Central limit theorem and almost sure central limit theorem for the product of some partial sums

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1 Proc. Idia Acad. Sci. Math. Sci. Vol. 8, No. 2, May 2008, pp Prited i Idia Cetral it theorem ad almost sure cetral it theorem for the product of some partial sums YU MIAO College of Mathematics ad Iformatio Sciece, Hea Normal Uiversity, Hea, Chia Departmet of Mathematics ad Statistics, Wuha Uiversity, Hubei, Chia yumiao728@yahoo.com.c MS received 26 Jauary 2007; revised 27 May 2007 Abstract. I this paper, we give the cetral it theorem ad almost sure cetral it theorem for products of some partial sums of idepedet idetically distributed radom variables. Keywords. Cetral it theorem; almost sure cetral it theorem; products of sums.. Itroductio Let X be a sequece of idepedet idetically distributed i.i.d. positive radom variables r.v.. Recetly there have bee several studies to the products of partial sums. It is well-kow that the products of i.i.d. positive, square itegrable radom variables are asymptotically log-ormal. This fact is a immediate cosequece of the classical cetral it theorem CLT. This poit, up to the kowledge of the author, was first argued by Arold ad Villaseñr [], who cosidered the itig properties of the sums of records. I their paper Arold ad Villaseñr obtaied the followig versio of the CLT for a sequece of i.i.d. expoetial r.v. s X with the mea equal to oe: log S k log + 2 L, as, where S k = k j= X j, k, ad is a stadard ormal r.v.. Rempała ad Wesołowski [8] have oted that this it behavior of a product of partial sums has a uiversal character ad holds for ay sequece of square itegrable, positive i.i.d. radom variables. Namely, they have proved the followig. Theorem RW. Let X be a sequece of i.i.d. positive square itegrable radom variables with EX = μ, VarX = σ 2 > 0 ad the coefficiet of variatio γ = σ/μ. The /γ S k L e 2!μ.. Recetly, Gochigdaza ad Rempała [4] discussed a almost sure it theorem for the product of the partial sums of i.i.d. positive radom variables as follows. 289

2 290 Yu Miao Theorem GR. Let X be a sequece of i.i.d. positive square itegrable radom variables with EX = μ>0, VarX = σ 2. Deote γ = σ/μ the coefficiet of variatio. The for ay real x, N log N = I /γ S k!μ x = F x, a.s..2 where F is the distributio fuctio of the r.v. e 2. For further discussios of the CLT, the author refers to [6,7]. Zhag ad Huag [0] obtaied the ivariace priciple of the product of sums of radom variables. It is perhaps worth to otice that by the strog law of large umbers ad the property of the geometric mea it follows directly that / S k a.s. μ.3! if oly the existece of the first momet is assumed. Throughout the preset paper let S,k = i= X i X k for all, k ad we are iterested i similar results as. ad Cetral it theorem Theorem 2.. Let X be a sequece of i.i.d. positive square itegrable radom variables with EX = μ, VarX = σ 2 > 0 ad the coefficiet of variatio γ = σ/μ. The /γ S,k L e μ, 2. where is a stadard ormal r.v. Proof. Let Y i = X i μ/σ, i =, 2,... The γ S,k μ = i k,i X i μ σ = Y k. 2.2 Therefore from the classical cetral it theorem ad EY i = 0, VarY i = for all i =, 2,..., we kow that γ S,k L μ. 2.3 Furthermore, let C,k = S,k / μ, k =, 2,... By the strog law of large umbers it follows that for ay δ>0, R such that P sup R, k C,k δ <δ.

