a limit theorem for random products of trimmed sums of i i d random variables

Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2011, Article ID 181409, 13 pages doi:10.1155/2011/181409 Research Article A Limit Theorem for Random Products of Trimmed Sums of i.i.d Random Variables Fa-mei Zheng School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China Correspondence should be addressed to Fa-mei Zheng, 16032@hytc.edu.cn Received 13 May 2011; Revised 25 July 2011; Accepted 11 August 2011 Academic Editor: Man Lai Tang Copyright q 2011 Fa-mei Zheng This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Let {X, Xi ; i ≥ 1} be a sequence of independent and identically distributed positive random variables with a continuous distribution function F, and F has a medium tail Denote Sn n n n max1≤i≤n Xi , i Xi , Sn a i Xi I Mn − a < Xi ≤ Mn and Vn i Xi − X , where Mn n X 1/n X , and a > is a fixed constant Under some suitable conditions, we show that i i nt k d t T k a /μk μ/Vn → exp{ W x /x dx} in D 0, , as n → ∞, where Tk a the trimmed sum and {W t ; t ≥ 0} is a standard Wiener process Sk − Sk a is Introduction n Let {Xn ; n ≥ 1} be a sequence of random variables and define the partial sum Sn i Xi n n and Vn X − X for n ≥ 1, where X 1/n X In the past years, the asymptotic i i i i behaviors of the products of various random variables have been widely studied Arnold and Villasenor considered sums of records and obtained the following form of the central ˜ limit theorem CLT for independent and identically distributed i.i.d exponential random variables with the mean equal to one, n k log Sk − n log n √ 2n n d −→ N as n −→ ∞ 1.1 d p a.s Here and in the sequel, N is a standard normal random variable, and → → , → stands for convergence in distribution in probability, almost surely Observe that, via the Stirling formula, the relation 1.1 can be equivalently stated as n k Sk k √ 1/ n d −→ e √ 2N 1.2 Journal of Probability and Statistics In particular, Rempała and Wesołowski removed the condition that the distribution is exponential and showed the asymptotic behavior of products of partial sums holds for any sequence of i.i.d positive random variables Namely, they proved the following theorem Theorem A Let {Xn ; n ≥ 1} be a sequence of i.i.d positive square integrable random variables with EX1 μ, Var X1 σ > and the coefficient of variation γ σ/μ Then, one has n k Sk n!μn √ 1/γ n d −→ e √ 2N as n −→ ∞ 1.3 Recently, the above result was extended by Qi , who showed that whenever {Xn ; n ≥ 1} is in the domain of attraction of a stable law L with index α ∈ 1, , there exists a numerical √ sequence An for α 2, it can be taken as σ n such that n k Sk n!μn μ/An d −→ e Γ α 1/α L , 1.4 ∞ as n → ∞, where Γ α xα e−x dx Furthermore, Zhang and Huang extended Theorem A to the invariance principle In this paper, we aim to study the weak invariance principle for self-normalized products of trimmed sums of i.i.d sequences Before stating our main results, we need to introduce some necessary notions Let {X, Xn ; n ≥ 1} be a sequence of i.i.d random variables with a continuous distribution function F Assume that the right extremity of F satisfies γF sup{x : F x < 1} ∞, 1.5 and the limiting tail quotient lim x→∞ F x F x a , 1.6 exists, where F x − F x Then, the above limit is e−ca for some c ∈ 0, ∞ , and F or X is said to have a thick tail if c 0, a medium tail if < c < ∞, and a thin tail if c ∞ Denote max1≤j≤n Xj For a fixed constant a > 0, we say Xj is a near-maximum if and only if Mn Xj ∈ Mn − a, Mn , and the number of near-maxima is Kn a : Card j ≤ n; X j ∈ Mn − a, Mn 1.7 These concepts were first introduced by Pakes and Steutel , and their limit properties have been