On the Borel-Cantelli Lemma

Transcription

1 On the Borel-Cantelli Lemma Alexei Stepanov, Izmir University of Economics, Turkey In the present note, we propose a new form of the Borel-Cantelli lemma. Keywords and Phrases: the Borel-Cantelli lemma, strong limit laws. AMS 2000 Subject Classification: 60G70, 62G30 Introduction Suppose A, A 2, is a sequence of events on a common probability space and that A c i denotes the complement of event A i. The Borel-Cantelli lemma (presented below as Lemma.) is used extensively for producing strong limit theorems. Lemma... If, for any sequence A, A 2, of events, P (A n ) <, (.) n= then P (A n i.o.) = 0, where i.o. is an abbreviation for infinitively often ; 2. If A, A 2, is a sequence of independent events and if then P (A n i.o.) =. P (A n ) =, (.2) n= The first part of the Borel-Cantelli lemma is generalized in Barndorff-Nielsen (96), and Balakrishnan and Stepanov (200). Department of Mathematics, Izmir University of Economics, 35330, Balcova, Izmir, Turkey;

2 2 The independence condition in the second part of the Borel-Cantelli lemma is weakened by a number of authors, including Chung and Erdos (952), Erdos and Renyi (959), Lamperti (963), Kochen and Stone (964), Spitzer (964), Ortega and Wschebor (983), and Petrov (2002), (2004). One can also refer to Martikainen and Petrov (990), and Petrov (995) for related topics. It should be noted that in all existing publications the sufficient condition in the second part of the Borel-Cantelli lemma is based on equality (.2) and some additional assumption. In our work, we prove the second part of the Borel-Cantelli lemma without any additional assumption. This allows us to derive a new and nice form of the Borel-Cantelli lemma. The rest of this paper is organized as follows. In Section 2 we present our main result. All technical results and their proofs are gathered in Appendix (Section 3). 2 Results Our main result is the following. Lemma 2.. Let A, A 2, be a sequence of events. Then. P (A n i.o.) = 0 iff (.) holds true, and 2. P (A n i.o.) = iff (.2) holds true. Proof The proof of this lemma consists of three parts.. In the first part, we state that This statement follows from Lemma 2.2. (.2) P (A n i.o.) =. Lemma 2.2. Let A, A 2,... be a sequence of events for which (.2) holds true. Then 2. In the second part, we state that This statement follows from Proposition 2. P (A n i.o.) =. P (A n i.o.) = (.2). Proposition 2.. Let A, A 2,... be a sequence of events such that P (A n i.o.) =. Then (.2) holds true.

3 3 3. To conclude the proof of Lemma 2., we analyze the above results. By parts., 2. of this proof and part. of the Borel-Cantelli lemma, we have P (A n i.o.) = (.2), (.) P (A n i.o.) = 0. It follows that Lemma 2. is proved. P (A n i.o.) = 0 (.). 3 Appendix Proof of Lemma 2.2 Observe that P {A n i.o.} = lim P ( ) A k (3.) and P {A n i.o.} = lim P ( To estimate the limit in (3.2) we need the following proposition. A c k ). (3.2) Proposition 3.. Let B, B 2,... be a sequence of events and p i >, q i > (i ) two number sequences such that p i + q i = (i ). Then for n 2 n P ( B i ) i= and [P (B )] p q [P (B 2 )] p 2 q... [P (B n )]...q n 2 p n [P (Bn )] i= i= q...q n (3.3) q P ( B i ) [P (B i )]...q i p i. (3.4) The proof of Proposition 3. will be given after the proof of Lemma 2.2. By (3.4), we have ( ) qn...q P [ P (A k )] k p k = Tn. A c k

4 4 Then log(t n ) = log( P (A k )) q n... q k p k P (A k ) q n... q k p k = K n. Our goal now is to find the conditions on the sequences p i and q i for K n to be valid. The following auxiliary proposition is well-known and given without proof. Proposition 3.2. Let A, A 2,... be a sequence of events for which (.2) holds true. Then [ ] P (A n ) n i= P (A =. (3.5) i) n= Let S n = n i= P (A i) and L n be the tail of the series in (3.5), i.e. L n = [ P (Ak ) S k ]. Suppose now that K n = L n, and all the terms in K n and L n are equal. Then we get the system of equations p n = S n q n p n+ = S n+ (3.6) q n q n+ p n+2 = S n+2..., where p i, q i are unknown variables and S i are known values. Choose n such that S n >. The solution of (3.6) is given by p n = S n q n = Sn S n p n+j = S n+j[s n...s n+j j j k=0 i=0,i k S n+i] S n S n+j (j ) q n+j = S n+j[s n...s n+j j j k=0 i=0,i k S n+i] (j ), where 0 k=0 0 i=0,i k S n+i =. Observe that S n S n+j j j k=0 i=0,i k S n+i p n+j S n+j and q n+j (j 0, n ). (3.7) The series in (3.5) is divergent, and K n = L n. Then K n provided that the sequences p i and q i (i n) in K n are determined by system (3.7). It follows from (3.2) that P {A n i.o.} = 0. The last observation concludes the proof of Lemma 2.2. Proof of Proposition 3. By Holder s inequality, for B and B 2, we have P (B B 2 ) [P (B )] p [P (B 2 )] q.

5 5 Replacing B and B 2 by C and C 2 C 3, respectively, and applying again the Holder inequality, we obtain P (C C 2 C 3 ) [P (C )] p q [P (C 2 )] p 2 q [P (C 3 )] q 2. By this argument one can come to (3.3). Since P ( lim n i= inequality (3.4) can be derived from (3.3). B i ) = lim P ( n B i ), i= Proof of Proposition 2. It follows from (3.) that P (A n i.o.) lim P (A k ). (3.8) The series in (3.8) can not be convergent under the condition P (A n i.o.) =. Otherwise, we would obtain a contradiction, because it would give us lim P (A k) = 0. We expect this work will be published soon in a statistical journal. References Balakrishnan, N., Stepanov, A. (200). Generalization of the Borel-Cantelli lemma. The Mathematical Scientist, 35 (), Barndorff-Nielsen, O. (96). On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables. Math. Scand., 9, Chung, K.L. and Erdos, P. (952). On the application of the Borel-Cantelli lemma. Trans. Amer. Math. Soc., 72, Erdos, P. and Renyi, A. (959). On Cantor s series with convergent /q n. Ann. Univ. Sci. Budapest. Sect. Math., 2, Kochen, S.B. and Stone, C.J. (964). A note on the Borel-Cantelli lemma. Illinois J. Math., 8, Lamperti, J. (963). Wiener s test and Markov chains. J. Math. Anal. Appl., 6, Martikainen, A.I., Petrov, V.V., (990). On the BorelCantelli lemma. Zapiski Nauch. Semin. Leningrad. Otd. Steklov Mat. Inst., 84, (in Russian). English translation in: (994). J. Math. Sci., 63, Petrov, V.V. (995). Limit Theorems of Probability Theory. Oxford University Press, Oxford.

6 6 Petrov, V.V. (2002). A note on the Borel-Cantelli lemma. Statist. Probab. Lett., 58, Petrov, V.V. (2004). A generalization of the BorelCantelli Lemma. Statist. Probab. Lett., 67, Ortega, J., Wschebor, M., (983). On the sequence of partial maxima of some random sequences. Stochastic Process. Appl., 6, Spitzer, F. (964). Principles of Random Walk. Van Nostrand, Princeton, New Jersey.