R E S E A R C H
Open Access
Limit properties of exceedance point
processes of strongly dependent normal
sequences
Fuming Lin
1,2*, Daimin Shi
1and Yingying Jiang
2*Correspondence:
1School of Statistics, Southwestern
University of Finance and Economics, Chengdu, Sichuan 611130, China
2School of Science, Sichuan
University of Science and Engineering, Zigong, Sichuan 643000, China
Abstract
In this paper, we define an in plane Cox process and prove the time-normalized point process of exceedances by a dependent normal sequence converging to the Cox process in distribution under some mild conditions. As some applications of the convergence result, two important joint asymptotic distributions for the order statistics are derived.
MSC: 60F05; 62E20
Keywords: Cox process; exceedance point process; strongly dependent normal sequences;kth maxima
1 Introduction
Let {ξi,i ≥ } be a standardized normal sequence with correlation coefficients rij = Cov(ξi,ξj) andM(nk)be thekth largest maxima of{ξi, ≤i≤n}. A conventional
assump-tion is thatrij→ asj–i→+∞at different rates according to which dependent normal
sequences are classified into two different types: ‘weakly dependent’ and ‘strongly depen-dent’, respectively. Leadbetteret al.[] considered the case:rij=r|j–i|andrnlogn→ as
n→+∞,i.e.{ξi,i≥}is a weakly dependent stationary normal sequence. By using
asymp-totic independence, they focused onM(nk)and its location (which is written asL(nk)) and
obtained the asymptotic behavior of the probabilitiesP(an(M()n –bn)≤x,L()n /n≤t), and
P(an(Mn()–bn)≤x,an(M()n –bn)≤x); here and in the sequel the standardized constants
anandbnare defined as
an= (logn)/, bn=an– (an)–(log logn+logπ). (.)
Mittal and Ylvisaker [] showed that ifrij=r|j–i|andrnlogn→γ > asn→+∞(the
strongly dependent stationary case) thenan(M()n –bn) tends in distribution to a
convo-lution ofexp(–e–x) and a normal distribution function, and further ifrnlogn→ ∞then
by a different normalization, the limiting distribution is normal. Recently, several impor-tant results for extremes of dependent normal sequences were established. Ho and Hsing [] and Tan and Peng [] investigated the joint asymptotic distributions of the maximum of{ξi, ≤i≤n}and
n
i=ξifor dependent Gaussian sequences. Hashorvaet al.[]
con-sidered the joint limit distributions of maxima of complete and incomplete samples, re-spectively,i.e.the Piterbarg theorem under some conditions on convergence rate of the
correlations. Leadbetteret al.[] developed an important tool: the weak convergence of exceedance point processes which is crucial to study the joint asymptotic distributions of some extremes. Many authors further studied the asymptotic behavior of exceedance point processes under different conditions. We refer to Piterbarg [], Huet al.[], Falket al.[], Penget al.[] and Hashorvaet al.[] for point processes of exceedances by weakly dependent stationary sequences including Gaussian ones and Wiśniewski [], Linet al.
[] for point processes of exceedances by strongly dependent Gaussian vector sequences. Throughout this paper, let {ξi,i≥} be a standardized strongly dependent
station-ary normal sequence with correlation coefficients rij=Cov(ξi,ξj). C stands for a
con-stant which may vary from line to line and ‘→’ for the convergence asn→ ∞. The re-mainder of the paper is organized as follows. In Section , we define an in plane Cox process and prove that the time-normalized point processNnof exceedances of levels
u()n,u()n , . . . ,un(r) by {ξi, ≤i≤n}converges in distribution to the in plane Cox process.
In Section , as the applications of our main result, the asymptotic results of the prob-abilitiesP(an(M()n –bn)≤x,L()n /n≤t), andP(an(M()n –bn)≤x,an(Mn()–bn)≤x) are
established.
