Estimation and bootstrap for stochastically monotone Markov processes

Neumann, Michael H.
A icle — Published Ve sion
Es ima ion and boo s ap o s ochas ically mono one
Ma ko p ocesses
Me ika
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Neumann, Michael H. (2023) : Es ima ion and boo s ap o s ochas ically
mono one Ma ko p ocesses, Me ika, ISSN 1435-926X, Sp inge , Be lin, Heidelbe g, Vol. 87, Iss. 1,
pp. 31-59,
h ps://doi.o g/10.1007/s00184-023-00903-7
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Me ika (2024) 87:31–59
h ps://doi.o g/10.1007/s00184-023-00903-7
Es ima ion and boo s ap o s ochas ically mono one
Ma ko p ocesses
Michael H. Neumann1
Recei ed: 9 May 2022 / Accep ed: 9 Feb ua y 2023 / Published online: 28 Feb ua y 2023
© The Au ho (s) 2023
Abs ac
The Ma ko p ope y is sha ed by se e al popula models o ime se ies such as
au o eg essi e o in ege - alued au o eg essi e p ocesses as well as in ege - alued
ARCH p ocesses. A na u al assump ion which is ul illed by co esponding pa ame ic
e sions o hese models is ha he andom a iable a ime ge s s ochas ically g ea e
condi ioned on he pas , as he alue o he andom a iable a ime −1 inc eases. Then
he associa ed amily o condi ional dis ibu ion unc ions has a ce ain mono onici y
p ope y which allows us o employ a nonpa ame ic an i onic es ima o . This es ima o
does no in ol e any uning pa ame e which con ols he deg ee o smoo hing and is
he e o e easy o apply. Ne e heless, i is shown ha i a ains a a e o con e gence
which is known o be op imal in simila cases. This es ima o o ms he basis o a new
me hod o boo s apping Ma ko chains which inhe i s he p ope ies o simplici y and
consis ency om he unde lying es ima o o he condi ional dis ibu ion unc ion.
Keywo ds Au o eg essi e p ocess ·Boo s ap ·INAR ·In ege - alued ARCH ·
Ma ko chain ·S ochas ic o de
Ma hema ics Subjec Classi ica ion P ima y 60G10; Seconda y 60J05
1 In oduc ion
We conside a ime-homogeneous Ma ko chain (X ) ∈N0d i en by a ansi ion ke -
nel which sa is ies a ce ain mono onici y p ope y: he condi ional dis ibu ion o he
andom a iable a ime ge s s ochas ically g ea e as he alue o he a iable a
ime −1 inc eases. Such a condi ion is ac ually sa is ied by se e al popula mod-
els o ime se ies such as au o eg essi e o in ege - alued au o eg essi e as well as
BMichael H. Neumann
[email p o ec ed]
1Ins i u ü Ma hema ik, F ied ich-Schille -Uni e si ä Jena, E ns -Abbe-Pla z 2, 07743 Jena,
Ge many
123
32 M. H. Neumann
in ege - alued ARCH p ocesses unde na u al assump ions on he in ol ed pa ame-
e s. To be speci ic, we assume ha , o each ixed z,Fx(z):= P(X ≤z|X −1=x)
is an i onic (mono onically non-inc easing) in x. This assump ion allows us o employ
a nonpa ame ic an i onic es ima o 
Fx(z)o he unc ion x→ Fx(z). Ou es ima o
does no in ol e any uning pa ame e which con ols he deg ee o smoo hing and is
he e o e easy o apply. Mo eo e , i s consis ency does no equi e smoo hness p op-
e ies o he unc ion x→ Fx(z); he pos ula ed mono onici y su ices. Theo em 2.1
s a es ha he es ima o 
Fx(z)con e ges in L1no m, weigh ed by he s a iona y dis-
ibu ion o he Ma ko chain, wi h a a e o n−1/3which is belie ed o be he op imal
one.
The es ima o o Fx(z)se es as a basis o a new boo s ap me hod o Ma ko
chains. Among se e al o he me hods, hose p oposed by Raja shi (1990) and Papa o-
di is and Poli is (2002) a e he closes ones o ou p oposal. While Raja shi’s boo s ap
p ocedu e is based on a nonpa ame ic es ima e o he one-s ep ansi ion densi y,
Papa odi is and Poli is (2002) used in hei so-called local boo s ap a local esam-
pling o he o iginal da a se . In bo h pape s, he p oo o consis ency o he espec i e
