This paper presents explicitly a survey of uniformly integrable sequences of random variables. We also study extensively several cases and conditions required for uniform integrability, with the establishment of some new conditions needed for the generalization of the earlier results obtained by many scholars and researchers, noting the links between uniform integrability and pointwise convergence of a class of polynomial functions on conditional based

Adeosun, S. A.

Edeki, S.O.

Covenant University Repository

 International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 1  ON A SURVEY OF UNIFORM INTEGRABILITY OF SEQUENCES OF RANDOM VARIABLES   S. A. Adeosun and S. O. Edeki 1Department of Mathematical Sciences, Crescent University, Abeokuta, Nigeria. 2Department of Industrial Mathematics, Covenant University, Cannanland, Ota, Nigeria.   ABSTRACT: This paper presents explicitly a survey of uniformly integrable sequences of random variables. We also study extensively several cases and conditions required for uniform integrability, with the establishment of some new conditions needed for the generalization of the earlier results obtained by many scholars and researchers, noting the links between uniform integrability and pointwise convergence of a class of polynomial functions on conditional based.  KEYWORDS: Uniform Integrability, Sequences, Boundedness, Convergence, Monotonicity.   INTRODUCTION  Uniform integrability is an important concept in functional analysis, real analysis, measure theory, probability theory, and plays a central role in the area of limit theorems in probability theory and martingale theory. Conditions of independence and identical distribution of random variables are basic in historic results due to Bernoulli, Borel and A.N. Kolmogorov[1]. Since then, serious attempts have been made to relax these strong conditions; for example, independence has been relaxed to pairwise independence.   In order to relax the identical distribution condition, several other conditions have been considered, such as stochastic domination by an integrable random variable or uniform integrability in the case of weak law of large number. Landers and Rogge [2] prove that the uniform integrability condition is sufficient for a sequence of pairwise independence random variables in verifying the weak law of large numbers.  Chandra [3] obtains the weak law of large numbers under a new condition which is weaker than uniform integrability: the condition of Ces ro uniform integrability. Cabrera [4], by studying the weak convergence for weighted sums of variables introduces the condition of uniform integrability concerning the weights, which is weaker than uniform integrability, and leads to Ces ro uniform integrabilty as a particular case. Under this condition, a weak law of large  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 2  numbers for weighted sums of pairwise independent random variables is obtained; this condition of pairwise independence can also be dropped at the price of slightly strengthening the conditions of the weights. Chandra and Goswami [5] introduce the condition of Ces ro -integrability ( ), and show that Ces ro -integrability for any  is weaker than uniform integrability. Under the Ces ro - integrability condition for some , they obtain the weak law of large numbers for sequences of pairwise independence random variables. They also prove that Ces ro -integrability for appropriate  is also sufficient for the weak law of large numbers to hold for certain special dependent sequences of random variables and h-integrability which is weaker than all these was later introduced by Cabrera [4].  As an application, the notion of uniform integrability plays central role in establishing weak law of large number. The new condition; Ces ro uniform integrability, introduced by Chandra [3] can be used in cases to prove –convergence of sequences of pairwise independent random variables, and in the study of convergence of Martingales. Other areas of application include the approximation of Green’s functions of some degenerate elliptic operators as shown by Mohammed [6].  Definition 1 The random variables  are independent if:   of measurable sets . A family of random variables  is independent if for each finite set , the family  is independent. Definition 2  A family of random variables ,  is pairwise independent if and  are independent whenever .  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 3  Definition 3  Let  be a probability space. A sequence of random variables   defined on  is said to converge in probability to a random variable  if for every ,  as .  This is expressed as :  Definition 4. A sequence of random variables   defined on  is said to converge in  to a random variable  if  for all , and    as .  This is expressed as :  Definition 5. Sequence of real valued random variables is uniformly integrable if and only if, for any   such that   where    is an indicator function of the event  i.e, the function which is equal one for  and zero otherwise, and  is an expectation operator. Expectation values are given by integrals for continuous random variables.  