2 Convergence of point processes of exceedances
Let {ξi,i ≥ } be a standardized normal sequence with correlation coefficients rij = Cov(ξi,ξj) satisfying the following assumptions:
rij=r|j–i| and rnlogn→γ ∈(,∞) asn→+∞. (.)
We concentrate on deriving the convergence of the time-normalized exceedance point process Nn of the levelsu()n,u()n , . . . ,u(nr) by {ξi, ≤i≤n} where u(nk)=xk/an+bn, k=
, , . . . ,r. In order to prove the main result in this section, we shall use the famous Berman’s inequality which is based on the early work of Slepian [], Berman [] and is polished up in Li and Shao []. The latest results related to Berman’s inequality are Hashorva and Weng [] and Lu and Wang []. The former gave a detailed introduction to Berman’s inequality and derived the inequality for some general scaling random vari-able and thus obtained a Berman inequality for non-normal random vector. The upper bound of Berman’s inequality gives an estimate of the difference between two standard-izedn-dimensional distribution functions by a convenient function of their covariances. According to Hashorva and Weng [], some results for normal sequences may be ex-tended to non-normal cases. The following lemmas are also needed in the proof of our result.
Lemma . Let d> andγ≥be constants,putρn=γ/logn and suppose that rnlogn→
γ as n→ ∞.Then,for any sequence{un}such that n( –(un))is bounded,we have
nd
[nd]
k=
|rk–ρn|exp
– u
n
+wk
→ as n→+∞,
where wk=max{|rk|,ρn}.
Leadbetteret al.[] defined a Cox processNwith intensityexp(–x–γ +√γ ζ), where ζ is a standard normal random variable,i.e.the process has the distribution determined by the following probability:
P
k
i=
N(Bi) =ki
=
∞
–∞ k
i=
(m(Bi)exp(–x–γ +
√ γz))ki
ki!
·exp–m(Bi)e–x–γ+ √
γzφ(z)dz, (.)
where m(·) is the Lebesgue measure and proved that a point process Nn of
time-normalized exceedances converges in distribution toNon (, +∞) which is summarized as Lemma . below.
Lemma . Suppose {ξi,i≥} is a standard stationary normal sequence with
covari-ances satisfying(.).Then the point process Nnof time-normalized exceedances of the level
un(un=x/an+bn)converges in distribution to N on(, +∞),where N is the Cox process
defined by(.).
Proof The proof can be found on p. in Leadbetteret al.[]. In Theorem . below, we extend Lemma . to the case of exceedances of several levels and study a vector of point processesNn= (Nn(),Nn(), . . . ,Nn(r)) which arises when
{ξi, ≤i≤n} exceeds the levelsu()n ,u()n , . . . ,un(r), whereu(nk)=xk/an+bn, ≤k≤r. For
clarity, we record the locations ofu()n ,u()n , . . . ,un(r)along fixed horizontal linesL,L, . . . ,Lr
in the plane. The structure of the process vector is the same as that of the exceedances process on pp.- in Leadbetteret al.[] where the authors presented a detailed and visualized introduction. According to Lemma ., each one-dimensional point process, on a givenLk,i.e. Nn(k) converges to a Cox process in distribution under appropriate
condi-tions. Before presenting Theorem ., we first give two definitions, one of which concerns a two-dimension Cox process,i.e.an in plane Cox process.
Definition . The locations of order statistics are the places where order statistics appear in the index set, for example the location of the maxima of{ξi, ≤i≤n}varying among
, . . . ,n.
Definition . Let{σj,j= , , . . .}be the points of a Cox processN(k)onLrwith
(stochas-tic) intensityexp(–xr–γ+
√
γ ζ), whereζ is a standard normal random variable,i.e. N(k)
has the distribution characterized in (.). Letβj,j= , , . . . be independent and identically
distributed (i.i.d.) random variables, independent also of the Cox process onLr, taking
val-ues , , . . . ,rwith conditional probabilities
P(βj=s|ζ=z) =
(τr–s+–τr–s)/τr, fors= , , . . . ,r– ,
τ/τr, fors=r,
i.e. P(βj≥s|ζ=z) =τr–s+/τrfors= , , . . . ,rwhereτi=e–xi–γ+ √
γz,i= , , . . . ,r. For each
aboveσjisP(βj≥|ζ =z) =τr–/τr and the deletions are conditionally independent, so
thatN(r–)is obtained as a conditionally independent thinning of the Cox processN(r).
