# A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

Authors: Fa-mei Zheng
Journal: Journal of Probability and Statistics
 (2011)

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## Abstract

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=∑i=1nXi,Sn(a)=∑i=1nXiI(Mn-a<Xi≤Mn) and Vn2=∑i=1n(Xi-X̅)2, where Mn=max1≤i≤nXi, X̅=(1/n)∑i=1nXi, and a>0 is a fixed constant. Under some suitable conditions, we show that (∏k=1[nt](Tk(a)/μk))μ/Vn→dexp{∫0t(W(x)/x)dx}inD[0,1], as n→∞, where Tk(a)=Sk-Sk(a) is the trimmed sum and {W(t);t≥0} is a standard Wiener process.


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## Body

## 1. Introduction

Let{Xn;n≥1} be a sequence of random variables and define the partial sum Sn=∑i=1nXi and Vn2=∑i=1n(Xi-X̅)2 for n≥1, where X̅=1/n∑i=1nXi. In the past years, the asymptotic behaviors of the products of various random variables have been widely studied. Arnold and Villaseñor [1] 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, (1.1)∑k=1nlog⁡Sk-nlog⁡(n)+n2n⟶dNasn⟶∞.
Here and in the sequel, 𝒩 is a standard normal random variable, and →d (→p,→a.s.) stands for convergence in distribution (in probability, almost surely). Observe that, via the Stirling formula, the relation (1.1) can be equivalently stated as (1.2)(∏k=1nSkk)1/n⟶de2N.
In particular, Rempała and Wesołowski [2] 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,VarX1=σ2>0 and the coefficient of variation γ=σ/μ. Then, one has
(1.3)(∏k=1nSkn!μn)1/γn⟶de2Nasn⟶∞.Recently, the above result was extended by Qi [3], who showed that whenever {Xn;n≥1} is in the domain of attraction of a stable law ℒ with index α∈(1,2], there exists a numerical sequence An (for α=2, it can be taken as σn) such that (1.4)(∏k=1nSkn!μn)μ/An⟶de(Γ(α+1)1/α)L,
as n→∞, where Γ(α+1)=∫0∞xαe-xdx. Furthermore, Zhang and Huang [4] 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 (1.5)γF=sup⁡{x:F(x)<1}=∞,
and the limiting tail quotient (1.6)lim⁡x→∞F¯(x+a)F¯(x),
exists, where F¯(x)=1-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 0<c<∞, and a thin tail if c=∞. Denote Mn=max1≤j≤nXj. For a fixed constant a>0, we say Xj is a near-maximum if and only if Xj∈(Mn-a,Mn], and the number of near-maxima is (1.7)Kn(a)∶=Card{j≤n;Xj∈(Mn-a,Mn]}.
These concepts were first introduced by Pakes and Steutel [5], and their limit properties have been widely studied by Pakes and Steutel [5], Pakes and Li [6], Li [7], Pakes [8], and Hu and Su [9]. Now, set (1.8)Sn(a)∶=∑i=1nXiI{Mn-a<Xi≤Mn},
where (1.9)I{A}={1,ω∈A,0,ω∉A,Tn(a)∶=Sn-Sn(a),
which are the sum of near-maxima and the trimmed sum, respectively. From Remark 1 of Hu and Su [9], we have that if F has a medium tail and EX≠0, then Tn(a)/n→a.s.EX, which implies that with probability one Card{k:Tk(a)=0,k≥1} is finite at most. Thus, we can redefine Tk(a)=1 if Tk(a)=0.


## 2. 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,VarX=σ2. Assume that F has a medium tail. Then, one has
(2.1)(∏k=1[nt]Tk(a)μk)μ/Vn⟶dexp⁡{∫0tW(x)xdx}inD[0,1],asn⟶∞,
where {W(t);t≥0} is a standard Wiener process.In particular, when we taket=1, it yields the following corollary.Corollary 2.2.
Under the assumptions of Theorem2.1, one has
(2.2)(∏k=1nTk(a)μk)μ/Vn⟶de2N,
as n→∞, where 𝒩 is a standard normal random variable.Remark 2.3.
Since∫01(W(x)/x)dx is a normal random variable with
(2.3)E∫01W(x)xdx=∫01EW(x)xdx=0,E(∫01W(x)xdx)2=∬01EW(x)W(y)xydxdy=∬01min⁡(x,y)xydxdy=2.
Corollary 2.2 follows from Theorem 2.1 immediately.