3 CLT ad almost sure CLT 29 Takig δ < /2, for ayx R,wehave P γ logc,k x ad γ logc,k x, + P γ logc,k x, := A + B sup R, k sup R, k C,k δ C,k <δ A δ. 2.4 Next we will cotrol the term B. By the followig logarithm: x 2 log + x = x + + θx 2, where θ 0, depeds o x,,wehave B γ logc,k x, sup { γ C,k + γ sup C,k <δ R, k } R, k C,k <δ C,k 2 + θ k C,k 2 x, { [ ] γ C,k + γ C,k 2 + θ k C,k 2 I sup R, k C,k <δ } x { P γ C,k x, sup := D + F, R, k C,k δ where θ k,k =,..., are 0, -valued radom variables ad F δ. To estimate the term D, by the followig elemetary iequality: for x < /2 ad ay θ 0, it }

4 292 Yu Miao follows that x 2 / + θx 2 4x 2. The we have [ ] γ C,k 2 + θ k C,k 2 I sup C,k <δ R, k 4 γ C,k 2 P 0, 2.5 as. Relatio 2.5 is a cosequece of the Markov iequality, sice for ay r>0, 4 P γ C,k 2 r 4 rγ VarC,k = 4 rγ γ 2 0. Therefore D x. For ay x R,wehave /γ S,k P log μ x γ log C,k x = A + D + F which implies our result. 3. Almost sure cetral it theorem I this sectio we will cosider the almost sure cetral it theorem as.2. Startig with Brosamler [3] ad Schatte [9], i the past decade several authors ivestigated the a.s. cetral it theorem ad related logarithmic it theorems for partial sums of idepedet radom variables. The simplest form of the a.s. cetral it theorem [3], [9], [5] states that if X,X 2,... are i.i.d. radom variables with mea 0, variace ad partial sums S = i= X i, the N log N k I Sk x = x a.s. x, 3. k where I deotes idicator fuctio. Berkes ad Csáki [2] exteded this theory ad showed that ot oly the cetral it theorem, but every weak it theorem for idepedet radom variables, subject to mior techical coditios, has a aalogous almost sure versio. However uder our model we oly eed the simplest versio of 3.. Theorem 3.. Let X be a sequece of i.i.d. positive square itegrable radom variables with EX = μ>0, VarX = σ 2. Deote γ = σ/μ the coefficiet of variatio. The for ay real x, N log N = where F is the distributio fuctio of the r.v. e. /γ I S,k μ x = F x, a.s. 3.2

5 CLT ad almost sure CLT 293 Proof. Let Y i = X i μ/σ, i =, 2,... The EY i = 0 ad VarY i = for all i =, 2,... ad from 2.2, for ay real x,wehave N log N I γ S,k μ x = = N log N = I Y k x = x a.s. 3.3 Note that i order to prove 3.2 it is sufficiet to show that for ay real x, N log N I γ S,k log μ x = x, a.s. 3.4 = To this ed let, as before, C,k = S,k / μ ad ote that by the law of the iterated logarithm we have for, log log /2 max C,k =O a.s. k Sice for x < wehavelog + x = x + Rx with x 0 Rx/x 2 = /2, log C,k C,k C,k 2 log log log log log a.s., where deote the iequality up to some uiversal costat. Hece for almost every ω ad ay ε>0there exists 0 = 0 ω,ε,xsuch that for 0, I γ S,k μ x ε Thus 3.3 implies 3.4. I γ S,k log μ x I γ S,k μ x + ε. Ackowledgemets The author is very grateful to the referee for his valuable report ad some helpful suggestios.

6 294 Yu Miao Refereces [] Arold B C ad Villaseñr J A, The asymptotic distributio of sums of records, Extremes [2] Berkes I ad Csáki E, A uiversal result i almost sure cetral it theory, Stoch. Proc. Appl [3] Brosamler G, A almost everywhere cetral it theorem, Math. Proc. Cambridge Philos. Soc [4] Gochigdaza K ad Rempała G A, A ote o the almost sure it theorem for the product of partial sums, Appl. Math. Lett [5] Lacey M ad Philipp W A, ote o the almost everywhere cetral it theorem, Statist. Probab. Lett [6] Lu X W ad Qi Y C, A ote o asymptotic distributio of products of sums, Statist. Probab. Lett [7] Qi Y C, Limit distributios for products of sums, Statist. Probab. Lett [8] Rempała G A ad Wesołowski J, Asymptotics for products of sums ad U-statistics, Electro. Comm. Probab [9] Schatte P, O strog versios of the cetral it theorem, Math. Nachr [0] Zhag L X ad Huag W, A ote o the ivariace priciple of the product of sums of radom variables, arxiv:math.pr/06055 v, 7 Oct. 2006