# Functional Limit Theorem for Products of Sums of Independent and Nonidentically Distributed Random Variables

Authors: Przemysław Matuła; Iwona Stępień
Journal: Journal of Mathematics
 (2013)

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## Abstract

We study the weak convergence in the spaceD[0,1]
of processes constructed from products of sums of independent
but not necessarily identically distributed random variables. The
presented results extend and generalize limit theorems known so
far for i.i.d. sequences.


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## Body

## 1. Introduction

Let(Xn)n∈ℕ be a sequence of independent, positive, and square-integrable random variables (r.v.’s) defined on some probability space (Ω,𝔉,P). For every n∈ℕ let us put
(1)μn=EXn,τn2=Var(Xn),Sn=∑k=1n‍Xk,σn2=Var(Sn)=∑k=1n‍τk2.
Arnold and Villaseñor [1] obtained the central limit theorem for sums of records. Rempała and Wesołowski [2] observed that this result was of general nature and proved that for a sequence of i.i.d. positive and square-integrable r.v.’s one has
(2)(∏i=1nSin!μn)1/γn→de2𝒩,asn→∞,
where 𝒩 is a standard normal variable and μ=μn and γ=τn/μn are the common mean and the coefficient of variation of the r.v.’s. The convergence in (2) was generalized and extended by many authors in different ways. For example, Zhang and Huang [3] studied the functional version of this result. They proved under general conditions, not involving the dependence structure of the sequence, that
(3)(∏i=1[nt]Sin!μn)1/γn→dexp(∫0t‍W(x)xdx)inD[0,1],asn→∞,
here and in the sequel W(x) denotes the Wiener process. The functional version of the convergence of products of sums was also studied by Kosiński [4], for i.i.d. sequences belonging to the domain of attraction of the α-stable law with α∈(1,2]. The first result for sequences of nonidentically distributed r.v.’s was obtained by Matuła and Stȩpień [5]. Similarly as in the invariance principle, the study of the non-i.i.d. sequences requires defining the processes in a different way than in (3). For t∈[0,1] and each n∈ℕ let us define the function mn(t)=max{i∈ℕ∪{0}:σi2≤tσn2}, where σ02=0. Let us introduce the process
(4)Vn(t):=(∏i=1mn(t)‍(SiESi)ai,n)1/σn
and Vn(t)=1 for t such that mn(t)=0, where ai,n=f(σi2/σn2)(τi+12/σn2)ESi. In [5] it was proved that for a sequence of independent, positive, and square-integrable random variables satisfying the Lindeberg condition and such that ∑i=1∞‍τi+12/(ESi)2<∞, the process Vn(t) converges weakly in the space D[0,1] to the process exp(∫0t‍f(x)W(x)dx). This result was proved in one theorem for the function f(x)=1/x and in the second for functions f∈C1[0,1].Our goal is to prove the aforementioned result for a large class of continuous functions on(0,1], which may be unbounded. Furthermore, we shall slightly change the process Vn(t) in order to avoid its artificial definition in the case mn(t)=0 and replace the strange shifted index in τi+12. In this setting, the imposed conditions and obtained results will be much more natural.