## 3. Proof of Theorem2.1

In this section, we will give the proof of Theorem2.1. In the sequel, let C denote a positive constant which may take different values in different appearances and [x] mean the largest integer ≤x.Note that via Remark 1 of Hu and Su [9], we have Ck∶=Tk(a)/μk→a.s.1. It follows that for any δ>0, there exists a positive integer R such that (3.1)P(sup⁡k≥R|Ck-1|>δ)<δ.
Consequently, there exist two sequences δm↓0(δ1=1/2) and Rm*↑∞ such that (3.2)P(sup⁡k≥Rm*|Ck-1|>δm)<δm.
The strong law of large numbers also implies that there exists a sequence Rm′↑∞ such that (3.3)sup⁡k≥Rm′|Ck-1|≤a.s.1m.
Here and in the sequel, we take Rm=max{Rm*,Rm′}, and it yields (3.4)P(sup⁡k≥Rm|Ck-1|>δm)<δmsup⁡k≥Rm|Ck-1|≤a.s.1m.
Then, it leads to (3.5)P(μVn∑k=1[nt]log⁡(Ck)≤x)=P(μVn∑k=1[nt]log⁡(Ck)≤x,sup⁡k≥Rm|Ck-1|>δm)+P(μVn∑k=1[nt]log⁡(Ck)≤x,sup⁡k≥Rm|Ck-1|≤δm)=∶Am,n+Bm,n,
and Am,n<δm. By using the expansion of the logarithm log(1+x)=x-x2/2(1+θx)2, where θ∈(0,1) depends on |x|<1, we have that (3.6)Bm,n=P(μVn∑k=1[nt]log⁡(Ck)≤x,sup⁡k≥Rm|Ck-1|≤δm)=P(μVn∑k=1Rm∧([nt]-1)log⁡(Ck)+μVn∑k=(Rm∧([nt]-1))+1[nt]log⁡(1+Ck-1)≤x,sup⁡k≥Rm|Ck-1|≤δm)=P(μVn∑k=1Rm∧([nt]-1)log⁡(Ck)+μVn∑k=(Rm∧([nt]-1))+1[nt](Ck-1)-μVn∑k=(Rm∧([nt]-1))+1[nt](Ck-1)22(1+θk(Ck-1))2≤x,sup⁡k≥Rm|Ck-1|≤δm)=P(μVn∑k=1Rm∧([nt]-1)log⁡(Ck)+μVn∑k=(Rm∧([nt]-1))+1[nt](Ck-1)-μVn∑k=(Rm∧([nt]-1))+1[nt](Ck-1)22(1+θk(Ck-1))2I(sup⁡k≥Rm|Ck-1|≤δm)≤x)-P(μVn∑k=1Rm∧([nt]-1)log⁡(Ck)+μVn∑k=(Rm∧([nt]-1))+1[nt](Ck-1)≤x,sup⁡k≥Rm|Ck-1|>δm)=:Dm,n-Em,n,
where θk (k=1,…,[nt]) are (0-1)-valued and Em,n<δm.Also, we can rewriteDm,n as (3.7)Dm,n=P(μVn∑k=1Rm∧([nt]-1)(log⁡(Ck)-Ck+1)+μVn∑k=1[nt](Ck-1)-μVn∑k=(Rm∧([nt]-1))+1[nt](Ck-1)22(1+θk(Ck-1))2I(sup⁡k≥Rm|Ck-1|≤δm)≤x).
Observe that, for any fixed m, it is easy to obtain (3.8)μVn∑k=1Rm∧([nt]-1)(log⁡(Ck)-Ck+1)⟶p0asn⟶∞,
by noting that Vn2→p∞.And ifRm≥[nt]-1, then we have (3.9)μVn(C[nt]-1)22(1+(C[nt]-1)θ[nt])2≤a.s.CVn⟶p0,
as n→∞. If Rm<[nt]-1, then Rm+1<[nt]. Denote (3.10)Fm,n∶=(μVn∑k=Rm+1[nt](Ck-1)22(1+θk(Ck-1))2)I(sup⁡k≥Rm⁡|Ck-1|≤δm),