The structure of the in plane Cox processNis very similar to that of the Poisson pro-cess on p. in Leadbetteret al.[], but the independent thinning is replaced with the conditionally independent thinning here.
Theorem . Suppose{ξi,i≥}is a standardized normal sequence satisfying the
condi-tions in Lemma..Let u(nk)=xk/an+bnsatisfy un()≥u()n ≥ · · · ≥un(r)(≤k≤r)where an
and bnare defined in(.).Then the time-normalized point process Nnof exceedances of
levels u()n,u()n , . . . ,un(r)by{ξi, ≤i≤n}converges in distribution to the above-mentioned in
plane Cox process.
Proof It is sufficient to show that whenngoes to∞:
(a) E(Nn(B))→E(N(B))for all setsBof the form(c,d]×(r,δ],r<δ, <c<d, where
d≤andE(·)is the expectation,
(b) P(Nn(B) = )→P(N(B) = )for all setsBwhich are finite unions of disjoint sets of this form.
Focus on (a) firstly. If B= (c,d]×(r,δ] intersects any of the lines, suppose these are
Ls,Ls+, . . . ,Lt(≤s≤t≤r). Then
Nn(B) = t
k=s
Nn(k)(c,d], N(B) =
t
k=s
N(k)(c,d]
and the number of pointsj/nin (c,d] is [nd] – [nc]. As in the proof of Theorem .. on p. in Leadbetteret al.[], we haveE(Nn(B)) = ([nd] – [nc])
t
k=s( –F(u (k)
n )), where
–Fu(nk)= –u(nk), ≤j≤n.
Obviously
n –u(nk)=n –(xk/an+bn)
∼e–xk asn→ ∞. (.)
Thus, we haveE(Nn(B))∼n(d–c)
t k=s(e
–xk
n +o(
n))→(d–c)
t
k=se–xk. Since
EN(B)=
t
k=s
E(d–c)exp(–xk–γ +
γ ζ)
=
t
k=s
(d–c)e–xk–γ·e(
√
γ)
=
t
k=s
(d–c)e–xk,
(a) follows. In order to prove (b), we must show thatP(Nn(B) = )→P(N(B) = ), where
does not intersect any of the linesL,L, . . . ,Lr. Because there are intersections and
differ-ences of the intervals (ck,dk], we may writeBin the form
s
k=(ck,dk]×Ek, where (ck,dk]
are disjoint andEkis a finite union of semi-closed intervals. It therefore follows that
Nn(B) = = s
k=
Nn(Fk) = , (.)
whereFk= (ck,dk]×Ek. Denote the lowestLjintersectingFkbyLlk. By the above thinning property, obviously
Nn(Fk) = =
N(lk)
n
(ck,dk]
= =Mn(ck,dk)≤u(nlk) , (.)
whereMn(ck,dk) stands for the maximum of{ηi,i≥}with indexk([cn] <k≤[dn]).
Con-sider the probabilities of (.) and (.) and obtain
PNn(B) =
=P s k=
Mn(ck,dk)≤u(nlk)
. (.)
It is convenient to firstly prove the following result. Let{¯ξi,i≥}be a standardized normal
sequence with the correlation coefficientρ.Mn(c,d;ρ) stands for the maximum of{¯ξk}
with indexk([cn] <k≤[dn]). It is well known thatMn(c,d;ρ), . . . ,Mn(ck,dk;ρ) have the
same distribution as ( –ρ)/M
n(c,d; ) +ρ/ζ, . . . , ( –ρ)/Mn(ck,dk; ) +ρ/ζ, where
c=c<d<· · ·<ck<dk=dandζ is a standard normal variable; see Leadbetteret al.[].
