Asymptotic distribution of products of sums of independent random variables
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1 Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege of Mathematics ad Iformatio Sciece, Hea Normal Uiversity, Hea, hia bigduckwyl@63.com; duhogxia4@gmail.com MS received 7 April 0; revised 0 October 0 Abstract. I the paper we cosider the asymptotic distributio of products of weighted sums of idepedet radom variables. Keywords. Asymptotic distributio; products of sums.. Itroductio Let X be a sequece of idepedet idetically distributed i.i.d. positive radom variables. It is well kow that the products of partial sums of i.i.d. positive, square itegrable radom variables r.v. are asymptotically log-ormal. This fact is a immediate cosequece of the classical cetral limit theorem LT. This poit, accordig to the authors, was first argued by Arold ad Villaseñor [], who cosidered the limitig properties of the sum of records. They obtaied the followig versio of the LT for a sequece of i.i.d. expoetial r.v. s X with the mea equal to oe: log S k log + d, as, where S k = k j= X j, k, ad is a stadard ormal r.v. Rempała ad Wesołowski [] have oted that this limit 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, Var X = σ > 0 ad the coefficiet of variatio γ = σ/μ. The /γ S k d e!μ. 83
2 84 Yalig Wag et al. Recetly, Gochigdaza ad Rempała [3] discussed a almost sure limit theorem for the product of the partial sums of i.i.d. positive radom variables as follows. Theorem GR. Let X be a sequece of i.i.d. positive square itegrable radom variables with EX = μ>0, Var X = σ. Deote γ = σ/μ the coefficiet of variatio. The for ay real x, N /γ lim N log N I S k!μ x = Fx, a.s., = where F is the distributio fuctio of the r.v. e. For further discussios of the LT, the authors refer to [5,8,]. Zhag ad Huag [3] obtaied the ivariace priciple of the product of sums of radom variables. It is perhaps worthy 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.! if oly existece of the first momet is assumed. Very recetly, Miao [6,7] obtaied LT ad ASLT for the product of some geeral partial sums. Miao ad Qia [0], Miao ad Mu [9] got the moderate deviatio of product of partial sums. I the preset paper we are iterested i the asymptotic distributio of products of weighted sums of idepedet radom variables, which is a simple ad iterestig model.. Mai results Theorem.. Let X k, k, be a triagular array of i.i.d. positive radom variables with idetical expectatio μ = EX, >0 ad variace σ = VarX,. Deote the coefficiet of variatio γ = σ/μ ad S k, = a, X, + a, X, + +a k, X k, for all k where {a k, } k, is a triagular array of positive real umbers with a k, = for all. Assume that E X, +δ < for some 0 <δ< ad S, μ 0 a.e.. Deote A = k ad assume that The we have lim e γ A k +δ/ A δ/ S k,k /μ γ A d e, where is a stadard ormal radom variable. = 0..
3 Asymptotic distributio of products of sums 85 Remark.. Sice k a i,k =, the from the iequality = a i,k k for k, we have That is to say A. k. Remark.3. Let us cosider the special case a i,k = /k for all i k. The the coditios. ad. are satisfied. It is easy to check that A log ad lim k +δ/ A δ/ = 0. Furthermore, by Markov s iequality, Rosethal s iequality see Lemma 3. ad c r - iequality, for ay r > 0, we have = S, P μ > r P X i, μ > rμ = = which implies.. rμ +δ E X i, μ +δ rμ +δ +δ rμ +δ = = +δ +δ +δ/ EX i, μ + +δ/ E X, μ +δ < E X i, μ +δ Throughout the paper, let be a positive costat which might ot be the same i each of its appearaces. 3. Several lemmas Before provig Theorem., we eed to state some lemmas.
4 86 Yalig Wag et al. Lemma 3. Rosethal s iequality [4]. If{X k, k } is a sequece of idepedet radom variables with EX k = 0, the for ay r, r r/ E X k c r EX k + E X k r, where c r is a positive costat, which depeds o r. Lemma 3.. Uder the assumptios of Theorem., for k ad i =,,...,k, let Y i,k = X i,k μ/μ. The we have +δ +δ/ E a i,k Y i,k a i,k. Proof. By Rosethal s iequality, we have +δ +δ/+ E a i,k Y i,k c +δ μ +δ E X i,k +δ E X i,k +δ where we used the iequality +δ/, a +δ i,k +δ/. Lemma 3.3. Uder the assumptios of Theorem., we have γ A Sk,k μ d. 3. Proof. Let Y i,k = X i,k μ/σ, i =,,...,k, k. The 3. becomes Now defie the A Z k, = A γ A d a i,k Y i,k. a i,k Y i,k, Sk,k μ = Z k,.