and, by observing that x2/(1+θx)2≤4x2, then we can obtain (3.11)Fm,n≤CVn∑k=Rm+1[nt](Ck-1)2=CVn∑k=Rm+1[nt](Sk-Sk(a)μk-1)2≤CVn∑k=Rm+1[nt](Skμk-1)2+CVn∑k=Rm+1[nt](Sk(a)μk)2=∶Hm,n+Lm,n.
For any ε>0, by the Markov’s inequality, we have (3.12)P(1n∑k=Rm+1[nt](Skμk-1)2>ε)≤CεnE(∑k=Rm+1[nt](Skμk-1)2)=Cεn∑k=Rm+1[nt]Var⁡(Skμk)=Cσ2εμ2n∑k=Rm+1[nt]1k⟶a.s.0.
Then, Hm,n→p0. To obtain this result, we need the following fact: (3.13)Vn2n→a.s.σ2,∑i=1n(Xi-X̅)2∑i=1n(Xi-μ)2⟶a.s.1,asn⟶∞.
Indeed, (3.14)∑i=1n(Xi-X̅)2∑i=1n(Xi-μ)2=∑i=1n(Xi-μ)2-n(μ-X̅)2∑i=1n(Xi-μ)2=1-(μ-X̅)2(∑i=1n(Xi-μ)2)/n.
Now, we choose two constants N>0 and 0<δ<1 such that P(|X-μ|>N)<δ. Hence, in view of the strong law of large numbers, we have for n large enough (3.15)(μ-X̅)2(∑i=1n(Xi-μ)2)/n≤(μ-X̅)2(∑i=1n(Xi-μ)2I(|Xi-μ|>N))/n≤(μ-X̅)2N2(∑i=1nI(|Xi-μ|>N))/n=o(1)N2(P(|X-μ|>N)+o(1))=a.s.o(1),
which together with (3.14) implies that (3.16)∑i=1n(Xi-X̅)2∑i=1n(Xi-μ)2=Vn2∑i=1n(Xi-μ)2⟶a.s.1,
as n→∞. Furthermore, in view of the strong law of large numbers again, we obtain (3.17)Vn2n=∑i=1n(Xi-X̅)2∑i=1n(Xi-μ)2⋅∑i=1n(Xi-μ)2n⟶a.s.σ2,
as n→∞, where σ2=Var(X)>0. For Lm,n, by noting that Sn(a)/Sn→a.s.0, as n→∞ (see Hu and Su [9]), thus we can easily get (3.18)Sn(a)n=Sn(a)Sn⋅Snn⟶a.s.0,
as n→∞. Then, for any 0<δ′<1, there exists a positive integer R′ such that (3.19)P(sup⁡k≥R′Sk(a)k≥δ′)<δ′.
Consequently, coupled with (3.18), we have (3.20)P(Lm,n>δ′)≤P(CVn∑k=1n(Sk(a)μk)2>δ′,sup⁡k≥R′Sk(a)k<δ′)+P(sup⁡k≥R′Sk(a)k≥δ′)≤P(CVn∑k=1nSk(a)k>δ′)+δ′.
Clearly, to show Lm,n→p0, as n→∞, it is sufficient to prove (3.21)1Vn∑k=1nSk(a)k⟶p0.
Indeed, combined with (3.17), we only need to show (3.22)1n∑k=1nSk(a)k⟶p0.
As a matter of fact, by the definitions of Sn(a) and Kn(a), we have (3.23)(Mn-a)Kn(a)<Sn(a)≤MnKn(a).
In view of the fact Mn↑∞(a.s.), we can get from Hu and Su [9] that (3.24)Sn(a)Mn~a.s.Kn(a),
and thus it suffices to prove (3.25)1n∑k=1nMkKk(a)k⟶p0.
Actually, for all ε,δ>0, and N1 large enough, we can have that (3.26)P(1n∑k=1nKk(a)Mkk>ε)=P(1n∑k=1nKk(a)k⋅Mkk>ε)≤P(1n∑k=1nKk(a)k⋅δ>ε,sup⁡k≥N1Mkk<δ)+P(sup⁡k≥N1Mkk≥δ).
