Table Of ContentElectronic Journal of Statistics
Vol.3(2009)25–40
ISSN:1935-7524
DOI:10.1214/08-EJS233
Functional asymptotic
confidence intervals for a common mean
of independent random variables
9
0
0 Yuliya V. Martsynyuk
∗
2
School of Mathematics and Statistics, Carleton University,
n
1125 Colonel By Drive, Ottawa, ON K1S5B6, Canada
a
e-mail: [email protected]
J
9 Abstract: Weconsiderindependentrandomvariables(r.v.’s)withacom-
2 mon mean µ that either satisfy Lindeberg’s condition, or are symmet-
ric around µ. Present forms of existing functional central limit theorems
] (FCLT’s)forStudentized partial sumsofsuchr.v.’son D[0,1] areseento
T be of some use for constructing asymptotic confidence intervals, or what
S we call functional asymptotic confidence intervals (FACI’s), for µ. In this
. paper we establish completely data-based versions of these FCLT’s and
h thusextendtheirapplicabilityinthisregard.Twospecialexamplesofnew
t FACI’sforµarepresented.
a
m
AMS 2000 subject classifications:Primary60F17, 60G50,62G15.
[ Keywordsand phrases:Lindeberg’scondition,symmetricrandomvari-
able,Studentstatistic,Studentprocess,Wienerprocess,functionalcentral
1 limit theorem, sup-norm approximation in probability, functional asymp-
v toticconfidence interval.
8
ReceivedApril2008.
2
6
4
Contents
.
1
0
1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . 25
9
1.1 Review of invariance principles for Student processes based on
0
: independent random variables . . . . . . . . . . . . . . . . . . . . 26
v
1.2 Main results: functional asymptotic confidence intervals for a
i
X common mean of independent random variables . . . . . . . . . . 29
r 2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
a Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1. Introduction and main results
Estimating an unknown mean of a population has been a prominent classical
problem in statistics. Perhaps, the most famous and influential work on this
∗ResearchsupportedbyNSERCCanadaDiscoveryGrantsofM.Cso¨rgo˝andB.Szyszkow-
iczatCarletonUniversity,andanNSERCPostdoctoral FellowshipofYu.V.Martsynyukat
UniversityofOttawa.
25
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 26
subject is “Student” (1908) that celebrated its centennial last year. Facing the
problemofanaccurateintervalestimation ofa meanof a smallrandomsample
drawnfromanormallydistributedpopulationwithanunknownvariance,W.S.
Gosset (“Student”), among other things, concluded the exact distribution of
the 1/√n 1 multiple of what is now known as the Student statistic, or the
−
so-called t random variable with n 1 degrees of freedom.
− −
Estimatingacommonmeanofseveralpopulationshasalsobeenanoutstand-
ing statistical problem. It is frequently posed in the context of finite samples.
A considerable amount of the literature in this regard treats the case of sev-
eralnormalpopulationswithunknownandpossiblyunequalvariances(cf.,e.g.,
Graybill and Deal (1959), Normwood and Hinkelmann (1977), Pal and Kim
(1997), and further references in these papers).
The present paper deals with asymptotic confidence interval estimation of a
commonmeanofunspecifiedpopulations.Let Z ,i 1 beasequenceofinde-
i
{ ≥ }
pendent, but not necessarily indentically distributed, random variables (r.v.’s)
with a common mean µ. We will consider two kinds of such r.v.’s, those with
finite positive variances that satisfy Lindeberg’s condition, and Z ’s that are
i
symmetric around µ and do not necessarily have finite variances.
The studies ofthe presentpaper weremotivated inpartby theproblems the
author faced in the context of linear error-in-variables models, when establish-
ing functional asymptotic confidence intervals for the slope in such models in
Martsynyuk (2008).
