In this paper, the concept of weak laws of large numbers for arrays (WLLNFA) is studied, and a new notion of uniform integrability referred to as h-integrability is introduced as a condition for WLLNFA in obtaining the main results

Adeosun, S. A.

Edeki, S.O.

Covenant University Repository

International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2, Issue 1, January- 2014, PP 44-50 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org  ©ARC                                                                                                                                                  Page | 44 The h-Integrability and the Weak Laws of Large Numbers for Arrays S. O. Edeki Department of Industrial Mathematics,  Covenant University, Cannanland,  Otta, Nigeria. soedeki@yahoo.com S. A. Adeosun Department of Mathematical Sciences,  Crescent University,  Abeokuta, Nigeria. adeosunsakiru@gmail.com  Abstract: In this paper, the concept of weak laws of large numbers for arrays (WLLNFA) is studied, and a new notion of uniform integrability referred to as  h-integrability is introduced as a condition for WLLNFA  in obtaining the main results. Keywords: Weak Laws of Large Numbers, Uniform Integrability, Boundedness, h-Integrability and Convergence. 1. INTRODUCTION The law of large numbers (LLN) as a theorem describes the result of performing the same experiment a large number of times. According to this, the average of the results obtained from a large number of trials should be close to the expected value, and have a tendency to become closer as more trials are performed. The LLN ensures stable long-term results for the averages of random events. It only applies when a large number of observations are considered.  While the weak law of large numbers (WLLN) states that the sample average converges in probability towards the expected value. The concept of uniform integrability has been a core aspect in applied probability with regard to the theory of convergence of weak law of large numbers.  Chandra [1] obtains the weak law of large numbers under a new condition known as the Cesa´ro uniform integrability, which is weaker than uniform integrability. Cabrera [2] studies the weak convergence of weighted sums of random variables and introduces the condition of uniform integrability concerning the weights, which is weaker than uniform integrability; this leads to a special case known as Cesa´ro uniform integrability.  On the condition of -uniform integrability, Cabrera [2] obtains the weak law of large numbers for weighted sums of pairwise independent random variables. Sung [3] introduces the concept of Cesa´ro type uniform integrability with exponent . Chandra and Goswami [4] introduce the concept of Cesa´ro α-integrability (α > 0), and show that Cesa´ro α-integrability for any α > 0 is weaker than Cesa´ro uniform integrability.  Cabrera and Volodin [5] introduce the notion of h-integrability for an array of random variables concerning an array of constant weights, and prove that this concept is weaker than Cesa´ro uniform integrability, - uniform integrability and Cesa´ro α-integrability, and also show that h-integrability concerning the weights is sufficient for the weak law of large numbers to hold for weighted sums of an array of random variables, when these random variables are subject to some special kind of rowwise dependence.  Sung et al [6] remarked that the main idea of notions of -uniform integrability introduced in Cabrera [2] and h-integrability with respect to the array of constants  introduced in Cabrera and Volodin [5] is to deal with weighted sums of random variables. Sung et al in [7] introduce a new concept of integrability which deals with usual normed sums of random variables.  Adeosun S. O. Edeki & S. A. Adeosun  International Journal of Scientific and Innovative Mathematical Research (IJSIMR)              Page | 45 and Edeki [8], on a survey of uniform integrability of sequences of random variables, noted some new conditions required for such. In this paper, we study and present the basic theorem regarding uniform integrability, in terms of weak laws of large numbers for arrays. 2. PRELIMINARY DEFINITIONS AND LEMMAS In this section, the needed technical definitions and lemmas for the main results will be discussed accordingly, with their corresponding proof, if need be. Definition 2.1 An array of random variables    is said to be Cesa´ ro uniform integrability if  where   is a sequence of positive integers such that  as n → ∞.  