Observe that if F has a medium tail, then we have Mn/n=(Mn/logn)(logn/n)→a.s.0 by noting that Mn/logn→a.s.1/c [9], where c is the limit defined in Section 1. Thus it follows (3.27)P(sup⁡k≥N1Mkk≥δ)⟶0,
as N1→∞. Further, by the Markov’s inequality and the bounded property of EKk(a) from Hu and Su [9], we have (3.28)P(1n∑k=1nKk(a)k⋅δ>ε,sup⁡k≥N1Mkk<δ)≤P(δn∑k=1nKk(a)k>ε)≤Cδεn∑k=1nEKk(a)k≤Cδεn∑k=1n1k≤Cδε,
and, hence, the proof of (3.22) is terminated. Thus Lm,n→p0 follows. Finally, in order to complete the proof, it is sufficient to show that (3.29)Yn(t)∶=μVn∑k=1[nt](Ck-1)⟶d∫0tW(x)xdx,
and, coupled with (3.21), we only need to prove (3.30)Yn(t)∶=μVn∑k=1[nt](Skμk-1)⟶d∫0tW(x)xdx.
Let (3.31)Hϵ(f)(t)={∫ϵtf(x)xdx,t>ϵ,0,0≤t≤ϵ,Yn,ϵ(t)={1Vn∑k=[nϵ]+1[nt]Sk-μkk,t>ϵ,0,0≤t≤ϵ.
It is obvious that (3.32)max⁡0≤t≤1|∫0tW(x)xdx-Hϵ(W)(t)|=sup⁡0≤t≤ϵ|∫0tW(x)xdx|⟶a.s.0,asϵ⟶0.
Note that (3.33)max⁡0≤t≤ϵ|Yn(t)-Yn,ϵ(t)|=max⁡0≤t≤ϵ1Vn∑k=1[nt]|Sk-μk|k≤1Vn∑k=1[nϵ]|Sk-μk|k,
and then, for any ϵ1>0, by the Cauchy-Schwarz inequality and (3.17), it follows that (3.34)lim⁡ϵ→0limsup⁡n→∞P(max⁡0≤t≤ϵ|Yn(t)-Yn,ϵ(t)|≥ϵ1)≤lim⁡ϵ→0limsup⁡n→∞P(1Vn∑k=1[nϵ]|Sk-μk|k≥ϵ1)≤lim⁡ϵ→0limsup⁡n→∞Cn∑k=1[nϵ]E|Sk-μk|k≤lim⁡ϵ→0limsup⁡n→∞Cn∑k=1[nϵ]1k(Var⁡(Sk-μkk))1/2=lim⁡ϵ→0limsup⁡n→∞Cn∑k=1[nϵ]1k≤lim⁡ϵ→0limsup⁡n→∞Cn[nϵ].
Furthermore, we can obtain (3.35)sup⁡ϵ≤t≤11Vn|∑k=[nϵ]+1[nt]Sk-μkk-∫nϵntS[x]-[x]μxdx|≤sup⁡ϵ≤t≤11Vn|∫[nϵ]+1[nt]+1S[x]-[x]μ[x]dx-∫nϵntS[x]-[x]μxdx|≤1Vn|∫nϵ[nϵ]+1S[x]-[x]μxdx|+sup⁡ϵ≤t≤11Vn|∫nt[nt]+1S[x]-[x]μxdx|+sup⁡ϵ≤t≤11Vn|∫[nϵ]+1[nt]+1(S[x]-[x]μ)(1x-1[x])dx|≤max⁡k≤n|Sk-μk|Vnsup⁡ϵ≤t≤1(2nϵ+2nt+1nϵ)≤Cmax⁡k≤n|Sk-μk|nVn≤Cmax⁡k≤n∑i=1k|Xi-μ|nVn=CVn∑i=1n|Xi-μ|n⟶a.s.0.
Therefore, uniformly for t∈[ϵ,1], we have (3.36)1Vn∑k=[nϵ]+1[nt]Sk-μkk=1Vn∫nϵntS[x]-[x]μxdx+oP(1)=∫ϵtWn(t)xdx+oP(1),
where Wn(t)∶=(S[nt]-[nt]μ)/Vn. Notice that Hϵ(·) is a continuous mapping on the space D[0,1]. Thus, using the continuous mapping theorem (c.f., Theorem 2.7 of Billingsley [10]), it follows that (3.37)Yn,ϵ(t)=Hϵ(Wn)(t)+oP(1)⟶dHϵ(W)(t),inD[0,1],asn⟶∞.
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.