1.1. Review of invariance principles for Student processes based on
independent random variables
Case of random variables satisfying Lindeberg’s condition
Suppose that µ = 0 and 0 < VarZ = σ2 < , i 1. Consider the Student
i i ∞ ≥
statistic
n Z /√n
T (Z ,...,Z ):= i=1 i , (1)
n 1 n ( n (Z Z)2/(n 1))1/2
i=1 Pi− −
with Z :=n−1 ni=1Zi, n≥1. In vPiew of (1), one can define a Student process
in D[0,1] space as follows:
P
Kn(t)Z /√n
Tt(Z ,...,Z ):= i=1 i , 0 t 1, (2)
n 1 n ( n (Z Z)2/(n 1))1/2 ≤ ≤
i=1Pi− −
where the time function Kn(P) is defined as
·
K (t):= sup s2 ts2 , 0 t 1, (3)
n m ≤ n ≤ ≤
0 m n
≤ ≤
(cid:8) (cid:9)
with
m
s2 :=0 and s2 := σ2, m 1. (4)
0 m i ≥
i=1
X
In (2), we put 0 Z :=0.
i=1 i
P
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 27
Cs¨orgo˝,SzyszkowiczandWang (2003,2004), amongother things,study self-
normalized partial sums and a corresponding process in D[0,1] for the above
Z ,i 1 , namely
i
{ ≥ }
n Z Kn(t)Z
V (Z ,...,Z ):= i=1 i and Vt(Z ,...,Z ):= i=1 i . (5)
n 1 n ( n Z2)1/2 n 1 n ( n Z2)1/2
Pi=1 i Pi=1 i
On assuming that thPe Lindeberg condition for Z ’s is satisfiedP, that is
i
n
for each ε>0, s−n2 EZi211{|Zi|≥εsn} →0, as n→∞, (6)
i=1
X
where 11 denotes the indicator function of set A, as a consequence of a well-
A
known result of Prohorov (1956, Theorem 3.1), they conclude (cf. Proposition
2.2 combined with Remark 2.6 in Cs¨orgo˝ et al. (2004)):
Vnt(Z1,...,Zn)→D W(t) on (D[0,1],ρ), n→∞, (7)
where W(t),0 t 1 isastandardWienerprocess,andρstandsforthesup-
{ ≤ ≤ }
norm metric on D[0,1]. The weak convergencein (7) is a weak invariance prin-
ciple, and it amounts to the following functional central limit theorem (FCLT)
(cf., e.g., Sections 3.3 and 3.4 in Cs¨orgo˝ (2002)):
h Vnt(Z1,...,Zn) →D h(W(t)), n→∞, (8)
(cid:0) (cid:1) D
for all functionals h : D[0,1] IR that are -measurable and ρ-continuous,
→
or ρ-continuous except at points forming a set of Wiener measure zero on
D D
(D[0,1], ), where is the sigma-field of subsets of D[0,1] generated by the
finite-dimensional subsets of D[0,1].
Cs¨orgo˝etal.(2004)alsoshow(cf.theirProposition2.3)thatonecanredefine
mean zero Z , i 1 as in (6) on a richer probability space together with a
i
{ ≥ }
sequence of independent standard normal r.v.’s Y , i 1 , such that
i
{ ≥ }
Kn(t)
sup (cid:12)Vnt(Z1,...,Zn)−s−n1 σiYi(cid:12)=oP(1), n→∞. (9)
0≤t≤1(cid:12)(cid:12) Xi=1 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Since s−n1 iK=n1(t(cid:12))σiYi =D W(s−n2 Ki=n1(t)σi2), 0 (cid:12)≤ t ≤ 1, and, on account of
(6), sup s 2 Kn(t)σ2 t s 2max σ2 0, n , then by
using th0e≤LPt≤´ev1y| m−noduil=u1s of cio−ntin|uP≤ity o−nf a Wi1e≤nei≤rnproice→ss (cf., e→.g.,∞Cs¨orgo˝and
P
R´ev´esz (1981)), sup W(s 2 Kn(t)σ2) W(t) = o (1), n . The
latter nearness comb0i≤nte≤d1w|ith t−nhe noi=ti1on oif an−FCLT|on (DP[0,1],ρ)→res∞ults in
P
Kn(t)
s−n1 σiYi →D W(t) on (D[0,1],ρ), n→∞. (10)
i=1
X
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 28
Consequently, in view of (10), the sup-norm approximation in probability in
(9) implies (8), that is the FCLT in (7). Moreover, although the main foci of
this paper are FCLT’s and their application to constructing asymptotic confi-
dence intervals, upcoming statements of the FCLT’s in Lemma 1 and our main
Theorem 1 will be accompanied by corresponding sup-norm approximations in
probability in (c) parts of Lemma 1 and Theorem 1, as by proving these ap-
proximations and using (10), we also establish the FCLT’s a` la (8).
