TUGraz OPEN Library

Stoeglehner, Gernot

Baaske, Wolfgang

Mitter, Hermine

Niemetz, Nora

Kettl, Karl-Heinz

Weiss, Michael

Lancaster, Bettina

Neugebauer, Georg

English

Sustainability appraisal of residential energy demand and supply - a life cycle approach including heating, electricity, embodied energy and mobility

Wriessnegger, Selina C.

Steyrl, David

Koschutnig, Karl

Müller-Putz, Gernot R.

Short time sports exercise boosts motor imagery patterns: implications of mental practice in rehabilitation programs

Xuan, Thanh Phan

Steinbach, Olaf

Boundary element methods for parabolic boundary control problems

Bazarova, Alina

Berkes, István

Horváth, Lajos

ON THE CENTRAL LIMIT THEOREM FOR MODULUS TRIMMED
SUMS
ALINA BAZAROVA1, ISTVAN BERKES2, AND LAJOS HORVATH3
Abstract. We prove a functional central limit theorem for modulus trimmed i.i.d. variables
in the domain of attraction of a nonnormal stable law. In contrast to the corresponding result
under ordinary trimming, our CLT contains a random centering factor which is inevitable
in the nonsymmetric case. The proof is based on the weak convergence of a two-parameter
process where one of the parameters is time and the second one is the fraction of truncation.
1. Introduction
Let X1; X2; : : : be independent, identically distributed random variables in the domain of
attraction of a stable law G with parameter 0 <  < 2. That is, assume that the partial
sums Sn =
Pn
k=1Xk satisfy
(1.1) (Sn   bn)=an d ! G
with suitable norming and centering sequences fang, fbng. The necessary and sucient
condition for (1.1) is that F , the distribution function of X1, satises
(1.2) 1  F (x) + F ( x) = x L(x); x > 0
and
(1.3)
1  F (x)
1  F (x) + F ( x) ! p;
F ( x)
1  F (x) + F ( x) ! q (x!1)
where L is a function slowly varying at 1 and p; q  0, p + q = 1. (See e.g. Feller [8].)
In contrast to the case of nite variances, the contribution of extremal terms in the partial
sums Sn is not negligible and dropping a single term can change the asymptotic behavior of
1Graz University of Technology, Institute of Statistics, Kopernikusgasse 24, 8010 Graz, Austria. e-mail:
bazarova@tugraz.at. Research supported by Austrian Science Fund (FWF) Grant W1230.
2Graz University of Technology, Institute of Statistics, Kopernikusgasse 24, 8010 Graz, Austria. e-mail:
berkes@tugraz.at. Research supported by Austrian Science Fund (FWF) Grant P24302-N18.
3University of Utah, Department of Mathematics, Salt Lake City, UT 84112-0090, USA. e-mail:
horvath@math.utah.edu. Research supported by NSF grant DMS 1305858.
1
the sum. Let Xn;1  Xn;2  : : :  Xn;n be the order statistics of (X1; X2; : : : ; Xn) and put
for d  1
(1.4) S(d)n =
n dX
j=d+1
Xn;j:
For xed d, Le Page, Woodrofe and Zinn [11] determined the asymptotic distribution of the
trimmed sum S
(d)
n and Csorg}o, Horvath and Mason [6] proved that under
(1.5) dn !1; dn=n! 0
the trimmed sum S
(dn)
n , suitably centered and normalized, is asymptotically normal. These
results give a remarkable picture on the partial sum behavior of i.i.d. sequences in the domain
of attraction of a non-normal stable law. They show that the contribution of dn extremal
terms under (1.5) already gives the stable limit distribution of the total partial sum Sn and
the contribution of the remaining elements will be an asymptotically normal variable with
magnitude negligible compared with Sn.
The previous results describe the eects of the extremal elements of an i.i.d. sample on
their partial sum. Note, however, that other kinds of trimming lead to dierent phenomena.
For 1  d  n let d;n denote the d-th largest of jX1j; : : : ; jXnj and let
(1.6) (d)Sn =
nX
k=1
XkIfjXkj  d;ng:
If the distribution of X1 is continuous, then jX1j; jX2j; : : : are dierent with probability 1,
and thus (d)Sn coincides with the usual modulus trimmed sum obtained by discarding from
Sn the d  1 elements with the largest moduli. Grin and Pruitt [9] showed that if X1 has
a symmetric distribution, then (dn)Sn is asymptotically normal for any dn !1, dn=n! 0,
but this is generally false in the nonsymmetric case. The purpose of this paper is to describe
the asymptotic distribution of (dn)Sn in the general case. Put
H(t) = P (jXj  t) and m(t) = EXIfjXj  tg;
and let H 1(t) = inffx : H(x)  tg (0 < t < 1) denote the generalized inverse of H. Our
main result is the following.
Theorem 1.1. Let X1; X2; : : : be i.i.d. random variables with distribution function F satis-
fying (1.2), (1.3) and assume that (1.5) holds. Then we have
(1.7)
1
An
[nt]X
i=1
(XiIfjXij  d;ng  m(d;n)) D[0;1] ! W (t)
where
(1.8) A2n =