widely studied by Pakes and Steutel , Pakes and Li , Li , Pakes , and Hu and Su Now, set n Sn a : i Xi I{Mn − a < Xi ≤ Mn }, 1.8 Journal of Probability and Statistics where I{A} ⎧ ⎨1, ω ∈ A, ⎩0, ω∈ / A, 1.9 T n a : S n − Sn a , which are the sum of near-maxima and the trimmed sum, respectively From Remark of a.s Hu and Su , we have that if F has a medium tail and EX / 0, then Tn a /n → EX, which 0, k ≥ 1} is finite at most Thus, we can implies that with probability one Card{k : Tk a if Tk a redefine Tk a Main Result Now we are ready to state our main results Theorem 2.1 Let {X, Xn ; n ≥ 1} be a sequence of positive i.i.d random variables with a continuous distribution function F, and EX μ, Var X σ Assume that F has a medium tail Then, one has nt k Tk a μk μ/Vn d −→ exp t W x dx x in D 0, , as n −→ ∞, 2.1 where {W t ; t ≥ 0} is a standard Wiener process In particular, when we take t 1, it yields the following corollary Corollary 2.2 Under the assumptions of Theorem 2.1, one has n k Tk a μk μ/Vn d −→ e √ 2N , 2.2 as n → ∞, where N is a standard normal random variable Remark 2.3 Since W x /x dx is a normal random variable with E E W x dx x W x dx x EW x dx x 0, 2.3 EW x W y dx dy xy Corollary 2.2 follows from Theorem 2.1 immediately x, y dx dy xy 4 Journal of Probability and Statistics Proof of Theorem 2.1 In this section, we will give the proof of Theorem 2.1 In the sequel, let C denote a positive constant which may take different values in different appearances and x mean the largest integer ≤ x a.s Note that via Remark of Hu and Su , we have Ck : Tk a /μk → It follows that for any δ > 0, there exists a positive integer R such that P sup|Ck − 1| > δ < δ 3.1 k≥R Consequently, there exist two sequences δm ↓ δ1 1/2 and R∗m ↑ ∞ such that P sup |Ck − 1| > δm < δm k≥R∗m 3.2 The strong law of large numbers also implies that there exists a sequence Rm ↑ ∞ such that a.s sup |Ck − 1| ≤ k≥Rm Here and in the sequel, we take Rm m 3.3 max{R∗m , Rm }, and it yields P sup |Ck − 1| > δm < δm k≥Rm 3.4 sup |Ck − 1| ≤ m k≥Rm a.s Then, it leads to P μ nt log Ck ≤ x Vn k μ nt log Ck ≤ x, sup |Ck − 1| > δm Vn k k≥Rm P P : Am,n μ nt log Ck ≤ x, sup |Ck − 1| ≤ δm Vn k k≥Rm Bm,n , 3.5 Journal of Probability and Statistics and Am,n < δm By using the expansion of the logarithm log x θ ∈ 0, depends on |x| < 1, we have that Bm,n μ nt log Ck ≤ x, sup |Ck − 1| ≤ δm Vn k k≥Rm ⎛ nt μ Rm ∧ nt −1 μ log Ck P⎝ Vn Vn k R ∧ nt −1 k x − x2 /2 θx , where P ⎞ log ⎛ μ Rm ∧ nt −1 P⎝ log Ck Vn k μ Vn k nt μ − Vn k nt Rm ∧ nt −1 ⎛ μ Rm ∧ nt −1 P⎝ log Ck Vn k μ Vn k nt μ − Vn k 12 ≤ x, sup |Ck − 1| ≤ δm ⎠ k≥Rm Ck − Rm ∧ nt −1 ⎞ θk Ck − ⎛ μ Rm ∧ nt −1 − P⎝ log Ck Vn k nt Ck − Rm ∧ nt −1 ⎞ θk Ck − 12 k≥Rm Ck − Rm ∧ nt −1 Ck − I sup |Ck − 1| ≤ δm ≤ x⎠ k≥Rm ⎞ nt μ Vn k Ck − ≤ x, sup |Ck − 1| ≤ δm ⎠ m Ck − ≤ x, sup |Ck − 1| > δm ⎠ Rm ∧ nt −1 k≥Rm : Dm,n − Em,n , 3.6 where θk k 1, , nt are 0-1 -valued and Em,n < δm Also, we can rewrite Dm,n as Dm,n P μ Rm ∧ nt −1 log Ck − Ck Vn k μ − Vn k nt Rm ∧ nt −1 Ck − 12 μ nt Ck − Vn k 1 θk Ck − I sup |Ck − 1| ≤ δm ⎞ 3.7 ≤ x⎠ k≥Rm Observe that, for any fixed m, it is easy to obtain μ Rm ∧ nt −1 log Ck − Ck Vn k p by noting that Vn2 → ∞ p −→ as n −→ ∞, 3.8 Journal of Probability and Statistics And if Rm ≥ nt − 1, then we have and, by observing that x2 / Fm,n ≤ ≤ Ck − k Rm C Vn k C Vn k θx nt 3.9 sup |Ck − 1| ≤ δm , I 3.10 k≥Rm ≤ 4x2 , then we can obtain Ck − nt Sk −1 μk Rm nt C Vn k Rm : Hm,n C p −→ 0, Vn θk Ck − 21 ≤ < nt Denote nt μ Vn a.s C nt − θ nt as n → ∞ If Rm < nt − 1, then Rm Fm,n : C nt − μ Vn Sk − Sk a −1 μk Rm nt C Vn k 3.11 Sk a μk Rm Lm,n For any ε > 0, by the Markov’s inequality, we have P √ nk nt Rm Sk −1 μk >ε nt C ≤ √ E ε n C √ k Rm nt