The results in (7) and (9) almost immediately yield Lemma 1 with the cor-
responding analogues for the Student process Tt(Z ,...,Z ) of (2).
n 1 n
Lemma 1. Let Z , i 1 be independent mean zero r.v.’s with finite positive
i
{ ≥ }
variances VarZ = σ2, i 1. Assume also the Lindeberg condition as in (6).
i i ≥
Then, as n ,
→∞
(a) Tnt0(Z1,...,Zn)→D N(0,t0), t0 ∈(0,1];
(b) Tnt(Z1,...,Zn)→D W(t) on (D[0,1],ρ);
(c) we can redefine Z , i 1 on a richer probability space together with a
i
{ ≥ }
sequence of independent standard normal r.v.’s Y , i 1 such that
i
{ ≥ }
Kn(t)
sup Tt(Z ,...,Z ) s 1 σ Y =o (1),
(cid:12) n 1 n − −n i i(cid:12) P
0≤t≤1(cid:12)(cid:12) Xi=1 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where Kn(t) and sn(cid:12) are as in (3) and (4). (cid:12)
Case of symmetric random variables
Consider now independent symmetric mean zero r.v.’s Z ,i 1 that do not
i
{ ≥ }
necessarily have finite variances. For such r.v.’s, Egorov (1996) proves that, as
n ,
→∞
max Z2
Vn(Z1,...,Zn)→D N(0,1) if and only if 1n≤i≤Zn2 i →P 0. (11)
i=1 i
Aimingatageneralizationof(11)foranappropriateD[P0,1]versionofV (Z ,...,
n 1
Z ), Cs¨orgo˝ et al. (2003) introduce the self-normalized partial sums process
n
Kn(t)Z
Vt(Z ,...,Z ):= i=1 i , 0 t 1, (12)
n 1 n ( n Z2)1/2 ≤ ≤
Pi=b1 i
b P
where the time function K (t) is a suitable analogue of K (t) of (3) for the
n n
r.v.’s Z ’s with not necessarily finite variances, namely
i
b
m n 0
K (t):= sup Z2 t Z2 , 0 t 1, Z2 :=0. (13)
n 0 m n( i ≤ i) ≤ ≤ i
≤ ≤ Xi=1 Xi=1 Xi=1
b
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 29
Cs¨orgo˝ et al. (2003, Theorem 2) show that, as n ,
→∞
max Z2
Vnt(Z1,...,Zn)→D W(t) on (D[0,1],ρ) if and only if 1n≤i≤Zn2 i →P 0.