2  d(H
 1(d=n))2
and W is the Wiener process.
2
Theorem 1.1 shows that allowing a random centering factor, the modulus trimmed CLT
holds for continuous i.i.d. variables under exactly the same conditions as under ordinary
trimming. If F is not continuous, the sample (X1; : : : ; Xn) may contain equal elements
with positive probability; according to the denition in Grin and Pruitt [9], 'ties' between
elements with equal moduli are broken according to the order in which the variables occur in
(X1; : : : ; Xn). But no matter how we break the ties, it may happen that from a set of sample
elements with equal moduli some are discarded and others are not, which is rather unnatural
from the statistical point of view, since trimming is mainly used to improve the performance
of statistical procedures by removing large elements from the sample. The denition of (d)Sn
in (1.6) resolves this diculty and leads to satisfactory asymptotic results in the general
case.
Theorem 1.1 enables one to give, among others, change point tests for heavy tailed pro-
cesses, while the standard CUSUM test fails under innite variances. A fairly precise char-
acterization of the modulus trimmed CLT with nonrandom centering and norming factors
was given in Berkes and Horvath [1].
Under additional technical assumptions on the distribution function of X1 and on the
growth speed of dn, Theorem 1.1 was proved by Berkes et al. in [2] with a fairly complicated
argument. The proof of Theorem 1.1 is much simpler and extends to dependent samples as
well, as we will show in a subsequent paper. Let
A^2n =
nX
i=1
X2i IfjXij  d;ng  
1
n
 
nX
i=1
XiIfjXij  d;ng
!2
:
Berkes et al. [2] showed that under the conditions of Theorem 1.1 we have that
A^n=An
P ! 1
and therefore Theorem 1.1 yields
1
A^n
0@ [nt]X
i=1
XiIfjXij  d;ng   [nt]
n
nX
i=1
XiIfjXij  d;ng
1A D[0;1] ! B(t);
where B(t) = W (t) tW (1) denotes a Brownian bridge. Hence standard CUSUM techniques
can be used to detect changes in the mean and/or location when in the case of observations
without second moments, observations with modulus larger than d;n are excluded from the
sample.
Let
Un(t; s) =
[nt]X
i=1
 
