A Note on the Kolmogorov-Feller Weak Law of Large Numbers

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1 Joural of Mathematical Research with Applicatios Mar., 015, Vol. 35, No., pp. 3 8 DOI: /j.iss: A Note o the Kolmogorov-Feller Weak Law of Large Numbers Yachu YI 1, Dehua QIU, 1. Departmet of Mathematics ad Computioal Sciece, Hegyag Normal Uiversity, Hua 41008,. R. Chia;. School of Mathematics ad Statistics, Guagdog Uiversity of Fiace ad Ecoomics, Guagdog 51030,. R. Chia Abstract I this paper, the Kolmogorov-Feller type weak law of large umbers are obtaied, which exted ad improve the related kow works i the literature. Keywords Kolmogorov-Feller type weak law of large umbers; egatively associated radom variables; idepedet idetically distributed radom variables MR010) Subject Classificatio 60F15 1. Itroductio The celebrated Kolmogorov-Feller weak law of large umbers WLLN) provides a ecessary ad sufficiet coditio i the i.i.d. case, the poit beig that the mea does ot exist. Theorem 1.1 [1, VII.7]) Let X, X 1, X,... be idepedet idetically distributed i.i.d.) radom variables with partial sums S = X i, 1. The S EXI X ) 0 as if ad oly if x X > x) 0 as x. Gut [] gave the followig example: Example 1. desity Suppose that X, X 1, X,... are idepedet radom variables with commo fx) = { 1 x, for x > 1, 0, otherwise. The mea does ot exist, i this case the Feller coditio becomes But X > ) = 1 dx = 1. x S 0 as. log Received December 13, 013; Accepted Jauary 16, 015 Supported by the Natioal Natural Sciece Foudatio of Chia Grat Nos ; ). * Correspodig author address: qiudhua@sia.com Dehua QIU)

2 4 Yachu YI ad Dehua QIU I other words, a weak law exists, but, with aother ormalizatio. Motivated by this example, Gut [] provided the followig geeral Kolmogorov-Feller weak law of large umbers. Theorem 1.3 Let X, X 1, X,... be i.i.d. radom variables with partial sums S, 1. Further, let bx) be a icreasig ad regular varyig fuctio at ifiity with idex 1/ρ for some ρ 0, 1]. Fially, set b = b), 1. The S EXI X b ) b if ad oly if X > b ) 0 as. 0 as Motivated by Theorem 1.3, we provide the followig more geeral Kolmogorov-Feller type weak law of large umbers Theorems.4 ad.5 which exted ad improve Theorem 1.3. A fiite family of radom variables {X i, 1 i } is said to be egatively associated NA) if for every pair of disjoit subsets A ad B of {1,,..., }, Covf 1 X i, i A), f X j, j B)) 0, wheever f 1 ad f are coordiatewise icreasig ad such that the covariace exists. A ifiite family of radom variables {X i, i 1} is NA if for every positive iteger, {X i, 1 i } is NA. This defiitio was itroduced by Alam ad Saxea [3] ad carefully studied by Block et al. [4] ad Joag-Dev ad roscha [5]. NA sequeces have may good properties ad extesive applicatios i multivariate statistical aalysis ad reliability theory. We refer to Joag-Dev ad roscha [5] for fudametal properties, Matula [6] for the Kolmogorov type strog law of large umbers ad the three series theorem, Su et al. [7] for a momet iequality, a weak ivariace priciple ad a example to show that there exists ifiite families of odegeerate o-idepedet strictly statioary NA radom variables, Shao [8] for the Rosethal type maximal iequality ad the Kolmogorov expoetial iequality, Qiu ad Yag [9] for strog laws of large umbers, ad so o. Throughout this paper, we assume that {X, X, 1} is a sequece of idetically distributed radom variables, {k, 1} is a sequece of positive itegers such that lim k =, S k = k X i, C always stads for a positive costat which may differ from oe place to aother.. Mai results ad proofs I order to prove the mai result of this paper, we preset the followig Lemmas: Lemma.1 Let {X, X, 1} be a sequece of idetically distributed NA radom variables. The for ay t > 0 roof Sice 1 x e x, we have X > t) l 1 max 1 j X j > t) )..1) max 1 j X j > t) = 1 max 1 j X j t) = 1 X 1 t, X t,..., X t)