i=1 i
(14)
b P
In view of Vt(Z ,...,Z ), one can define and study the Student process
n 1 n
b Kn(t)Z /√n
Tt(Z ,...,Z ):= i=1 i , 0 t 1, (15)
n 1 n ( n (Z Z)2/(n 1))1/2 ≤ ≤
i=1Pib− −
b P
with K (t) of (13). It is not hard to see that
n
Tbt(Z ,...,Z )= Vnt(Z1,...,Zn) , 0 t 1. (16)
n 1 n (n V2(Z ,...,Z ))/(n 1) ≤ ≤
− nb 1 n −
b p
Hence,ifTt(Z ,...,Z )orVt(Z ,...,Z )hasanasymptoticdistribution,then
n 1 n n 1 n
sodoestheother,andthesedistributionscoincide.Consequently,(14)alsoholds
true for Tbt(Z ,...,Z ). b
n 1 n
Lemma 2. Let Z ,i 1 be independent mean zero symmetric r.v.’s. Then,
b { i ≥ }
Tnt(Z1,...,Zn)→D W(t)on(D[0,1],ρ)ifandonlyif max1≤i≤nZi2/ ni=1Zi2→P 0,
as n .
→∞ P
b
1.2. Main results: functional asymptotic confidence intervals for a
common mean of independent random variables
Case of random variables satisfying Lindeberg’s condition
Consider independent r.v.’s Z ,i 1 with a common mean µ and finite pos-
i
{ ≥ }
itive variances that satisfy Lindeberg’s condition. As a consequence of the (a)
part with t0 = 1 of Lemma 1, Tn(Z1 µ,...,Zn µ) D N(0,1), n .
− − → → ∞
SincetheStudentstatisticT (Z µ,...,Z µ)doesnotcontainthetypically
n 1 n
unknown variances σ2, as compar−ed to the ex−pression s 1 n (Z µ) in the
i −n i=1 i−
usual statement of the classical Lindeberg-Feller central limit theorem (CLT),
P
theaboveCLTforT (Z µ,...,Z µ)canbeusedforasymptoticconfidence
n 1 n
− −
interval (CI) estimation of the mean µ.
The data-based Studentized FCLT in the (b) part of Lemma 1 provides a
source of further asymptotic CI’s, or what we call functional asymptotic CI’s
(FACI’s), for µ. For example, since the sup-functional sup on D[0,1]is
ρ-continuous, from the (b) part of Lemma 1 we conclude 0≤t≤1|·|
sup Tnt(Z1−µ,...,Zn−µ) →D sup |W(t)|, n→∞, (17)
0 t 1 0 t 1
≤ ≤ (cid:12) (cid:12) ≤ ≤
(cid:12) (cid:12)
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 30
and the latter convergence in distribution yields a 1 α size FACI for µ as
−
follows:
n k Z a n ni=1(Zi−Z)2 k Z +a n ni=1(Zi−Z)2
i=1 i− n 1 i=1 i n 1
r P − , r P − , (18)
P k P k
k=1
\
whereP sup W(t) >a =α,0<α<1.Thedistributionfunctionofthe
r.v. sup 0≤Wt≤(1t)| can|be found, for example, in Cs¨orgo˝ and R´ev´esz (1981),
as well a0s≤(cid:0)ti≤n1C|s¨orgo˝|and Horv(cid:1)a´th (1984) where it is also tabulated.
In construction of the FACI (18) for µ, due to the very nature of the sup-
functional sup on D[0,1], the time function K (t) of (3) of Tt(Z
µ,...,Z µ)0≤wta≤s1e|m·|ployed only to the extent of usingn its values 0,1n,...1,n−.
n
−
However, when dealing with some other appropriate functionals in regard of
constructing FACI’s for µ from the FCLT in (b) of Lemma 1, the jump points
s2/s2 ofthe step-function K (t) may alsoenter the picture. These jump points
k n n
are typically unknown, unless Z ’s have equal variances and s2/s2 = k/n. To
i k n
resolve this problem, we replace K (t) with its “empirical”, data-based version
n
m n
K (t):= sup (Z Z)2 t (Z Z)2 , 0 t 1, (19)
n i i
0 m n( − ≤ − ) ≤ ≤
≤ ≤ Xi=1 Xi=1
wheree 0 (Z Z)2 := 0, and establish our main Theorem 1, an analogue of
i=1 i−
Lemma 1 for the Student process
P
Kn(t)Z /√n
Tt(Z ,...,Z ):= i=1 i , 0 t 1. (20)
n 1 n ( n (Z Z)2/(n 1))1/2 ≤ ≤
i=1Pie− −
Withoutloessofgenerality,ThPeorem1isstatedundertheassumptionthatµ=0.