NOTION OF USEFUL INEQUALITIES AND RESULTS OF UNIFORM INTEGRABILITY  In this section, we introduce the basic inequalities and results needed for the conditions and applicability of uniformly integrable sequences of random variables.  Markov Inequality If   is a random variable such that  for some positive real number  which may or may not be a whole number, then for any ,  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 4   .  Chebyshev’s Inequality Let  be a random variable with finite mean and finite variance . Then   This follows immediately by putting and  in the Markov’s inequality above. Chebyshev’s Sum Inequality If  and , then  Proof: Assume  and , then by rearrangement of inequality, we have that:                             Now adding these  inequalities gives:         Hence,                 RESULTS OF UNIFORM INTEGRABILITY Lemma 2.1 Uniform integrability implies - boundedness  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 5  Let  be uniformly integrable. Then  is - bounded. Proof: Choose  so large such that .  Then     Remark: The converse of Lemma 2.1 is not true, i.e boundedness in  is not enough for uniform integrability. For a counter example, we present the following: Let  be uniformly distributed on  such that:  Then  so  is bounded. But for , , so  is not uniformly integrable. Theorem 2.2 Let  be a random variable and  be a sequence of random variables. Then the following are equivalent: i)      ii)      is uniformly integrable and  in probability. Proof:  Suppose i) holds. By Chebychev’s inequality, for ,   So  in probability. Moreover, given , there exists  such that  whenever . Then we can find  so that  implies , , . Then for  and ,  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 6   Hence,  is uniformly integrable. We have shown that i) implies ii). Suppose on the other hand that ii) holds, then there is a subsequence  such that  almost surely. So, by Fatou’s Lemma,  Now, given , there exists  such that, for all , , . Consider the uniformly bounded sequence  and set  Then  in probability, so by bounded convergence, there exists  such that, for all , . But then, for ,  . Therefore, ii) implies i) since  was arbitrary   Other authors like [7], [8], [9] have also shown that uniform integrability of functions were related to the sums of random variables.  UNIFORM INTEGRABILITY OF A CLASS OF POLYNOMIALS ON A UNIT INTERVAL Let  denote the probability of exactly  successes in an  independent Bernoulli trials with probability  success by any trial. In other words:   International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 7  and, for integers , we define the family of functions   by           (1) The family of polynomials arise in the context of statistical density estimation based on Bernstein polynomials. Specifically, the case  has been considered by many authors [10], [11], & [12] while the case  and  was considered by [13]. These same authors have considered issues linked to uniform integrability and pointwise convergence of  and . However, the generalization to any  has not been considered. In this section, we will establish the following results.  Theorem 3.1 Let  be fixed positive integers. Then i)      for  and  ii)      is uniformly integrable on . iii)     for  as . For the case , Babu et al [10, Lemma 3.1] contains the proof of iii). Leblanc et al [13,Lemma 3.1] considered when  and . The proof here generalizes (but follows the same line as) these previous results. In establishing Theorem 3.1, we first show that for all  and ,          (2)  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 8  The proof of this inequality is based on a class of completely monotonic functions and hence of general interest [14]. Using completely different methods, Leblanc and Johnson [13] previously showed that  is decreasing in  and hence, (2) is a generalization of the earlier result. Lemma 3.1 Let  and  be real numbers such that  and  and let  denote the digamma function. Define   If  and , then  is completely monotonic on  and hence  is increasing and concave on , see[14] & [15] . Proof: Let , and . Then the integral representation of  is:  Therefore, for ,                 (3) The assumption  yields .       (4)  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 9  where  Calculus shows that, for ,  is strictly decreasing on  and hence, for every , is decreasing [we note that, if , there is no difficulty in taking , since these terms vanish in (3)]. Since  is also decreasing, Chebyshev’s inequality for sums yields:  We see that, if , the integrand in (4) is non-negative and hence  on . We conclude that  is completely monotonic on  and, in particular,  is increasing and concave on  whenever  and .            