3 A ote o the Kolmogorov-Feller weak law of large umbers 5 1 X j t) = 1 { X t)} j=1 = 1 {1 X > t)} 1 exp{ X > t)}, therefore X > t) l1 max 1 j X j > t)). Replacig the X j by X j ad repeatig the above argumet will establish Hece,.1) holds. X > t) l 1 max 1 j X j > t) ). Lemma. Let X, X 1, X,..., X be idetically distributed NA radom variables. The for ay t > 0 max 1 j S j > t) 1 e 1 X >t)..) roof By Lemma.1 ad max 1 j S j > t) max 1 j X j > t),.) holds. Lemma.3 Let X, X 1, X,..., X be symmetric i.i.d. radom variables. The for ay t > 0 S > t) 1 1 e 1 X >t))..3) roof Note that idepedet radom variables are NA radom variables, by Lemma.1 ad 5.7.b of [10],.3) holds. Now we preset the mai result of this paper. Theorem.4 Let {X, X, 1} be a sequece of idetically distributed NA radom variables, {b, 1} be a sequece of icreasig positive reals. i) The followig statemets are equivalet: ii) If k b k X > b ) 0 as,.4) max 1 j k X j b 0 as..5) = o1), k b k i 1 = O1),.6) where b 0 = 0, k 0 = 1, the.4),.5) ad the followig statemet are equivalet: max 1 j k S j jexi X b ) b 0 as..7) roof i).4)=.5) is obvious. By Lemma.1, we have that.5)=.4). ii).4)=.7). For 1 j k, 1, set Note that for ε > 0 Y ) j = b IX j < b ) + X j I X j b ) + b IX j > b ). max 1 j k S j jexi X b ) > εb )

4 6 Yachu YI ad Dehua QIU = max S j jexi X b ) > εb ad X i b for all i k ) 1 j k max S j jexi X b ) > εb ad X i > b for at least oe i {1,,..., k }) 1 j k j ) max Y i EY ) i b X i < b ) + b X i > b ) ) > εb ) 1 j k k ) X i > b ). By.4) k ) X i > b ) k X i > b ) = k X > b ) 0 as. Sice Y ) 1 EY ) 1, Y ) EY ),..., Y ) k EY ) k are NA radom variables for every 1, by.4), Theorem of Shao [8],.6) ad Toeplitz Lemma [11], for large eough, we have j max 1 j k 4ε b = C k b C k b C k b = C k b Y ) i j max 1 j k j max 1 j k k EY ) i b X i < b ) + b X i > b ) ) > εb ) Y ) i Y ) i EY ) ) ) i + k b X > b ) > εb EY ) ) i > εb /) E Y ) i C k { E X b I X b ) + b X > b ) } E X Ib i 1 < X b i ) + Ck X > b ) b i { X > b i 1 ) X > b i )} + Ck X > b ) b i b i 1) X > b i 1 ) + Ck X > b ) Therefore,.7) holds..7)=.4). costat a ad ε > 0, we have k i 1 k i 1 X > b i 1 ) + Ck X > b ) 0,. By Lemma. ad X mx) > ε) 4 X a > ε/) for every max 1 j k S j jexi X b ) > εb ) 1 e 1 k X EXI X b ) >εb ) 1 e 1 8 k X >4εb + mx) ), where mx) deotes the media of X. Therefore,.4) holds by.7).