XiIfjXij  sH 1(d=n)g   EXiIfjXij  sH 1(d=n)g

(s  0; t  0):
We will deduce (1.7) from the following two-dimensional limit theorem.
3
Theorem 1.2. Let X1; X2; : : : be i.i.d. random variables with distribution function F satis-
fying (1.2), (1.3) and assume that (1.5) holds. Then
1
An
Un(t; s)  !W (t; s2 ) weakly in D[0; 1]D[1=2; 3=2];
where An is dened by (1.8) and fW (x; y); x  0; y  0g is a two-parameter Wiener process.
Note that by Kiefer [10] we have
d;n
H 1(d=n)
P ! 1:
Since the limit process in Theorem 1.2 has continuous trajectories a.s., Billingsley [4], p.
144-145 implies that
1
An
Un(t; d;n=H
 1(d=n))
D[0;1] ! W (t; 1)
which is exactly the functional CLT in (1.7), sinceW (t; 1) is a Wiener process. Thus Theorem
1.1 is a consequence of Theorem 1.2.
2. Proof of Theorem 1.2
Given a process Y (s; t) dened on a rectangle H = [a; b]  [c; d], let Y (H) denote the
increment of Y over H.
Lemma 2.1. Let fYn(t; s); n  1g be processes dened on a rectangle [a; b] [c; d]  [0;1)2
and assume that for some  > 0
(2.1) EjYn(B)jjYn(C)j  (B)(C);
where  denotes area and B and C are rectangles of the form [t1; t2]  [s1; s2] having one
common edge, but otherwise disjoint. Then the sequence fYn(t; s); n  1g is tight. If every
Yn(t; s) is piecewise constant in t, i.e. there exists a nite set Hn  [a; b] such that Yn(t; s)
is constant in the left closed intervals determined by the elements of Hn [ fag [ fbg, then it
suces to verify (2.1) for rectangles [t1; t2] [s1; s2] where t1; t2 2 Hn.
This is a special case of a general tightness condition due to Bickel and Wichura, see [3],
Theorem 3.
As is shown in Csorg}o et al. [7], Proposition A.3, the conditions of Theorem 1.2 imply
that H 1(t) = t 1=`(t) (0 < t < 1), where ` is slowly varying at 0. Then by (1.8) we have
(2.2) A2n 

2  d (n=d)
2=`2(d=n) as n!1
where an  bn means an=bn ! 1 as n!1.
4
Lemma 2.2. If the conditions of Theorem 1.2 are satised, then for any p  2 and any
xed 0  a < b <1 we have
(2.3) EjX1jpIfaH 1(d=n) < jX1j  bH 1(d=n)g  
p  (b
p    ap )`p(d=n)(n=d)(p )=
as n!1. Also, if b > 0, then
(2.4) EjX1jIfjX1j  bH 1(d=n)g =
8><>:
O((n=d)(1 )=`(d=n)) if  < 1;
O((n=d)") if  = 1;
O(1) if  > 1
for any " > 0.
Proof. Assume rst p  2, 0 < a < b <1. Clearly the left hand side of (2.3) equals
 