Theorem1. Let Z , i 1 beindependent meanzeror.v.’s withfinitepositive
i
{ ≥ }
variances VarZ = σ2, i 1. Assume also the Lindeberg condition as in (6).
i i ≥
Then, as n ,
→∞
(a) Tnt0(Z1,...,Zn)→D N(0,t0), t0 ∈(0,1];
((bc)) Twenet(Zca1n,.r.e.d,eZfinn)e→DZW,(it) 1onon(Da[0ri,c1h]e,rρ)p;robability space together with a
i
{ ≥ }
seequence of independent standard normal r.v.’s Y , i 1 such that
i
{ ≥ }
Kn(t)
sup Tt(Z ,...,Z ) s 1 σ Y =o (1),
(cid:12) n 1 n − −n i i(cid:12) P
0≤t≤1(cid:12)(cid:12) Xi=1 (cid:12)(cid:12)
(cid:12)e (cid:12)
where K (t) and s(cid:12) are as in (3) and (4). (cid:12)
n n(cid:12) (cid:12)
To illustrate when construction of FACI’s for a not necessarily zero µ call
for the FCLT as in the (b) part of Theorem 1, we consider convergence in
distribution of two special functionals of Tt(Z µ,...,Z µ) in Examples 1
n 1− n−
and 2.
e
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 31
Example 1. Fora fixedt (0,1],we considerTt0(Z µ,...,Z µ),one of
0 ∈ n 1− n−
thesimplestρ-continuousfunctionalsofTt(Z µ,...,Z µ).Asaconsequence
n 1− n−
of the FCLT in (b) of Theorem 1, or directly bye(a) of Theorem 1, we obtain
the following 1 α size FACI for µ: e
−
Kn(t0)Z z √t n ni=1(Zi−Z)2 Kn(t0)Z +z √t n ni=1(Zi−Z)2
i=1 i− α/2 0 n 1 i=1 i α/2 0 n 1
r P − , r P − ,
Pe Kn(t0) Pe Kn(t0)
e e (21)
wherez is the 100(1 α/2)th percentile ofthe standardnormaldistribution.
α/2
−
The FACI in (21) is completely data-based,as K (t ) is computable. Indeed, if
n 0
t =1,then K (t )=n,while for t (0,1)andagivensample Z ,...,Z ,we
0 n 0 0 1 n
∈
can find k , 0 k n 1, such that k0 (Ze Z)2/ n (Z Z)2 t <
0 ≤ 0 ≤ − i=1 i− i=1 i− ≤ 0
k0+1(Z Ze)2/ n (Z Z)2, and consequently, K (t )=k . We also note
i=1 i− i=1 i− P nP0 0
that(21)iswell-defined,asby(42)below,max (Z Z)2/ n (Z Z)2 P
0Pand hence, (Z PZ)2/ n (Z Z)2 < t 1(≤oir≤Kn (eit−) = 0) wii=th1 prio−babili→ty
1− i=1 i− 0 n 0 6 P
approaching one, as n .
