. Lett., 724):29 298, 25. [3] S. H. Sung, S. Lisawadi, and A. Volodin. Weak laws of large numbers for arrays under a condition of uniform integrability. J. Korean Math. Soc., 45):289 3, 28. ) 6