boo s ap me hod is based on he assump ion o a smoo h ansi ion densi y. In con as ,
ou app oach does no equi e any smoo hness assump ion on he ansi ion mecha-
nism; i is me ely based on he mono onici y assump ion on he Ma ko ke nel. We
show i s applicabili y o Ma ko chains wi h s a e space N0={0,1,2,...}. Consis-
ency o boo s ap can be shown in a mos anspa en way by a so-called coupling o
he o iginal p ocess and i s boo s ap coun e pa , i.e. we de ine e sions (
X ) ∈N0and
(
X∗
) ∈N0o hese p ocesses on a common p obabili y space (
, 
A,
P)such ha he
co esponding andom a iables 
X and 
X∗
a e equal wi h a high p obabili y. Some-
wha su p isingly, his na u al app oach was a ely used in s a is ics. Using Mallows
me ic o measu e he dis ance be ween a iables om he o iginal and he boo s ap
p ocess, i was implici ly employed in he con ex o independen andom a iables by
Bickel and F eedman (1981) and F eedman (1981). A mo e explici use o coupling
was made, in he con ex o U- and V-s a is ics, bu again in he independen case, by
Dehling and Mikosch (1994) and Leuch and Neumann (2009). Fo dependen da a,
his app oach was adop ed by Leuch and Neumann (2013), Leuch e al. (2015), and
Neumann (2021). Ou second main esul , Theo em 3.1, desc ibes he esul s o ou
coupling app oach. The s a iona y dis ibu ion P∗
X∗o he boo s ap p ocess con e ges
in o al a ia ion no m and in p obabili y o ha o he o iginal p ocess. The coupled
p ocess is φ-mixing wi h coe icien s decaying a an exponen ial a e and he co e-
sponding alues 
X0
and 
X∗,0
o a s a iona y e sion o he coupled p ocess coincide
wi h a p obabili y con e ging o 1. These gene al esul s can hen be used o p o e
boo s ap consis ency o speci ic s a is ics. The p oo s o ou main heo ems and some
auxilia y esul s a pos poned o a inal Sec .4.
2 An es ima o o a mono one amily o dis ibu ion unc ions
Suppose ha we obse e andom a iables X0,X1,...,Xn, whe e X=(X ) ∈N0is
a s ic ly s a iona y Ma ko chain wi h s a e space D⊆R, de ined on a p obabili y
123
Es ima ion and boo s ap o s ochas ically mono one… 33
space (, A,P). We deno e he s a iona y dis ibu ion by PXand he co esponding
dis ibu ion unc ion by FX.Le (Fx)x∈Rde ined as Fx(z)=P(X ≤z|X −1=x)
be he co esponding amily o condi ional dis ibu ion unc ions. We impose he
ollowing as ou key assump ion.
(A1) Fo each z∈R, he unc ion x→ Fx(z)is mono onically non-inc easing, i.e. i
x1<x2, hen P(X ≤z|X −1=x1)≥P(X ≤z|X −1=x2).
In addi ion we suppose ha
(A2) X=(X ) ∈N0is s ong mixing wi h exponen ially decaying coe icien s αX(k),
i.e.
αX(k)=Oρk,
o some ρ∈[0,1).
Assump ion (A1) may be pa aph ased as ollows. I x1<x2and i Y1and Y2a e
andom a iables ollowing he espec i e condi ional dis ibu ions PX |X −1=x1and
PX |X −1=x2, hen Y2is s ochas ically no smalle han Y1. I u ns ou ha his assump-
ion is ac ually sa is ied by popula classes o Ma ko chain models unde na u al
assump ions. He e is a lis o models we ha e in mind:
(1) Nonlinea au o eg essi e p ocesses wi h non-dec easing link The p ocess X=
(X ) ∈N0is assumed o obey he model equa ion
X = (X −1)+ε ∀ ∈N,
whe e (ε ) ∈Nis a sequence o i.i.d. andom a iables and ε is independen o
X −1,...,X0. I he unc ion :R→Ris mono onically non-dec easing, hen,
o x1<x2,
PX ≤z|X −1=x1=Pε ≤z− (x1)≥Pε ≤z− (x2)
=PX ≤z|X −1=x2.
Fu he mo e, i ε has an e e ywhe e posi i e densi y and i
 (x)≤γ|x|−∀x≥K,
o some γ<1, >0, and K<∞, hen he p ocess Xhas a unique s a iona y
dis ibu ion and sa is ies (A2); see e.g. Doukhan (1994).
(2) B anching p ocesses wi h immig a ion Le X0,(Z ,k) ,k∈Nand (ε ) ∈Nbe mu ually
independen andom a iables aking alues in N0. We assume ha (Z ,k) ,k∈Nas
well as (ε ) ∈Na e sequences o iden ically dis ibu ed andom a iables. Then
he p ocess X=(X ) ∈N0gi en by
X =
X −1