→P∞
e
Example 2. The integral functional 1 dt on D[0,1] is ρ-continuous, as for
0 ·
1 1
any f(t) and g(t) in D[0,1], f(t)dt g(t)dt sup f(t) g(t). In
view of this and the FCLT in|(0b) partRo−f T0heorem|1≤, as n0≤t≤1|, − |
R R →∞
1 1
Tnt(Z1−µ,...,Zn−µ)dt→D W(t)dt=D N(0,1/3). (22)
Z0 Z0
By noting thate
1Tt(Z µ,...,Z µ)dt=n−1ν ki=1(Zi−µ)/√n ,
n 1− n− k+1( n (Z Z)2/(n 1))1/2
Z0 kX=1 i=P1 i− −
e P
with
(Z Z)2
k
ν := − , 1 k n, (23)
k n (Z Z)2 ≤ ≤
i=1 i−
we obtain a 1 α size FACIPfor µ with the lower and upper bounds given by
−
kn=−11νk+1 ki=1Zi∓ z√α/32 n ni=n1(Z1i−Z)2
P P kn=−11νk+1kr P − , (24)
where P N(0,1/3) >z /√3 P=α.
α/2
| |
It wou(cid:0)ld naturally be desira(cid:1)ble to investigate individual and comparative
performances,suchasthe expectedlengthsforexample,oftheobtainedFACI’s
for µ in (18), (21) and (24).
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 32
Case of symmetric random variables
Considerindependent symmetric r.v.’s Z with a common,notnecessarilyzero,
i
mean µ. Lemma 2 remains true for such Z ’s if Z ,...,Z are replaced with
i 1 n
Z µ,...,Z µ in its statement. However, in the thus stated Lemma 2, the
1 n
tim−e function o−f Tt(Z µ,...,Z µ) becomes sup m (Z µ)2
t n (Z µ)2 ,na fu1n−ction of ann−unknown µ. Henc0≤e,ms≤unc{h ani=1FCLiT−is no≤t
i=1 i − } P
necessarily of immbediate use for construction of various FACI’s for µ, just like
P
the FCLT of (b) of Lemma 1 when it is applied to independent mean µ r.v.’s
Z satisfying Lindeberg’s condition (cf. the lines preceding (19)). To extend the
i
applicabilityoftheFCLTofLemma2inthisregard,wefirstestablishourmain
Theorem 2 for the Student process Tt(Z µ,...,Z µ) as in (20), a data-
n 1− n−
based version of Lemma 2 that uses K (t) of (19) instead of the above noted
n
time function of Tt(Z µ,...,Z eµ).
n 1− n−
e
Theorem 2. Let Z ,i 1 be independent symmetric r.v.’s with a common
i
b{ ≥ }
meanµ.Then, for Tnt(Z1,...,Zn)as in(20), Tnt(Z1−µ,...,Zn−µ)→D W(t)on
(D[0,1],ρ) if and only if max (Z µ)2/ n (Z µ)2 P 0, as n .
e 1≤i≤n i− e i=1 i− → →∞
ItisappealingtoreplacetheconvergencemaPx (Z µ)2/ n (Z µ)2 P
1≤i≤n i− i=1 i− →
0inTheorem2withthedata-basedoneofmax (Z Z)2/ n (Z Z)2 P
1≤i≤n i− Pi=1 i− →
0,asn ,especiallywhenconcludingFACI’sforµviaappropriatefunction-
als of T→t(Z∞ µ,...,Z µ) of Theorem 2. Hence we presentPCorollary 1 that
n 1− n−
amounts to a completely data-based version of Theorem 2.
e
Corollary 1. Let Z ,i 1 be independent symmetric r.v.’s with a common
i
{ ≥ }
mean µ. Then Tnt(Z1 −µ,...,Zn −µ) →D W(t) on (D[0,1],ρ) if and only if
max (Z Z)2/ n (Z Z)2 P 0, as n .
1≤i≤n i−e i=1 i− → →∞
We note that, in vPiew of Corollary 1, the FACI’s for µ in (21) and (24) also
hold true for independent symmetric r.v.’s Z with a common mean µ and not
i
necessarilyfinitevariances,providedthatmax (Z Z)2/ n (Z Z)2 P
1≤i≤n i− i=1 i− →
0, as n .
→∞ P
2. Proofs
Hereafter,notations o (1) and O (1) stand for sequences of r.v.’s that, respec-
P P
tively, converge to zero and are bounded in probability, as n .