## 2. Main Results

Letℋ be the family of functions f:[0,1]→ℝ which are continuously differentiable on (0,1],that is,f∈𝒞1(0,1], furthermore, such that
(5)∫01|f(x)|xlog(1x)dx<∞,(6)f(x)x3/2→0,asx→0+.
Let us observe that from (5) it follows that f(x)x is integrable, if this function is also monotonic then, in consequence, (6) is satisfied. It is obvious that 𝒞1[0,1]⊂ℋ. The family ℋ also contains functions of the form f(x)=x-p with p∈(0,3/2) and f(x)=1/(x3/2logr(1/x∧e)) with r>3/2 and the value at 0 is arbitrary.For anyn∈ℕ let us define the process
(7)Tn(t):=(∏i=1mn(t)+1‍(SiESi)αi,n)1/σn,
where αi,n=f(σi2/σn2)(τi2/σn2)ESi and we also take mn(1)=n-1. In this way we obtain a sequence of processes in D[0,1] such that
(8)Tn(t)=(∏i=1k‍(SiESi)αi,n)1/σn,fort∈〈σk-12σn2,σk2σn2),k=1,2,…,n-1,Tn(t)=(∏i=1n‍(SiESi)αi,n)1/σn,fort∈〈σn-12σn2,1〉,
recall that we have defined σ02=0. Under our assumptions the trajectories of the process Tn(t) are almost surely positive functions.Our main result concerning weak convergence ofTn(t) in the space D[0,1] is as follows.Theorem 1.
Let(Xn)n∈ℕ be a sequence of independent, positive, and square-integrable r.v.’s satisfying the Lindeberg condition:
(9)limn→∞1σn2∑i=1n‍E((Xi-EXi)2I[|Xi-EXi|≥εσn])=0,∀ε>0,
and such that
(10)ESn→∞,asn→∞,∑i=1∞‍τi2(ESi)2<∞.
Then for any f∈ℋ(11)Tn(t)→dexp(∫0tf(x)W(x)dx)inD[0,1],asn→∞.In order to illustrate our result let us present two examples.Example 2.
Let(ξk)k∈ℕ be a sequence of independent r.v.’s with Poisson distribution Po(λk) with parameter λk=k. Let us take Xk=ξk+1, then Xk≥1 almost surely, EXk=k+1 and τk2=Var(Xk)=k. Moreover, ESn=2+⋯+(n+1)=n(n+3)/2 and σn2=VarSn=1+⋯+n=n(n+1)/2. Let us observe that E|Xk-EXk|3=E|ξk-Eξk|3 but ξk may be considered as a sum of i.i.d. (Poisson) r.v.’s. Thus, from the well-known moment bounds, there exists a constant C>0 not depending on k such that E|ξk-Eξk|3≤C·k3/2, for every k∈ℕ. Therefore, the Lyapunov condition may be easily verified. The condition (10) is also satisfied. We apply our Theorem 1 with f(x)=1/x, then αi,n=(τi2/σi2)ESi=i(i+3)/(i+1) and we have the following weak convergence in the space D[0,1], as n→∞(12)(∏i=1mn(t)+1‍(2Sii(i+3))i(i+3)/(i+1))1/n(n+1)/2→dexp(∫0t‍W(x)xdx).It may be easily checked (see page  42 in [6] for details) that the integral ∫01‍xpW(x)dx has Gaussian distribution 𝒩(0,σ2) with
(13)σ2=∬01min(x,y)xpypdxdy=2(p+2)(2p+3),
provided p>-3/2. Therefore, we have the following example.Example 3.
Let(Xk)k∈ℕ be a sequence of i.i.d. r.v.’s with the standard exponential distribution. Then EXk=Var(Xk)=1 and σn2=n. We apply Theorem 1 with t=1 and the function f(x)=xp, where p>-3/2. In this case αi,n=(i/n)p+1 and we get
(14)(∏i=1n‍(Sii)ip+1)1/n(p+3/2)→dexp(2(p+2)(2p+3)𝒩),asn→∞.