## 4. Application toU-Statistics

A useful notion of aU-statistic has been introduced by Hoeffding [11]. Let a U-statistic be defined as (4.1)Un=(nm)-1∑1≤i1<⋯<im≤nh(Xi1,…,Xim),
where h is a symmetric real function of m arguments and {Xi;i≥1} is a sequence of i.i.d. random variables. If we take m=1 and h(x)=x, then Un reduces to Sn/n. Assume that Eh(X1,…,Xm)2<∞, and let (4.2)h1(x)=Eh(x,X2,…,Xm),Ûn=mn∑i=1n(h1(Xi)-Eh)+Eh.
Thus, we may write (4.3)Un=Ûn+Rn,
where (4.4)Rn=(nm)-1∑1≤i1<⋯<im≤nH(Xi1,…,Xim),H(x1,…,xm)=h(x1,…,xm)-∑i=1m(h1(xi)-Eh)-Eh.
It is well known (cf. Resnick [12]) that (4.5)Cov(Ûn,Rn)=0,nVar⁡((nm)-1Rn)⟶0,asn⟶∞.
Theorem 2.1 now is extended to U-statistics as follows.Theorem 4.1.
LetUn be a U-statistic defined as above. Assume that Eh2<∞ and P(h(X1,…,Xm)>0)=1. Denote μ=Eh>0 and σ2=Var(h1(X1))≠0. Then,
(4.6)(∏k=m[nt]Ukμ(nm))μ/mVn⟶dexp⁡{∫0tW(x)xdx},inD[0,1],asn⟶∞,
where W(x) is a standard wiener process, and Vn2=∑i=1n(Xi-X̅)2.In order to prove this theorem, by (3.17), we only need to prove (4.7)μmn∑k=m[nt](Ukμ(nm)-1)⟶dσ∫0tW(x)xdx,inD[0,1],asn⟶∞.
If this result is true, then with the fact that (nm)-1Un→a.s.Eh=μ deduced from E|h|<∞ (see Resnick [12]) and (4.3), Theorem 4.1 follows immediately from the method used in the proof of Theorem 2.1 with Sk/k replaced by (nm)-1Un. Now, we begin to show (4.7). By (4.3), we have (4.8)μmn∑k=m[nt](Ukμ(nm)-1)=μmn∑k=m[nt](Ûkμ(nm)-1)+μmn∑k=m[nt]Rkμ(nm).
By applying (3.30) to random variables mh1(Xi) for i≥1, we have (4.9)μmn∑k=m[nt](Ûkμ(nm)-1)=μn(∑k=1n(∑i=1kh1(xi)μk-1)-∑k=1m-1(∑i=1kh1(xi)μk-1))⟶dσ∫0tW(x)xdx,
in D[0,1], as n→∞, since the second expression converges to zero a.s. as n→∞. Therefore, for proving (4.7), we only need to prove (4.10)μmn∑k=m[nt]Rkμ(nm)⟶p0,asn⟶∞,
and it is sufficient to demonstrate (4.11)R̃n∶=μmn∑k=mnRkμ(nm)⟶p0,asn⟶∞.
Indeed, we can easily obtain ER̃n2→0 as n→∞ from Hoeffding [11]. Thus, we complete the proof of (4.7), and, hence, Theorem 3.1 holds.



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