→∞
Proof of Lemma 1. In view of (10), the proof reduces to establishing the (c)
part of Lemma 1. On account of (7), sup Vt(Z ,...,Z ) = O (1) and
V (Z ,...,Z ) = O (1). Moreover, we a0l≤sot≤1ha|vne (91) withnZ| ,i P 1 and
n 1 n P i
{ ≥ }
Y ,i 1 defined on the same probability space. Combining all this with a
i
{ ≥ }
representation for Tt(Z ,...,Z ) a` la (16), as n , we arrive at
n 1 n →∞
Kn(t)σ Y Kn(t)σ Y
sup Tt(Z ,...,Z ) i=1 i i sup Vt(Z ,...,Z ) i=1 i i
0≤t≤1(cid:12)(cid:12) n 1 n −P sn (cid:12)(cid:12)≤0≤t≤1(cid:12)(cid:12) n 1 n −P sn (cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 33
1
+ sup Vt(Z ,...,Z ) 1
=oP0(≤1t)≤+1(cid:12)(cid:12)OnP(11)oP(1)=no(cid:12)(cid:12)P(cid:12)(cid:12)(cid:12)(cid:12)(p1)(.n−Vn2(Z1,...,Zn))/(n−1) − (cid:12)(cid:12)(cid:12)(cid:12) (25)
(cid:12) (cid:12)
Next,wespelloutaspecialcaseofRaikov’stheorem(cf.Theorem4onp.143
in Gnedenko and Kolmogorov (1954)), and then, establish auxiliary Lemma 4
that is required for the proof of Theorem 1.
Lemma 3. Let Z , i 1 be independent mean zero r.v.’s with finite positive
i
{ ≥ }
variances VarZ = σ2, i 1. Suppose also that the Lindeberg condition in (6)
i i ≥
is satisfied. Then,
n
s 2 Z2 P 1, n , (26)
−n i → →∞
i=1
X
with s2 of (4).
n
Lemma 4. For Z , i 1 as in Lemma 3,
i
{ ≥ }
sup Vt(Z ,...,Z ) Vt(Z ,...,Z ) =o (1), n , (27)
n 1 n − n 1 n P →∞
0 t 1
≤ ≤ (cid:12) (cid:12)
where the self-n(cid:12)(cid:12)ormalized partial seums processes V(cid:12)(cid:12) t(Z ,...,Z ) and
n 1 n
Vt(Z ,...,Z ) are defined as follows: for 0 t 1,
n 1 n ≤ ≤
e Kn(t)Z Kn(t)Z
Vt(Z ,...,Z )= i=1 i and Vt(Z ,...,Z )= i=1 i , (28)
n 1 n ( n Z2)1/2 n 1 n ( n Z2)1/2
Pi=1 i Pi=e1 i
P e P
with the time functions K (t) and K (t) as in (3) and (19).
n n
Proof of Lemma 4. The scheme of this proofis motivated by the lines of the
e
proof of Theorem 2 in Raˇckauskas and Suquet (2001).
LetUt andUt be therespectiveC[0,1]“Donskerized”versionsoftheD[0,1]
n n
processes Vt(Z , ,Z ) and Vt(Z , ,Z ). Namely, Ut and Ut, as contin-
n 1 ··· n n 1 ··· n n n
uous functionseof t, are linear respectively on the intervals [s2/s2,s2 /s2]
k n k+1 n
and [ k (Z Z)2/ n (Z e Z)2, k+1(Z Z)2/ n (Z eZ)2] for each
i=1 i− i=1 i− i=1 i− i=1 i−
k =0,1, ,n 1, and both take values k Z /( n Z2)1/2 respectively at
P ··· − P P i=1 i i=P1 i
s2/s2 and k (Z Z)2/ n (Z Z)2, k =0,1, ,n, where s2 is defined in
k n i=1 i− i=1 i− P ·P·· k
(4), and 0 (Z Z)2 :=0 and 0 Z :=0. Clearly,
Pi=1 i− P i=1 i
P P max Z
sup Vnt(Z1,...,Zn)−Unt ≤ ( n1≤iZ≤n2)|1/i2|,
0≤t≤1(cid:12) (cid:12) maxi=1 i Z (29)
sup (cid:12)Vnt(Z1,...,Zn)−Unt(cid:12)≤ (Pn1≤iZ≤n2)|1/i2|.