Z bH 1(d=n)
aH 1(d=n)
tpdH(t) =
Z H(aH 1(d=n))
H(bH 1(d=n))
H 1(u)pdu:(2.5)
(Note that H is non-increasing and thus the left hand side of (2.5) is nonnegative.) Since H
is regularly varying with exponent  , we have
H(aH 1(d=n))  a (d=n); H(bH 1(d=n))  b (d=n) as n!1:
Thus using the uniform convergence theorem for regularly varying functions (see e.g. [5],
Theorem 1.5.2; note that we actually need the analogous result for regular variation at 0),
we see that for n ! 1 we have, uniformly for all u in the interval of integration of the
second integral in (2.5),
H 1(u) = u 1=`(u)  u 1=`(d=n):
Thus the integral equals
(2.6) (1 + o(1))
Z (1+o(1))a (d=n)
(1+o(1))b (d=n)
u p=`p(d=n) du;
which yields the right hand side of (2.3) after a simple calculation, since p 6= . If a = 0,
then the upper limit in the integral on the right hand side of (2.5) and thus also in (2.6)
becomes H(0) = 1 and by using Theorem 1.5.11 of [5] we get the right hand side of (2.3)
with a = 0.
In the case of (2.4), instead of the integral in (2.6) we get
(2.7)
Z 1
(1+o(1))b (d=n)
u 1=`(u) du:
By Proposition 1.3.6(i) in [5], p. 16 we have `(u) = O(u ") as u ! 0 for any " > 0 which
shows that for  > 1 the integral
R 1
0
u 1=`(u) du converges and thus the expression (2.7)
is O(1). Using the same estimate for `(u) for  = 1 we get the second bound in (2.4).
5
Finally, for  < 1 Theorem 1.5.11 of [5] yields the rst bound in (2.4), completing the proof
of Lemma 2.2.
Proof of Theorem 1.2. Let  (t; s) denote the limit process in Theorem 1.2 and put
Qn =
1
An
MX
m=1
JX
j=1
m;jUn([tm 1; tm] [sj 1; sj])
and
Z =
MX
m=1
JX
j=1
m;j ([tm 1; tm] [sj 1; sj])
for all M  1; J  1, real coecients m;j, 1=2  s1 < s2 < : : : < sJ  3=2, 0 < t1 < : : : <
tM = 1, t0 = s0 = 0. Clearly, Z is a centered normal r.v. and
(2.8) EZ2 =
MX
m=1
JX
j=1
2m;j(s
2 
j   s2 j 1 )(tm   tm 1):
We claim that
(2.9) Qn
d ! Z for all considered values of M;J; m;j; tm; sj:
Since the processes Un and   are equal to 0 on the boundary of the rst quadrant, we have
Un(tm; sj) =
MX
m=1
JX
j=1
Un([tm 1; tm] [sj 1; sj])
and the same relation holds for  . Thus (2.9) implies
1
An
MX
m=1
JX
j=1
m;jUn(tm; sj)
d !
MX
m=1
JX
j=1
m;j (tm; sj)
for arbitrary real coecients m;j and this, by the Cramer-Wold device, implies the conver-
gence of the nite-dimensional distributions in Theorem 1.2.
Relation (2.9) can be written equivalently as
(2.10)
1
An
nX
k=1
(zk;n   Ezk;n) d ! N(0; EZ2);
where
zk;n =
JX
j=1
m;jXkIfsj 1H 1(d=n) < jXkj  sjH 1(d=n)g; [ntm 1] + 1  k  [ntm]:
6
Since the terms in the last sum are random variables with disjoint support, we get from
Lemma 2.2
Ez2k;n = (1+on(1))

2  (n=d)
(2 )=`2(d=n)
JX
j=1
2m;j(s
2 
j  s2 j 1 ); [ntm 1]+1  k  [ntm]
and similarly
Ez4k;n = (1+on(1))

4  (n=d)
(4 )=`4(d=n)
JX
j=1
4m;j(s
4 
j  s4 j 1 ); [ntm 1]+1  k  [ntm]:
Thus using d = dn !1 we get by a simple calculation
(2.11) lim
n!1
Pn
k=1 Ez
4
k;n Pn
k=1 Ez
2
k;n
2 = 0:
On the other hand, the previous asymptotics for Ez2k;n and the statement of Lemma 2.2 for
p = 1 imply
E2jzk;nj = on(1)Ez2k;n; 1  k  n
and thus by Minkowski's inequality
(2.12) Ejzk;n   Ezk;nj2 = (1 + on(1))Ez2k;n; Ejzk;n   Ezk;nj4 = (1 + on(1))Ez4k;n:
Thus (2.11) remains valid if we replace zk;n with zk;n   Ezk;n. Further by (2.2) and (2.8)
nX
k=1
Ez2k;n = (1 + on(1))