Next we must estimate the bound of P s k=
Mn(ck,dk)≤u(nlk)
–P s k=
Mn(ck,dk,ρn)≤u(nlk)
, (.)
whereρn=γ/logn.
Using Berman’s inequality, the bound of (.) does not exceed
π
|rij–ρn|
–ρn–/exp
–
((u
(i)
n)+ (u(nj)))
+ωij
, (.)
where the sum is carried out overi<jandi,j∈sk=([ckn], [dkn]],u(ni) oru(nj) stands for
xi/an+bnorxj/an+bn, andωij=max{|rij|,ρn}. Furthermore, (.) does not exceed
C
≤i<j≤n
|rij–ρn|exp
–
((xi/an+bn) + (x
j/an+bn))
+ωij
<Cn
n
k=
|rk–ρn|exp
–((min≤i≤nxi)/an+bn)
+ωk
→.
Notingn( –((min≤i≤nxi)/an+bn)) is bounded, the last ‘→’ attributes to Lemma .. So
it suffices to prove
P
s
k=
Mn(ck,dk,ρn)≤u(nlk)
→P s k=
N(B) =
Noting the definition ofMn(ck,dk,ρn), clearly, it follows that P s k=
Mn(ck,dk,ρn)≤u(nlk)
=P s k= ( –ρn)
Mn(ck,dk, ) +ρ
nζ≤u(nlk)
= +∞ –∞ P s k=
Mn(ck,dk, )≤( –ρn)–
u(lk)
n –ρ
nz
φ(z)dz,
where the proof of the last ‘=’ can be completed by using the argument on the first line from the bottom on p. in Leadbetteret al.[]. Sincean= (logn)
,bn=an+O(a–
n log logn),
andρn=γ/logn, it is easy to show
( –ρn)–
u(lk)
n –ρ
nz
=xlk+γ – √
γz an
+bn+o
a–n ,
see also the proof of Theorem .. on p. in Leadbetteret al.[]. Furthermore, we may obtain the following result:
P
s
k=
Mn(ck,dk, )≤( –ρn)–
u(lk)
n –ρ
nz =P s k= ˜
ζ[ckn]+≤( –ρn)
–u(lk)
n –ρ
nz , . . . , ˜
ζ[dkn]≤( –ρn)
–u(lk)
n –ρ
nz → s k=
exp–(dk–ck)e–xlk–γ+ √
γz
,
whereζ˜kis a sequence of independent standard normal variables and we used the same
arguments as (.) for the last step. Using the dominated convergence theorem, it follows that +∞ –∞ P s k=
Mn(ck,dk, )≤( –ρn)–
u(lk)
n –ρ
nz
φ(z)dz
→ +∞ –∞ s k=
exp–(dk–ck)e–xlk–γ+ √
γz
φ(z)dz
=PN(B) = .
The proof of (b) is completed.
Corollary . Suppose{ξi,i≥}satisfies the conditions of Theorem..Let B, . . . ,Bsbe
integers m(jk),
PNn(k)(Bj) =mj(k),j= , , . . . ,s;k= , , . . . ,r
→PN(k)(Bj) =mj(k),j= , , . . . ,s;k= , , . . . ,r
.
Proof Combining Theorem . and the proof of Corollary .. in Leadbetteret al.[], we
can complete the proof.
Theorem . Let the levels u(nk)(≤k≤r)satisfy
Pmax
≤i≤nξi≤u (k) n
→
+∞
–∞
exp–e–xk–γ+√γzφ(z)dz, n→ ∞,
with u()n ≥u()n ≥ · · · ≥un(r).Let Sn(k)be the number of exceedances of u(nk)by{ξi, ≤i≤n}.
Then for k≥,k≥, . . . ,kr≥,
PS()n =k,Sn()=k+k, . . . ,Sn(r)=k+k+· · ·+kr
→τ
k
(τ–τ)k· · ·(τr–τr–)kr
k!k!· · ·kr!