0 t 1 i=1 i
≤ ≤ (cid:12) (cid:12)
(cid:12)(cid:12)e e (cid:12)(cid:12) P
Yu. V. Martsynyuk/Functional asymptotic confidence intervals 34
From Lindeberg’s condition in (6), for any ε>0, as n ,
→∞
n n
P max Z εs P(Z εs ) (εs ) 2 E Z211 0.
1 i n| i|≥ n ≤ | i|≥ n ≤ n − i {|Zi|≥εsn} →
(cid:18) ≤≤ (cid:19) Xi=1 Xi=1 (cid:16) (cid:17)
(30)
On combining (26) and (30),
max Z
1 i n i
( n≤Z≤2)|1/2| =oP(1), n→∞. (31)
i=1 i
In view of (29) and (31)P, in order to show (27), it suffices to prove that
sup Ut Ut =o (1), n . (32)
n− n P →∞
0 t 1
≤ ≤ (cid:12) (cid:12)
Let θ (t) be the random(cid:12)(cid:12)elemenet(cid:12)(cid:12)of C[0,1] that is linear in t on the inter-
n
vals [s2/s2,s2 /s2] for each k = 0,1,...,n 1, with θ (s2/s2) = k (Z
k n k+1 n − n k n i=1 i−
Z)2/ n (Z Z)2,k =0,1,...,n.Sincesup Ut Ut =sup Uθn(t)
Uθn(t) ai=n1d Uiθ−n(t) =Ut, 0 t 1, then (32)0≤reta≤d1s|ans− n| 0≤Pt≤1| n −
n P| n n ≤ ≤ e
e e sup Uθn(t) Ut =o (1), n . (33)
n − n P →∞
0 t 1
≤ ≤ (cid:12) (cid:12)
(cid:12) (cid:12)
For f(t) C[0,1], let ω(f(cid:12)(t);δ) := sup(cid:12) f(t ) f(t ) be the modulus
of continu∈ity of f(t). For any λ>0 and|t01−<t2|δ≤δ|1, w1e−have2 |
≤
P sup Uθn(t) Ut λ
| n − n|≥
(cid:18)0≤t≤1 (cid:19)
P sup Uθn(t) Ut λ
≤ 0≤t≤1,|t−θn(t)|≤sup0≤t≤1|t−θn(t)|| n − n|≥ !
P ω(Ut;δ) λ +P sup t θ (t) >δ . (34)
≤ n ≥ | − n |
(cid:18)0≤t≤1 (cid:19)
(cid:0) (cid:1)
ByTheorem3.1inProhorov(1956)and(26),Ut D W(t)on(C[0,1],ρ),n ,
n → →∞
and therefore, for the continuous functional ω(;δ) on C[0,1] and any λ>0,
·
P ω(Ut;δ) λ P (ω(W(t);δ) λ), n . (35)
n ≥ → ≥ →∞
In view of, for e(cid:0)xample, the L(cid:1)´evy modulus of continuity of a Wiener process
(cf., e.g., Cs¨orgo˝ and R´ev´esz (1981)), for any ε>0 there is δ (0,1] such that
∈
P (ω(W(t);δ) λ)<ε. (36)
≥
Takinginto account(34)–(36),to completethe proofofLemma 4,weonlyneed
to verify that
sup t θ (t) =o (1), n . (37)
n P
| − | →∞
0 t 1
≤ ≤