5 A ote o the Kolmogorov-Feller weak law of large umbers 7 Theorem.5 Let {X, X, 1} be a sequece of i.i.d. radom variables, {b, 1} be a sequece of icreasig positive reals. The i).4) ad.5) are equivalet. ii) If.6) holds, the.4),.5),.7) ad the followig statemet are equivalet: S k k EXI X b ) b 0 as..8) roof From the proof of Theorem.4, it is eough to prove that.7)=.8) ad.8)=.4)..7)=.8) is obvious. We prove that.8)=.4). By the weak symmetrizatio iequalities [10] ad Lemma.3, we have S k k EXI X b ) > εb ) Sk S > εb ) 1 1 e 1 k XS >εb ) ) 1 1 e 1 4 k X >εb+ mx) )), where X S deotes the symmetrized versio of X, Sk S = X1 S + X S + + Xk S, mx) deotes the media of X. Therefore,.4) holds by.8). Remark.6 Suppose that bx) is a icreasig ad regular varyig fuctio at ifiity with idex 1/ρ for some ρ 0, 1], ad set b = b), k =, 1. The.6) holds. Therefore, Theorem 1.3 is obtaied from Theorem.5. We preset two examples to illustrate Theorem.5. Example.7 I Example 1., we take b =, k = [ ], where [x] deotes the greatest iteger ot exceedig x. Thus, ad k b = o1) ad k b k X > b ) = [ ] [ ] { k i Therefore, by Theorem.5, we have Example.8 desity 1 x dx = [ ] 0 as i= max 1 j [ ] S j jexi X ) i } [ 4 [ ] i 1] i 4. 0 as. Suppose that X, X 1, X,... are idepedet radom variables with commo fx) = { l 3) xl x), 3 for x > 3, 0, otherwise. The mea does ot exist. Let k =. If we take b = b), 1, where bx) is a arbitrary icreasig ad regular varyig fuctio with idex 1/ρ for some ρ 0, 1], the X > b ) = l 3) b xl x) 3 dx = C as, l b )

6 8 Yachu YI ad Dehua QIU therefore, i this case, by Theorem 1.3 But, if we take bx) = exp x), b = b), the S EXI X b ) b 0 as. X > b ) = l 3) b xl x) 3 dx = C 1 0 as ad k b = e = o1). Sice fx) = e x /x, x [, ) is a icreasig fuctio, we have k b k i 1 = b Therefore, by Theorem.5, we have {b + b b 1 1) ) + b } 1 4 { b b + b b 1 1) + b } 1 4 {b b + b } < max 1 j S j jexi X e ) e 0 as. I other words, a weak law exists, but, with aother ormalizatio. Ackowledgemets We thak the referees for their time ad commets. Refereces [1] W. FELLER. A itroductio to probability theory ad its applicatios II). Secod editio, Joh Wiley & Sos, Ic., New York-Lodo-Sydey, [] A. GUT. A extesio of the Kolmogorov-Feller weak law of large umbers with a applicatio to the St. etersburg game. J. Theoret. robab., 004, 173): [3] K. ALAM, K. M. L. SAXENA. ositive depedece i multivariate distributios. Comm. Statist. A Theory Methods, 1981, 101): [4] H. W. BLOCK, T. H. SAVITS, M. SHAKED. Some cocepts of egative depedece. A. robab., 198, 103): [5] K. JOAG-DEV, F. ROSCHAN. Negative associatio of radom variables, with applicatios. A. Statist., 1983, 111): [6]. MATULA. A ote o the almost sure covergece of sums of egatively depedet radom variables. Statist. robab. Lett., 199, 153): [7] Chu SU, Licheg ZHAO, Yuebao WANG. Momet iequalities ad weak covergece for egatively associated sequeces. Sci. Chia Ser. A, 1997, 40): [8] Qima SHAO. A compariso theorem o maximal iequalities betwee egatively associated ad idepedet radom variables. J. Theoret. robab., 000, 13): [9] Dehua QIU, Xiagqu YANG. Strog laws of large umbers for weighted sums of idetically distributed NA radom variables. J. Math. Res. Expositio, 006, 64): [10] Zhegya LIN, Zhidog BAI. robability Iequalities. Sciece ress, Beijig, 010. [11] M. LOÈVE. robability Theory I). 4th ed. Spriger-Verlag, New York, 1977.