ε nk Rm Sk −1 μk 3.12 Cσ √ εμ2 n k Sk Var μk nt Rm a.s −→ k p Then, Hm,n → To obtain this result, we need the following fact: n i Vn2 a.s → σ , n n i Xi − X Xi − μ a.s −→ 1, as n −→ ∞ 3.13 Indeed, n i Xi − X n i Xi − μ n i Xi − μ n i 1− −n μ−X Xi − μ μ−X n i 3.14 Xi − μ 2 n Journal of Probability and Statistics Now, we choose two constants N > and < δ < such that P |X − μ| > N < δ Hence, in view of the strong law of large numbers, we have for n large enough μ−X n i Xi − μ n μ−X ≤ ≤ 2 n i Xi − μ I Xi − μ > N μ−X n i N2 3.15 I Xi − μ > N o1 P X−μ >N N2 n n a.s o1 o 1, which together with 3.14 implies that n i n i Xi − X Xi − μ Vn2 n i a.s Xi − μ −→ 1, 3.16 as n → ∞ Furthermore, in view of the strong law of large numbers again, we obtain Vn2 n n i n i Xi − X Xi − μ · n i Xi − μ n a.s −→ σ , 3.17 a.s as n → ∞, where σ Var X > For Lm,n , by noting that Sn a /Sn → 0, as n → ∞ see Hu and Su , thus we can easily get Sn a n Sn a Sn a.s · −→ 0, Sn n 3.18 as n → ∞ Then, for any < δ < 1, there exists a positive integer R such that P sup k≥R Sk a ≥δ k δ ≤P ≤P C n Sk a Vn k μk C n Sk a >δ Vn k k > δ , sup k≥R δ Sk a 0, and N1 large enough, we can have that P n Kk a Mk >ε √ k nk P ≤P n Kk a Mk · √ >ε √ √ nk k k Mk n Kk a · δ > ε, sup √ < δ √ √ nk k k k≥N1 Mk P sup √ ≥ δ k k≥N1 3.26 √ √ a.s Observe that if F has a medium tail, then we have Mn / n Mn / log n log n/ n → by a.s noting that Mn / log n → 1/c , where c is the limit defined in Section Thus it follows Mk P sup √ ≥ δ k k≥N1 −→ 0, 3.27 Journal of Probability and Statistics as N1 → ∞ Further, by the Markov’s inequality and the bounded property of EKk a from Hu and Su , we have n Kk a Mk · δ > ε, sup √ < δ √ √ nk k k k≥N1 P ≤P δ n Kk a >ε √ √ nk k δ n EKk a ≤C √ √ ε nk k 3.28 δ n δ ≤C √ √ ≤C , ε ε nk k p and, hence, the proof of 3.22 is terminated Thus Lm,n → follows Finally, in order to complete the proof, it is sufficient to show that μ nt d Ck − −→ Vn k Yn t : t W x dx, x 3.29 and, coupled with 3.21 , we only need to prove Yn t : μ nt Sk −1 Vn k μk d −→ t W x dx x 3.30 Let ⎧ ⎪ ⎨ H f t Yn, t t ⎪ ⎩0, ⎧ ⎪ ⎪ ⎨ Vn k ⎪ ⎪ ⎩0, nt n f x dx, x t> , 0≤t≤ , Sk − μk , k 3.31 t> , 0≤t≤ It is obvious that t max 0≤t≤1 W x dx − H W t x t sup 0≤t≤ W x a.s dx −→ 0, x as −→ 3.32 Note that max|Yn t − Yn, t | 0≤t≤ nt max 0≤t≤ Vn k n Sk − μk Sk − μk ≤ , k Vn k k 3.33 10 Journal of Probability and Statistics and then, for any > 0, by the Cauchy-Schwarz inequality and 3.17 , it follows that lim lim sup P max|Yn t − Yn, t | ≥ →0 0≤t≤ n→∞ n Vn k ≤ lim lim sup P →0 n→∞ Sk − μk ≥ k n E Sk − μk C ≤ lim lim sup √ →0 n→∞ k nk n Sk − μk C ≤ lim lim sup √ √ Var √ →0 n→∞ nk k k 1/2 n C lim lim sup √ √ →0 n→∞ nk k C ≤ lim lim sup √ →0 n→∞ n n 3.34 Furthermore, we can obtain V ≤t≤1 n nt sup k n Sk − μk − k 1 V ≤t≤1 n ≤ sup ≤ n Vn nt n Sx − x μ dx x nt n 1 n Sx − x μ dx − x Sx − x μ dx x nt V ≤t≤1 n sup ≤ Therefore, uniformly for t ∈ n ≤t≤1 Vn nt nt 1 − x x nt Sx − x μ dx x 3.35 dx n maxk≤n Sk − μk maxk≤n ki Xi − μ ≤C nVn nVn C Vn nt n Sx − x μ dx x sup Sx − x μ maxk≤n Sk − μk sup Vn n ≤t≤1 ≤C Vn k n nt Sk − μk k n i Xi − μ a.s −→ n , , we have Vn nt n Sx − x μ dx x t oP Wn t dx x oP , 3.36 Journal of Probability and Statistics 11 where Wn t : S nt − nt μ /Vn Notice that H · is a continuous mapping on the space D 0, Thus, using the continuous mapping theorem c.f., Theorem 2.7 of Billingsley 10 , it follows that Yn, t H Wn d oP −→ H W t , t in D 0, , as n −→ ∞ 3.37 Hence, 3.32 , 3.34 , and 3.37 coupled with Theorem 3.2 of Billingsley 10 lead to 3.30 The proof is now completed Application to U-Statistics A useful notion of a U-statistic has been introduced by Hoeffding 11 Let a U-statistic be defined as −1 n Un h Xi1 , , Xim , m 4.1 1≤i1