Definition 2.2 Let  be an array of random variables and  an array of constants with   for all  ∈  and some constant . The array  is -uniform integrable if  -uniform integrability tends to Cesaro uniform integrability when ,  ,  Definition 2.3   Let } be an array of random variables and  >0. The array  is said to be Cesa´ro type uniformly integrable with exponent  if  We note that the conditions of Cesa´ro uniform integrability and Cesa´ro type uniformly integrable with exponent  are equivalent when , , and . Sung [3] obtains the weak law of large numbers for an array  satisfying Cesa´ro type uniform integrability with exponent  for some . Definition 2.4  For any , a sequence   of random variables is said to be Cesa´ro α-integrable if  Definition 2.5  Let  be an array of random variables and  an array of constants with  for all  ∈  and some constant . Let moreover  be an increasing sequence of positive constants with  ↑ ∞ as n ↑ ∞. The array  is said to be h-integrable with respect to the array of constants   if The H-Integrability and the Weak Laws of Large Numbers for Arrays.  International Journal of Scientific and Innovative Mathematical Research (IJSIMR)              Page | 46  Definition 2.6 Let  be an array of random variables and . Let moreover,  be an increasing sequence of positive constants with ↑ ∞ as n ↑ ∞. The array   is said to be h-integrable with exponent r  if  Lemma 2.1 [6] If the array  satisfies the condition of Cesa´ro type uniform integrability with exponent r > 0, then it satisfies the condition of h-integrability with exponent r. Proof: We note that the first condition of the Cesa´ro type uniform integrability with exponent r and the first condition of the h-integrability with exponent r are the same. Hence,  it suffices to show that the second condition of Cesa´ro type uniform integrability with exponent r implies the second condition of h-integrability with exponent r. If  satisfies the Cesa´ro type uniform integrability with exponent r, then there exist A > 0 such that if . Since  ↑ ∞ as ,   such that  if . For ,  . Hence the second condition of h-integrability with exponent r is satisfied      Note 2.1. As noted by [6], the concept of h-integrability with exponent r is strictly weaker than the concept of Cesa´ro type uniform integrability with exponent r, i.e., there exists an array  which is h-integrability with exponent r, but not Cesa´ro type uniform integrability with exponent r. Remark 2.1 In considering an array   of random variables defined on a probability space  and setting ,  and ,  authors like [10], [9], [7], [5], [3] established weak laws of large numbers.  Gut [9] proved that, for some  ,  if  is an array of Cesa´ro uniformly integrable random variables, where   if  and  if  .  Lemma 2.2[3] Suppose that  is an array of h-integrability with exponent r for some , ,  ↑ ∞, and → 0. Then the following statements hold. (i)  if  , (ii)   if  . 3. UNIFORM LAWS OF LARGE NUMBERS AND CONVERGENCE S. O. Edeki & S. A. Adeosun  International Journal of Scientific and Innovative Mathematical Research (IJSIMR)              Page | 47 This section deals with uniform laws of large numbers as condition for convergence of a sample mean of random variables on a compact space, and the general WLLNFA.  3.1 Uniform Laws of Large Numbers (ULLN) and a Compact Euclidean Space  Let  and  be two Euclidean sets with a Cartesian product  . Define on  a real-valued function  such that  is Lebesgue measurable for every . Then for any fixed , the sequence  will be a sequence of independent and identically distributed (iid) random variables on  with its sample mean converges in probability to . A uniform weak law of large numbers therefore defines a set of conditions on which    Theorem 3.1:  If (a)  is compact, (b)  is continuous at each  with probability one, (c) , and (d) . Then:   Proof: Let . So   as  since (i)  a.s. by condition (b). (ii)  by condition (c), and condition (d). Hence, for all  and , there exists  such that  . Obviously, the whole parameter space ,  is covered by . So since,  is compact, we can find a finite subcover, such that  is covered by . We recall that                                                                         ,                                                                             Note.  The first equality holds by the WLLN, since , and the last inequality follows the definition of , with   arbitrarily chosen. Whence, .                     Remark 3.1  The uniform law of large numbers states the condition under which the convergence happens uniformly. The H-Integrability and the Weak Laws of Large Numbers for Arrays.  International Journal of Scientific and Innovative Mathematical Research (IJSIMR)              Page | 48  Uniform strong law of large numbers defines such set of conditions if the convergence satisfies almost surely instead of in probability.  