5 It is easy to check that Asymptotic distributio of products of sums 87 EZ k, = 0, VarZ k, = A, k =,..., ad Var Z k, = A =. I order to complete the proof we eed to check that the Lideberg coditio see p. 530 i [] is satisfied for the triagular array {Z k, }. For ay r > 0, from Lemma 3., we have EZk, I Z k, > r A +δ/ A +δ/ A +δ/ +δ E a i,k Y i,k max k +δ/ δ/ 0, k where we used the fact that max k. Hece the Lideberg coditio holds. Lemma 3.4. Uder the assumptios of Theorem., for k ad i =,,...,k, let Y i,k = X i,k μ/μ. The we have ad Proof. Deote A A a i,k Y i,k γ P P a i,k Y i,k U k = a i,k Y i,k γ,
6 88 Yalig Wag et al. the we have EU k = 0. Let U k = U k I U k A. By Lemma 3., we get P U k = U k, for some k P U k > A A +δ/ A +δ/ A +δ/ A +δ/ E U k +δ/ +δ E a i,k Y i,k + max k +δ/ δ/ Furthermore, for ay r > 0, by usig Lemma 3. agai, we have P U k EU k A > r A A δ/ A δ/ EU k EU k A E U k +δ/ EU k +δ/ δ/ 0, 3.5 where we used the coditio.. Note that EU k = 0. The we have EU k = EU k I U k > A E U k +δ/ A δ/ Hece implies3.. Next we prove 3.3. Let V k = a i,k Y i,k ad V k = V k I V k A.
7 Asymptotic distributio of products of sums 89 By a similar proof of 3., P V k = V k, for some k P V k > A A +δ/ ad A A +δ/ E V k +δ +δ/ E V k 3 A δ/ E V k +δ A δ/ +δ/ The 3.3 ca be obtaied easily. 4. Proof of Theorem. Here we will use the delta-method expasio to prove our results. Deote k = S k,k /μ. By the coditio., for ay δ>0, there exists a umber R such that for ay r > R, P sup k >δ k r δ. osequetly, there exist two sequeces {δ m } 0δ = / ad R m such that P sup k >δ m δ m. Takig ow ay real x ad ay m,wehave P γ log k + γ x A = P γ log k + γ x, sup k δ m A + P γ log k + γ x, sup k <δ m A := A m + B m, where A m δ m. Next we will cotrol the term B m. By the followig equality for the logarithm: log + x = x x + x θ x 3,
8 90 Yalig Wag et al. where θ 0, depeds o x,, wehave { R m B m = P γ log k + γ + A γ A log[ + k ]+ γ x, sup k <δ m k=r m+ R m = P γ log k + γ + A γ k A k=r m+ [ ] k γ γ A k=r m+ + 3γ k 3 x, sup A + θ k=r k k 3 k <δ m + m R m = P γ log k + γ + A γ k A k=r m+ [ ] γ k γ A k=r m+ + 3γ k 3 A + θ k=r k k 3 I sup k <δ m x + m R m P γ log k + γ + A γ k A k=r m+ [ ] γ k γ x, sup k δ m A := D m F m, k=r m+ where θ k, k =,..., are 0, -valued radom variables ad F m δ m. To estimate the term D m, we rewrite it as { D m = P γ R m [log k k + k ] A [ ] + γ k A γ k γ A + 3γ k 3 A + θ k=r m + k k 3 } I sup k <δ m x.
9 For ay fixed m, Asymptotic distributio of products of sums 9 γ R m [log k k + k ] P 0 A as. From Lemma 3.4, we get γ A [ k γ ] P 0. Furthermore, ote the followig elemetary iequality: for x < / ad ay θ 0, it follows that x 3 / + θ x 3 8 x 3. The by Lemma 3.4, 3γ A 8 3γ A k=r m + k 3 + θ k k 3 I sup k δ m k 3 P 0. Sice by Lemma 3.3, it follows that γ A k d which implies from the above discussio that D m x, where deotes the stadard ormal distributio fuctio. Fially, by observig that we have := P log = P γ A e γ A S kk /μ log k + γ = A m + D m F m, D m F m A m + D m. γ A x x Sice A m δ m 0 ad F m δ m 0asm, ad lim D m x for every m, the desired result ca be obtaied.
10 9 Yalig Wag et al. Ackowledgemets The authors are very grateful to the referees for their helpful commets ad very helpful suggestios, which have resulted i a improved presetatio of the paper. This work is supported by Hea Provice Foudatio ad Frotier Techology Research Pla Refereces [] Arold B ad Villaseñor J A, The asymptotic distributio of sums of records, Extremes [] Feller W, A itroductio to probability theory ad its applicatios, Secod editio 97 New York Lodo Sydey: Joh Wiley ad Sos Ic. vol. [3] Gochigdaza K ad Rempała G A, A ote o the almost sure limit theorem for the product of partial sums, Appl. Math. Lett [4] Li Z Y ad Bai Z D, Probability iequalities 00 Beijig: Sciece Press Beijig; Spriger: Heidelberg [5] Lu X W ad Qi Y, A ote o asymptotic distributio of products of sums, Statist. Probab. Lett [6] Miao Y, etral limit theorem ad almost sure cetral limit theorem for the product of some partial sums, Proc. Idia Acad. Sci. Math. Sci [7] Miao Y, A extesio of almost sure cetral limit theory for the product of partial sums, J. Dy. Syst. Geom. Theor [8] Miao Y ad Li J F, Asymptotic distributio of products of weighted sums of depedet radom variables, Austral. J. Math. Aal. Appl Article, 7 pp. [9] Miao Y ad Mu J Y, Moderate deviatios priciple for products of sums of radom variables, Sci. hia Math [0] Miao Y ad Qia B, A ote o the expoetial covergece rate for products of sums, Mat. Vesik [] Qi Y, Limit distributios for products of sums, Statist. Probab. Lett [] Rempała G A ad Wesołowski J, Asymptotics for products of sums ad U-statistics, Electro. omm. Probab [3] Zhag L X ad Huag W A ote o the ivariace priciple of the product of sums of radom variables, Electro. omm. Probab
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