k=1
Z ,k+ε ∀ ∈N
123
34 M. H. Neumann
is a b anching p ocess wi h immig a ion. In he special case o Z ,k∼Bin(1,α)
we ob ain a so-called i s -o de in ege - alued au o eg essi e (INAR(1)) p ocess
which was p oposed by McKenzie (1985) and Al-Osh and Alzaid (1987). Since
he Z ,ka e non-nega i e andom a iables, i is ob ious ha (A1) is ul illed. I
in addi ion Eε <∞and EZ ,k<1, hen Xhas a unique s a iona y dis ibu ion
and sa is ies (A2); see Pakes (1971).
(3) Poisson-INARCH p ocesses The p ocess X=(X ) ∈N0is an in ege - alued
ARCH p ocess o o de 1 wi h Poisson inno a ions (Poisson-INARCH(1)) i
X |F −1∼Poisson (X −1),
whe e Fsdeno es he σ-algeb a gene a ed by X0,...,Xs.I is mono onically
non-dec easing, hen we ob ain, o x1<x2and Y1∼Poisson( (x1)),Y2∼
Poisson( (x2)),
P(X ≤z|X −1=x1)=P(Y1≤z)≥P(Y2≤z)=P(X ≤z|X −1=x1),
i.e., (A1) is ul illed. Fu he mo e, i in addi ion
(x)≤γx−∀x≥K,
o some γ<1, >0, and K<∞, hen Xhas a unique s a iona y dis ibu ion
and sa is ies (A2); see e.g. Theo em 2 in Doukhan (1994, Sec. 2.4, p. 90).
We conside an es ima o o Fx(z)=PX ≤z|X −1=xwhich akes in o accoun
ha he unc ion x→ Fx(z)is mono onically non-inc easing unde (A1). Nonpa a-
me ic es ima o s o mono one unc ions ha e a long his o y and we e p oposed e.g.
by B unk (1955) and Aye e al. (1955). Deno e by 1(·) he indica o unc ion. Fo
z∈Dand x∈{X0,...,Xn−1}, we de ine

F(max −min)
x(z):= max
: ≥xmin
u:u≤xn
=11X ≤z,X −1∈[u, ]
#{ ≤n:X −1∈[u, ]} (2.1a)
and

F(min −max)
x(z):= min
u:u≤xmax
: ≥xn
=11X ≤z,X −1∈[u, ]
#{ ≤n:X −1∈[u, ]} .(2.1b)
I is well-known ha 
F(max −min)
x(z)=
F(min −max)
x(z) o all x∈{X0,...,Xn−1},
see e.g. Theo em 1 in B unk (1955) and Theo em 1.4.4 in Robe son, W igh , and
Dyks a (1988, p. 23). As poin ed ou by Deng and Zhang (2020), (2.1a) and (2.1b)
ha e o be modi ied o x/∈{X0,...,Xn−1}. Since i could well happen ha an in e al
wi h x∈[u, ]does no con ain any poin om he collec ion {X0,...,Xn−1}we
se nu, =#{ ≤n:X −1∈[u, ]},nu,∗=#{ ≤n:u≤X −1},n∗, =#{ ≤
123

Es ima ion and boo s ap o s ochas ically mono one… 35
n:X −1≤ }, and de ine

F(max −min)
x(z):= max
: ≥x,n∗, >0min
u:u≤x,nu, >0n
=11X ≤z,X −1∈[u, ]
#{ ≤n:X −1∈[u, ]} (2.2a)
and