Theorem 3.2 Suppose that  is an array of h-integrability with exponent  random variables, ,   and    . Then  in  and, hence, in probability as , where  if   and  if   . Proof:  Let  and  . First we consider the case . Since , it follows that            . By the Cr-inequality, Jensen’s inequality, and Lemma 2.2 with   and , we have                                                                                                    , which completes the proof for  the case   . Now considering the case of  , it is observed that:                                             . Hence, using the Cr-inequality, Burkholder’s and Davis’ inequalities [12]; [13] for    and,  respectively),  Jensen’s inequality, and Lemma 2.2 ( i.e. Lemma 1, Sung [3]) with  and , we obtain                                                               .                                                   S. O. Edeki & S. A. Adeosun  International Journal of Scientific and Innovative Mathematical Research (IJSIMR)              Page | 49           where  is a constant depending only on . Hence, the proof of the case   is done  Corollary 3.1 Suppose that  is an array of random variables satisfying the Cesa´ro type uniform integrability with exponent 0 < r < 2 and  → ∞. Then    in    and, hence, in probability as n → ∞, where  = 0 if    and   if . Proof: By lemma2.1 [7] the condition of Cesa´ro type uniform integrability with exponent r implies the condition of h-integrability with exponent r, and so the result follows from Thrm 3.2.   4. CONCLUDING REMARKS In conclusion, weak laws of large numbers for the array of dependent random variables satisfying the condition of h-integrability with exponent r is obtained. The work extends and sharpens previous results with regard to applied probability, in terms of theorems and conditions for boundedness, and convergence of laws of large numbers with applications to stationary data.  REFERENCES [1]  T.K. Chandra, ‘‘Uniform integrability in the Cesro sense and the weak law of large numbers’’, Sankhya Ser. A 51 (3),(1989) 309- 317. [2]  M. Cabrera, ‘‘Convergence of weighted sums of random variable and uniform integrability concerning the weights’’, Collect. Math. 45 (2), (1994), 121 - 132. [3]  S.H. Sung, ‘‘Weak law of large numbers for arrays of random variables’’, Statist. Probab. Lett 42 (3), (1999), 293-298. [4]  T.K. Chandra and A. Goswami, ‘‘Ces ro α- integrability and laws of large numbers Int.,J. Theoret. Prob. 16 (3), (2003), 655- 669. [5]  M. Cabrera and A. 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. 305(2), (2005), 644 -658. [6]  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 (1),(2008), 289-300. [7]  S.H. Sung, T.C. Hu, and A. Volodin, ‘‘On the weak laws for arrays of random variables’’, Statist. Probab. Lett. 72 (4),(2005),291-298. [8]  S. A. Adeosun and S. O. Edeki ‘‘On a Survey of Uniform Integrability of Sequences of Random Variables’’ Int. J. Math & Stat Stud.  2 (1),(2014),1-14.  [9]  A. Gut. ‘‘The weak law of large numbers for arrays’’, Statist. Probab. Lett.14 (1), (1992),49-52. [10] D. H. Hong and K.S. Oh, On the weak law of large numbers for arrays, Statist. Probab. Lett. 22 (1),(1995),55-57. [11] D. Burkholder, ‘‘Martingales transforms’’, Ann. Math. Statist. 37 (1966), 1497-1504. [12] B. Davis, ‘‘On the integrability of the martingale square function’’, Israel J. Math. 8 (1970),187-190. [13] A. Leblanc and B. C. Johnson, ‘‘A family of inequalities related to binomial prob- abilities’’. Department of Statistics, University of Manitoba. Tech. Report, 2006. [14] Newey W.K. ‘‘Uniform Convergence in Probability and Stochastic Equicontinuity’’ Econometrica  59(4) (1991), 1161-1167. [15] B.M. Brown, ‘‘Moments of a stopping rule related to the central limit theorem’’. Ann. Math. Statist. 40 (2004), 1236 - 1249. [16] D. Li, A. Rosalsky, and A. Volodin, ‘‘On the strong law of large numbers for sequences of pairwise negative quadrant dependent random variables’’, Bull. Inst. Math. Acad. Sin. (N.S) 1 (2), (2006), 281-305. The H-Integrability and the Weak Laws of Large Numbers for Arrays.  International Journal of Scientific and Innovative Mathematical Research (IJSIMR)              Page | 50 [17] Jennrich,R.I.  ‘‘Asymptotic Properties of non-linear least squares Estimators’’,The Annals of Mathematical Statistics 40(1969), 633-643. [18] J. Bae, D. Jun, and S. Levental, ‘‘The uniform clt for martingale diff erence arrays under the uniformly integrable entropy’’, Bull. Korean Math. Soc. 47(1), (2010),39-51.  

The h-Integrability and the Weak Laws of Large Numbers for Arrays

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The h-Integrability and the Weak Laws of Large Numbers for Arrays

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