· +∞
–∞
exp(γz–γ)k+k+···+kr·exp–e–xk–γ+√γzφ(z)dz. (.)
Proof By Corollary ., the left-hand side of (.) converges to
PS()=k,S()=k+k, . . . ,S(r)=k+k+· · ·+kr
, (.)
whereS(i)=N(i)([, ]). In our paper, the definition of the Cox process is similar to that
of the in plane Poisson process in Leadbetter et al.[]. So we can refer to the proof of Theorem .. in Leadbetteret al.[] and find that (.) equals
(k+k+· · ·+kr)!
k!k!· · ·kr!
τ
τr
kτ
–τ
τr
k · · ·
τr–τr–
τr
kr
·PN(r)(, ]=k
+k+· · ·+kr
.
The proof is completed since
PN(r)(, ]=k+k+· · ·+kr
= +∞
–∞
(exp(–xr–γ +
√
γz))k+k+···+kr (k+k+· · ·+kr)! ·
exp–e–xr–γ+√γzφ(z)dz
=(exp(–xr))
k+k+···+kr
(k+k+· · ·+kr)!
+∞
–∞
exp(–γ +γz)k+k+···+kr
·exp–e–xr–γ+√γzφ(z)dz.
3 The joint distributions of some order statistics
Theorem . Suppose{ξi,i≥}is a standard normal sequence satisfying the conditions of
Theorem..Let u(nk)=xk/an+bnand M()n ,L()n be the second largest maxima ofξ,ξ, . . . ,ξn
and its location.Then for x>x,as n→ ∞,
Pan
Mn()–bn
≤x,an
M()n –bn
≤x
→ +∞
–∞
exp(–x–γ+
γz) –exp(–x–γ +
γz) +
·exp–e–x–γ+√γzφ(z)dz (.)
and
P
an
Mn()–bn
≤x,
nL
() n ≤t
→
x
–∞H(y,t)dy, (.)
where
H(y,t) = +∞
–∞ ( –t)
exp(–y–γ+γz)exp–( –t)e–y–γ+√γzφ(z)dz
· +∞
–∞ t
exp(–y–γ +γz)exp–te–y–γ+√γzφ(z)dz
+ +∞
–∞ exp
–( –t)e–y–γ+√γzφ(z)dz
· +∞
–∞ t
exp(–y– γ + γz)exp–te–y–γ+√γzφ(z)dz.
Proof Clearly the left-hand side of (.) is equal to
Pan
Mn()–bn
≤x,an
M()n –bn
≤x
=PSn()= +PSn()= ,S()n = ,
whereS(ni)is the number of exceedances ofun(i)byη,η, . . . ,ηn. Using Theorem . in the
special case yields (.). In order to prove (.), writeIandJfor intervals{, , . . . , [nt]} and{[nt] + , . . . ,n}, respectively.M()(I),M()(I),M()(J),M()(J) stand for the maxima and second largest maxima ofξ,ξ, . . . ,ξnin the intervalsI,J, respectively. LetHn(x,x,x,x)
be the joint d.f. of the normalized r.v.
Xn()=an
M()n (I) –bn
, Xn()=an
M()n (I) –bn
,
Yn()=an
M()n (J) –bn
, Yn()=an
M()n (J) –bn
.
Generally letx>xandx>x, that is,
Hn(x,x,x,x)
=PM()n (I)≤u()n ,M()n (I)≤un(),Mn()(J)≤u()n ,M()n (J)≤u()n
whereI = (,t] andJ = (t, ]. By using Corollary . withB=I andB=J, we obtain
lim
n→∞Hn(x,x,x,x)
= lim n→∞P
Nn()I= ,Nn()I≤·PNn()J= ,Nn()J≤ =
+∞
–∞
e–x–e–xtexp(γ –γ) + exp–te–x–γ+√γzφ(z)dz ·
+∞
–∞
e–x–e–x( –t)exp(γ –γ) + exp–( –t)e–x–γ+√γzφ(z)dz =Ht(x,x)H–t(x,x) =H(x,x,x,x).