Lemma 3.2 [14] Let  be integers such that  and and define    . Then  is decreasing in  and  . Proof: The limit is easily verified using Stirling’s formula, thus we need only show that  is decreasing in . Treating  as a continuous function in  and differentiating we obtain:   International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 10  where . Now, taking    an , and  in Lemma 3.1, we have that  is increasing on  and hence  for all  since  always     Corollary 3.1 Let . Then  is decreasing in  for every fixed . Proof:   if and only if,  and we have, by Lemma 3.2.    which completes the proof.  Proof of Theorem 3.1. First we note that (i) holds since                                                                                                                                                       with equality if and only if . Similarly, (ii) holds since  is uniformly integrable on  and, by Corollary 3.1, we have  for all .  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 11  To prove (iii), let  and  be two sequences of independent random variables such that  is Binomial  and  is Binomial . Now, define  so that  has a lattice distribution with span gcd  [17]. We can write  in terms of the  as:       . Now, define the standardized variable  so that Var  and note that these also have a lattice distribution, but with span gcd  . Theorem 3 of Section XV.5 of Feller [16] leads to          , where  corresponds to the standard normal probability density function. The result now follows from the fact that                Remark 3.1  Since  is decreasing, it is obvious that:  And since  for . we see that the sequence  define by:                      (5)  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 12  is increasing. Also, Corollary 3.1 trivially leads to a similar family of inequalities for “number of failure”-negative binomial probabilities. As such, let  be the probability of exactly  failures  before the th success  in a sequence of i.i.d. Bernoulli trials with success probability  so that, for     . Hence, as a direct consequence of Corollary 3.1, we have that  is also decreasing. As a consequence of Theorem 3.1, we have, for any function f  bounded on [0,1],           (6) In particular, Kakizawa [12], establish (6) for the case   CONCLUSION   Generally, we conclude by pointing out the usefulness of the results to some other interesting areas such as combinatorial and discrete probability inequalities in terms of monotonicity. In addition, the consequence of Theorem 3.1 is a key tool in assessing the performance of nonparametric density estimators based on Bernstein polynomials.   The work complements previous results in the literature with significance in computational analysis and in applied probability. Also, the result is of special interest in the study of uniform integrability of martingales in terms of pointwise boundedness, and equicontinuity of a certain class of functions.   REFERENCES  [1] O.O.Ugbebor, Probability distribution and Elementary limit Theorems, University of Ibadan External studies programme (1991),ISBN 978-2828-82-3 [2] D. Landers and L. Rogge, ‘‘Laws of large numbers for pairwise independent uniformly integrable random variables’’, Math. Nachr. 130 (1987), 189 - 192.  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 13  [3] T.K. Chandra, ‘‘Uniform integrability in the Ces ro sense and the weak law of large numbers’’, Sankhya Ser. A 51 (3),(1989) 309- 317. [4] M. O. Cabrera, ‘‘Convergence of weighted sums of random variable and uniform integrability concerning the weights’’, Collect. Math. 45 (2), (1994), 121 - 132. [5] T.K. Chandra and A. Goswami, ‘‘Ces ro α- integrability and laws of large numbers Int.,J. Theoret. Prob. 16 (3), (2003), 655- 669. [6] A. Mohammed, ‘‘Uniform Integrability of Approximate Green Functions of some Degenerate Elliptic Operators’’, Pacific Journal of Mathematics,199,(2),2001. [7] B.M. 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Probab. 13 (2), (2000) 343-356.  International Journal of Mathematics and Statistics Studies Vol.2, No.1, pp. 1-13, March 2014           Published by European Centre for Research Training and Development UK (www.ea-journals.org) 14  [20] V. Kruglov, ‘‘Growth of sums of pairwise-independent random variables with finite means’’, Teor. Veroyatn. Primen. 51 (2), (2006),382-385. [21] 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. [22] 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. [23] S.H. Sung, S. Lisa wadi, 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. [24] J. Bae, D. Jun, and S. Levental, ‘‘The uniform clt for martingale difference arrays under the uniformly integrable entropy’’, Bull. Korean Math. Soc. 47(1), (2010),39-51.   ACKNOWLEDGEMENT We would like to express our sincere thanks to the anonymous reviewers and referees for their helpful suggestions and valuable comments. 

On a Survey of Uniform Integrability of Sequences of Random Variables

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On a Survey of Uniform Integrability of Sequences of Random Variables

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