## 3. Proofs and Auxiliary Results

In the proof of our main result, we shall use the following lemma.Lemma 4.
LetdD[0,1] be the metric making the space D[0,1] to be a complete and separable metric space (Polish space) and let dU(x(t),y(t))=sup0≤t≤1|x(t)-y(t)| be the uniform metric. Furthermore, denote by dLPD[0,1] the Lévy-Prohorov metric in the space of probability measures on D[0,1] and by dKFD[0,1] the Ky-Fan metric. Let X, Y, and Xn be random elements with values in D[0,1], then (a)
ifEdU(X,Y)≤a, then dKFD[0,1](X,Y)≤a,(b)
dLPD[0,1](X,Y)≤dKFD[0,1](X,Y),(c)
ifdU(Xn,X)→0 almost surely, then dKFD[0,1](Xn,X)→0.Proof.
The proof is based on the inequalitydD[0,1]≤dU, which may be found on page  150 in [7]. The inequality (b) is proved, for example, in Theorem 11.3.5 on page  397 in [8].Proof of Theorem1.
In the first step we shall prove that(15)logTn(t)=1σn∑i=1mn(t)+1‍f(σi2σn2)τi2σn2(Si-ESi)+An(t),
where An→P0 in D[0,1], as n→∞. We shall apply the expansion of the logarithm log(1+x)=x+R(x), for |x|≤1/2, where |R(x)|≤2x2. Let us put Ci=(Si-ESi)/ESi, then we easily get
(16)log(SiESi)=log(Ci+1)=Ci+R(Ci)I[|Ci|≤12]+[log(1+Ci)-Ci]×I[|Ci|>12],logTn(t)=1σn∑i=1mn(t)+1‍f(σi2σn2)τi2ESiσn2log(SiESi)=1σn∑i=1mn(t)+1‍f(σi2σn2)τi2ESiσn2Ci+1σn∑i=1mn(t)+1‍f(σi2σn2)τi2ESiσn2R(Ci)I[|Ci|≤12]+1σn∑i=1mn(t)+1‍f(σi2σn2)τi2ESiσn2[log(1+Ci)-Ci]×I[|Ci|>12]=1σn∑i=1mn(t)+1‍f(σi2σn2)τi2σn2(Si-ESi)+An′(t)+An′′(t),say.
Let us observe that
(17)Emax0≤t≤1|An′(t)|≤E1σn∑i=1n‍|f(σi2σn2)|τi2ESiσn2|R(Ci)|I[|Ci|≤12]≤2σn∑i=1n‍|f(σi2σn2)|τi2ESiσn2E(Si-ESiESi)2=2∑i=1n‍|f(σi2σn2)|σiσnτi2σn2σiESi.
From the assumption (10) and the Kronecker lemma it follows that (1/(ESn)2)∑i=1nτi2=σn2/(ESn)2→0. It is easy to see that from the Lindeberg condition (9), the Feller condition
(18)limn→∞max1≤i≤nτi2σn2=0
holds; therefore,
(19)∑i=1n‍|f(σi2σn2)|σiσnτi2σn2→∫01‍|f(x)|xdx<∞.
Moreover, by our assumption (6) we get
(20)|f(σi2σn2)|σiσnτi2σn2≤|f(σi2σn2)|(σiσn)3→0.
By the above remarks and Toeplitz theorem on the transformation of sequences into sequences (Problem 2.3.1 in [9]) we obtain
(21)∑i=1n‍|f(σi2σn2)|σiσnτi2σn2σiESi→0.
Thus from (17) we have An′→P0 in D[0,1], as n→∞.
From (10) the strong law of large numbers holds, that is, Cn=(Sn-ESn)/ESn→0 almost surely. Thus, for almost all ω∈Ω and sufficiently large i≥i0(ω) there holds I[|Ci(ω)|>1/2]=0. Since that
(22)max0≤t≤1|An′′(t)|≤1σn∑i=1n‍|f(σi2σn2)|τi2ESiσi2|log(1+Ci)-Ci|×I[|Ci|>12]→0,a.s.
Consequently An′′→P0 in D[0,1], and (15) is proved.
It remains to prove that(23)Yn(t):=1σn∑i=1mn(t)+1‍f(σi2σn2)τi2σn2(Si-ESi)→d∫0t‍f(x)W(x)dx
in D[0,1], as n→∞.
Let0<ε<1 be given and let us define a process
(24)Yn,ε(t)={0,0≤t<σmn(ε)2σn2,1σn∑i=mn(ε)+1mn(t)+1‍f(σi2σn2)τi2σn2(Si-ESi),σmn(ε)2σn2≤t≤1.
Let us observe that(25)Emax0≤t≤1|Yn(t)-Yn,ε(t)|≤E1σn∑i=1mn(ε)‍|f(σi2σn2)|τi2σi2|Si-ESi|≤∑i=1mn(ε)‍|f(σi2σn2)|τi2σn2σiσn→∫0ε‍|f(x)|xdx,asn→∞,forfixedε>0.
Since from the definition ofmn(t) we have σmn(ε)2/σn2→ε, therefore, by the integrability of the function f(x)x and Lemma 4(a) we get
(26)limε→0limsupn→∞dKFD[0,1](Yn(t),Yn,ε(t))=0.