Now the left-hand side of (.) is equal to
PMn()(I)≤u()n ,M()n (I)≥Mn()(J)
+PMn()(I)≤u()n ,M()n (J) >M()n (I)≥M()n (J). (.) ObviouslyHis absolutely continuous and the boundaries of sets inRsuch as{(w
,w,w,
w) :w≤x,w>w}and{(w,w,w,w) :w≤x,w>w≥w}have zero Lebesgue
measure. Thus using Corollary ., it follows that (.) converges to
P(X≤x,X≥Y) +P(X≤x,Y>X≥Y).
According to the joint distributionH(x,x,x,x) ofX,X,Y, andY, a simple evaluation
completes the proof.
Remark . We may obtain the joint asymptotic distribution ofM()n ,M()n , . . . ,M(nk)by
us-ing the same method as in Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgements
The research of the second author was supported by the National Natural Science Foundation of China, Grant No. 71171166. The research of other authors was supported by the Scientific Research Fund of Sichuan Provincial Education Department under Grant 12ZB082, the Scientific research cultivation project of Sichuan University of Science and Engineering under Grant 2013PY07, the Scientific Research Fund of Sichuan University of Science and Engineering under Grant 2013KY03, and the Science Research Programs for Doctors in Southwestern University of Finance and Economics, Grant No. JBK1207085.
Received: 28 September 2014 Accepted: 2 February 2015 References
1. Leadbetter, MR, Lindgren, G, Rootzen, H: Extremes and Related Properties of Sequences and Processes. Springer, New York (1983)
2. Mittal, Y, Ylvisaker, D: Limit distributions for the maxima of stationary Gaussian processes. Stoch. Process. Appl.3, 1-18 (1975)
3. Ho, HC, Hsing, T: On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. J. Appl. Probab.33, 138-145 (1996)
5. Hashorva, E, Peng, Z, Weng, Z: On Piterbarg theorem for maxima of stationary Gaussian sequences. Lith. Math. J.53, 280-292 (2013)
6. Piterbarg, VI: Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs, vol. 148. Am. Math. Soc., Providence (1996)
7. Hu, A, Peng, Z, Qi, Y: Limit laws for maxima of contracted stationary Gaussian sequences. Metrika70, 279-295 (2009) 8. Falk, M, Hüsler, J, Reiss, RD: Laws of Small Numbers: Extremes and Rare Events, 3rd edn. DMV Seminar, vol. 23.
Birkhäuser, Basel (2010)
9. Peng, Z, Tong, J, Weng, Z: Joint limit distributions of exceedances point processes and partial sums of Gaussian vector sequence. Acta Math. Sin. Engl. Ser.28, 1647-1662 (2012)
10. Hashorva, E, Peng, Z, Weng, Z: Limit properties of exceedances point processes of scaled stationary Gaussian sequences. Probab. Math. Stat.34, 45-59 (2014)
11. Wi´sniewski, M: Multidimensional point processes of extreme order statistics. Demonstr. Math.27, 475-485 (1994) 12. Lin, F, Shi, D, Jiang, Y: Some distributional limit theorems for the maxima of Gaussian vector sequences. Comput.
Math. Appl.64, 2497-2506 (2012)
13. Slepian, D: The one-sided barrier problem for Gaussian noise. Bell Syst. Tech. J.41, 463-501 (1962)
14. Berman, SM: Limit theorems for the maximum term in stationary sequences. Ann. Math. Stat.35, 502-516 (1964) 15. Li, WV, Shao, QM: A normal comparison inequality and its applications. Probab. Theory Relat. Fields122, 494-508
(2002)
16. Hashorva, E, Weng, Z: Berman’s inequality under random scaling. Stat. Interface7, 339-349 (2014)