2  n(n=d)
(2 )=`2(d=n)
MX
m=1
JX
j=1
2m;j(s
2 
j   s2 j 1 )(tm   tm 1)
= (1 + on(1))A
2
nEZ
2:
The last relation, together with (2.11), (2.12) and Ljapunov's CLT for triangular arrays,
implies (2.10).
Next we prove tightness in Theorem 1.2. Consider two pairs of sets B11 = [t1; t] [s1; s],
B12 = [t1; t]  [s; s2] and B11 = [t1; t]  [s1; s], B21 = [t; t2]  [s1; s], where t1 < t < t2,
s1 < s < s2. In view of Lemma 2.1, it suces to show that
(2.13) E
 1AnUn(B11)
2  1AnUn(Bij)
2  C(B11)(Bij);
holds for each ij 2 f12; 21g with some constant C > 0. Moreover, since Un(t; s) is constant
on intervals k=n  t < (k + 1)=n, by the last statement of Lemma 2.1 we may assume that
nt; nt1 and nt2 are all integers. Using the independence of the Xj's, relation (2.2), Lemma
2.2 and the fact that the function x2  has a bounded derivative on [1=2; 3=2], we get
E
 1AnUn(B11)
2  1AnUn(B21)
2
7
= E
 
1
An
ntX
i=nt1+1
 
XiIfs1H 1(d=n) < jXij  sH 1(d=n)g  mi
!2

 
1
An
nt2X
i=nt+1
 
XiIfs1H 1(d=n) < jXij  sH 1(d=n)g  mi
!2
= E
 
1
An
ntX
i=nt1+1
 
XiIfs1H 1(d=n) < jXij  sH 1(d=n)g  mi
!2
 E
 
1
An
nt2X
i=nt+1
 
XiIfs1H 1(d=n) < jXij  sH 1(d=n)g  mi
!2
 1
A4n
 
ntX
i=nt1+1
EX2i Ifs1H 1(d=n) < jXij  sH 1(d=n)g
!
(2.14)

 
nt2X
i=nt+1
EX2i Ifs1H 1(d=n) < jXij  sH 1(d=n)g
!
 C1(t  t1)(t2   t)(s2    s2 1 )2  C2(t  t1)(t2   t)(s  s1)2
= C2(B11)(B21);
where
mi = mi(s1; s) = EXiIfs1H 1(d=n) < jXij  sH 1(d=n)g
and C1; C2 are positive constants. On the other hand,
E
 1AnUn(B11)
2  1AnUn(B12)
2
=
1
A4n
E
 