Let us define a continuous mappingHε:D[0,1]→C[0,1](27)Hε(g)(t)={0,0≤t<ε,∫εt‍g(x)f(x)dx,ε<t≤1.
The above integral is properly defined since g is bounded and f∈𝒞1[ε,1]. In the next step we shall prove that
(28)Hε(Wn)(t)=Yn,ε(t)+Bn(t),
where Bn→P0 in D[0,1], as n→∞ and
(29)Wn(t)=∑k=1mn(t)+1‍(Xk-μk)σn.
At first let us observe that(30)Emax0≤t≤ε|Hε(Wn)(t)-Yn,ε(t)|=1σn|f(σmn(ε)+12σn2)|τmn(ε)+12σn2E|Smn(ε)+1-ESmn(ε)+1|≤σmn(ε)+1σnsupε≤x≤1|f(x)|max1≤i≤nτi2σn2→0,asn→∞,
by the Feller condition (18).
In the caseε<t≤1, let us calculate the integral
(31)Hε(Wn)(t)=∫εt‍f(x)Wn(x)dx=-∫σmn(ε)2/σn2εf(x)Wn(x)dx-∫tσmn(t)+12/σn2f(x)Wn(x)dx+∑i=mn(ε)+1mn(t)+1‍∫σi-12/σn2σi2/σn2f(x)Wn(x)dx=Tn′(t)+Tn′′(t)+Tn′′′(t),say.
In factTn′(t) does not depend on t, thus
(32)Emaxε≤t≤1|Tn′(t)|=E|-∫σmn(ε)2/σn2εf(x)Wn(x)dx|≤|ε-σmn(ε)2σn2|×supσmn(ε)2/σn2≤x≤1|f(x)|E|Smn(ε)+1-ESmn(ε)+1|σn→0,
as n→∞, since σmn(ε)2/σn2→ε and E|Smn(ε)+1-ESmn(ε)+1|≤σmn(ε)+1≤σn.
Let us find the bounds for the second termTn′′(t):
(33)|Tn′′(t)|=|∫tσmn(t)+12/σn2f(x)Wn(x)dx|≤max1≤i≤nτi2σn2supε≤x≤1|f(x)||Smn(t)+1-ESmn(t)+1|σn.
Further, for any δ>0, we have
(34)P(maxε≤t≤1|Tn′′(t)|>δ)≤P(maxτi2σn3supε≤x≤1|f(x)|maxε≤t≤1|Smn(t)+1-ESmn(t)+1|>δ)≤P(maxτi2σn3supε≤x≤1|f(x)|max1≤i≤n|Si-ESi|>δ)≤(maxτi2σn2)2(supε≤x≤1|f(x)|)21δ2→0,asn→∞,
by the Kolmogorov inequality and the Feller condition (18).
Finally, letn be sufficiently large, that is, such that σmn(ε)2/σn2≥ε/2, then
(35)maxε≤t≤1|Tn′′′(t)-Yn,ε(t)|≤maxε≤t≤11σn∑i=mn(ε)+1mn(t)+1‍|Si-ESi|×|∫σi-12/σn2σi2/σn2f(x)dx-f(σi2σn2)τi2σn2|≤max1≤i≤n|Si-ESi|σn×maxε≤t≤1∑i=mn(ε)+1mn(t)+1‍|∫σi-12/σn2σi2/σn2|f(x)-f(σi2σn2)|dx|≤max1≤i≤n|Si-ESi|σn∑i=1n‍cf,ε(τi2σn2)2≤max1≤i≤n|Si-ESi|σnmaxτi2σn2cf,ε,
where cf,ε=maxε/2≤x≤1|f′(x)|. Thus, again by the Kolmogorov inequality and the Feller condition, we get
(36)P(maxε≤t≤1|Tn′′′(t)-Yn,ε(t)|>δ)≤P(max1≤i≤n|Si-ESi|σnmaxτi2σn2cf,ε>δ)≤(maxτi2σn2δcf,ε)2→0,asn→∞.
Consequently, from (30), (32), (34), and (36) the statement (28) follows.
To end the proof, let us note that for the processWn(t) defined by (29), we have Wn(t)→dW(t) in D[0,1], by the result of Prohorov (see Problem 1, page  77 in [7]). In fact, Prohorov considered a broken-line process Xn(t) and its convergence in C[0,1], but
(37)sup0≤t≤1|Wn(t)-Xn(t)|≤max1≤i≤n|Xi-EXi|σn→P0,
from the Lindeberg condition (9). Thus, by the continuity of Hε, we get
(38)Hε(Wn)(t)→dHε(W)(t),inD[0,1],asn→∞.
Furthermore,(39)sup0≤t≤1|Hε(W)(t)-∫0t‍f(x)W(x)dx|=sup0≤t≤ε|∫0t‍f(x)W(x)dx|≤sup0<x≤ε|W(x)|xlog(1/x)·∫0ε‍|f(x)|xlog(1x)dx→0,almostsurelyasε→0+,
by (5) and the properties of the modulus of continuity of the Wiener process, that is,
(40)limsupε→0+sup0<x≤ε|W(x)|xlog(1/x)≤2almostsurely;
see formula (7) on page  534 in [6].
For short, let us putH(W)(t)=∫0t‍f(x)W(x)dx. Then, with the notation introduced in Lemma 4, we get
(41)dLPD[0,1](Yn;H(W))≤dLPD[0,1](Yn;Yn,ε)+dLPD[0,1](Yn,ε;Hε(Wn))+dLPD[0,1](Hε(Wn);Hε(W))+dLPD[0,1](Hε(W);H(W))≤dKFD[0,1](Yn;Yn,ε)+dKFD[0,1](Yn,ε;Hε(Wn))+dLPD[0,1](Hε(Wn);Hε(W))+dKFD[0,1](Hε(W);H(W)).
Now, (23) follows from (26), (28), (38), and (39) and the proof is completed.



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