ntX
i=nt1+1

XiIfs1H 1(d=n) < jXij  sH 1(d=n)g  m(s1;s)i
!2

 
ntX
i=nt1+1

XiIfsH 1(d=n) < jXij  s2H 1(d=n)g  m(s;s2)i
!2
=
1
A4n
E
 
ntX
i=nt1+1
(X
(s1;s)
i  m(s1;s)i )
!2 ntX
i=nt1+1
(X
(s;s2)
i  m(s;s2)i )
!2
(2.15)
where we put
X
(u;v)
i = XiIfuH 1(d=n) < jXij  vH 1(d=n)g; m(u;v)i = EX(u;v)i :
Expanding the product expectation in (2.15), we get the sum of all expressions
(2.16) E(X
(s1;s)
i  m(s1;s)i )(X(s1;s)j  m(s1;s)j )(X(s;s2)k  m(s;s2)k )(X(s;s2)`  m(s;s2)` );
8
where nt1 + 1  i; j; k; `  nt. By the independence of the X 's, the product expectation in
(2.16) equals 0 if one of the i; j; k; ` diers from the other three. Thus it suces to estimate
the contribution of the terms where i; j; k; ` are pairwise equal, or all are equal. Assume rst
that i = j, k = ` and i 6= k; the other cases i = k; j = `, i 6= j and i = `; j = k, i 6= j can be
handled similarly as the case i = j = k = ` below. Then Xi and Xk are independent, and
thus using Lemma 2.2, the product expectation (2.16) becomes
E
h
(X
(s1;s)
i  m(s1;s)i )2(X(s;s2)k  m(s;s2)k )2
i
= E(X
(s1;s)
i  m(s1;s)i )2E(X(s;s2)k  m(s;s2)k )2(2.17)
 E(X(s1;s)i )2E(X(s;s2)k )2 
2
(2  )2 (s
2    s2 1 )(s2 2   s2 )`4(d=n)(n=d)(4 2)=
 C3(s  s1)(s2   s)`4(d=n)(n=d)(4 2)=:
The number of such pairs (i; k) is at most (nt   nt1)2 and thus dividing by A4n and using
(2.2) we get that the contribution of such terms (2.16) is not greater than
C4(t  t1)2(s  s1)(s2   s) = C4(B11)(B12):
Consider now the case i = j = k = `. In this case (2.16) becomes, expanding and introducing
new letters to lighten the notations,
E
h
(X
(s1;s)
i  m(s1;s)i )2(X(s;s2)i  m(s;s2)i )2
i
= E(  m(1))2(  m(2))2(2.18)
= E22   2m(2)E2 + (m(2))2E2   2m(1)E2 + 4m(1)m(2)E
  2m(1)(m(2))2E + (m(1))2E2   2(m(1))2m(2)E + (m(1))2(m(2))2;
where
 = X
(s1;s)
i ;  = X
(s;s2)
i ; m
(1) = E; m(2) = E:
Clearly  and  have disjoint support and thus  = 0, showing that the rst, second, fourth
and fth term of the last sum in (2.18) are equal to 0. Thus the sum equals
(m(2))2E2   2m(1)(m(2))2E + (m(1))2E2   2(m(1))2m(2)E + (m(1))2(m(2))2
= (m(2))2E2   2(m(1))2(m(2))2 + (m(1))2E2   2(m(1))2(m(2))2 + (m(1))2(m(2))2:
By the Cauchy-Schwarz inequality we have (m(1))2  E2, (m(2))2  E2 and thus the
absolute value of the last sum is at most 7E(2)E(2), which, apart from the coecient, is
exactly the third expression in (2.17), leading to the same estimate as there. The number
of choices for i in (2.18) is nt   nt1  (nt   nt1)2, so for the contribution of all terms in
(2.18) we get the same estimate as for (2.17), i.e. C5(B11)(B12). Thus we proved (2.13)
for Bij = B12 and the proof of Theorem 1.2 is completed.
9
References
[1] Berkes, I. and Horvath, L.: The central limit theorem for sums of trimmed variables with heavy tails.
Stochastic Process. Appl. 122 (2012), 449{465.
[2] Berkes, I., Horvath, L. and Schauer, J.: Asymptotics of trimmed CUSUM statistics. Bernoulli 17 (2011),
1344{1367.
[3] Bickel, P. J. and Wichura, M. J.: Convergence criteria for multiparameter stochastic processes and some
applications. Ann. Math. Statist. 42 (1971), 1656{1670.
[4] Billingsley, P.:. Convergence of probability measures. Wiley, New York 1968.
[5] Bingham, N.H., Goldie, C. M., Teugels, J.L.: Regular variation. Cambridge University Press, 1987.
[6] Csorg}o, S., Horvath, L. and Mason, D.: What portion of the sample makes a partial sum asymptotically
stable or normal ? Z. Wahrschein. verw. Gebiete 72 (1986), 1{16.
[7] Csorg}o, S., Csorg}o, M., Horvath, L. and Mason, D.: Normal and stable convergence of integral functions
of the empirical distribution function. Ann. Probability 14 (1986), 86-118.
[8] Feller, W.: An Introduction to probability theory and its applications, Vol II, Second Edition. Wiley,
1971.
[9] Grin, P. S. and Pruitt, W. E.: The central limit problem for trimmed sums. Math. Proc. Cambridge
Philos. Soc., 102 (1987), 329{349.
[10] Kiefer, J.: Iterated logarithm analogues for sample quantiles when pn ! 0. In: Proceedings of the
Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. I, pp. 227{244. University
of California Press, 1972.
[11] Le Page, R., Woodrofe, M. and Zinn, J.: Convergence to a stable distribution via order statistics. Ann.
Prob. 9 (1981), 624{632.
10


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