F(min −max)
x(z):= min
u:u≤x,nu,∗>0max
: ≥x,nu, >0n
=11X ≤z,X −1∈[u, ]
#{ ≤n:X −1∈[u, ]} .
(2.2b)
The es ima o s 
F(max −min)
x(z)and 
F(min −max)
x(z)a e bo h non-inc easing in xas he
maxima a e aken o e non-inc easing classes indexed by xand he minima o e non-
dec easing classes. Fu he mo e, o ixed x∈D, he mappings z→ 
F(max −min)
x(z)
and z→ 
F(min −max)
x(z)a e non-dec easing which ollows om he iso onici y o
he unc ions z→ 1(X ≤z,X −1∈[u, ]). Fu he mo e, i X[1],...,X[n]is an
enume a ion o he alues in {X1,...,Xn}in non-dec easing o de , hen i ollows ha ,
again o ixed x∈D, he mappings z→ 
F(max −min)
x(z)and z→ 
F(min −max)
x(z)
a e cons an on he hal -open in e als [X[k],X[k+1])(k=1,...,n−1), and a ain
he espec i e alues 0 and 1 on (−∞,X[1])and [X[n],∞). Hence, hese es ima o s
a e genuine p obabili y dis ibu ion unc ions.
We choose as ou es ima o o Fx(z)

Fx(z):= 
F(max −min)
x(z)+
F(min −max)
x(z)/2.
I ollows ha all o he abo e p ope ies o 
F(max −min)
x(z)and 
F(min −max)
x(z)a e
inhe i ed by 
Fx(z). I s pe o mance is cha ac e ized by he ollowing heo em.
Theo em 2.1 Suppose ha (A1) and (A2) a e ul illed. Then
sup
zED
Fx(z)−Fx(z)dP
X(x)=On−1/3.
The a e o con e gence n−1/3is known o be op imal in ela ed p oblems o es ima -
ing a mono one unc ion on he basis o independen andom a iables; see e.g. Du o
(2002, Theo em 1) and Zhang (2002, Theo em 2.3). We belie e ha his a e canno
be imp o ed in ou mo e delica e case o ime se ies da a. No e ha Mösching and
Dümbgen (2020) conside ed a nonpa ame ic an i onic es ima o o Fxin a eg ession
con ex whe e he dependen a iables, condi ional on he eg esso s, a e indepen-
den . They de i ed unde addi ional Hölde condi ions a es o uni o m and poin wise
con e gence o his es ima o .
Ou app oach o p o e his esul can be mos easily explained i he dis ibu ion
unc ion FXis con inuous. We spli he domain Din o kn=n1/3in e als Ik=
[xk−1,xk), whe e x0=−∞i D=R,x0=0i D=N0and, in bo h cases,
123
36 M. H. Neumann
xk=F−1
X(k/kn)=sup{x:FX(x)≥k/kn} o k=1,...,kn−1, xkn=∞.(As
usual, adeno es he la ges in ege less han o equal o a.) We can expec a a o able
beha io o 
Fx(z)i Nk(ω) := #{ ≤n:X −1(ω) ∈Ik}is su icien ly la ge o all k.
Le
An=ω:Nk(ω) ≥n/(2kn) o all k=1,...,kn.
I ollows om Lemma 4.2 ha P(Ac
n)=O(n−1/3). Since D
Fx(z)−Fx(z)dP
X(x)
≤1 holds wi h p obabili y 1, we ob ain ha
ED
Fx(z)−Fx(z)dP
X(x)1Ac
n≤PAc
n=On−1/3.(2.3)
To es ima e ED
Fx(z)−Fx(z)+dP
X(x)1Anwe p oceed as ollows. Fo x∈Ik,
k∈{2,...,kn}, we use he es ima e

Fx(z)−Fx(z)+1An≤
Fxk−1(z)−Fxk(z)+1An
≤max
: ≥xk−1n
=11(X ≤z)−FX −1(z)1(X −1∈[xk−2, ])
#{ ≤n:X −1∈[xk−2, ]}
1An
+Fxk−2(z)−Fxk(z).
We ob ain om Lemma 4.3 ha
Emax
: ≥xk−1n
=11(X ≤z)−FX −1(z)1(X −1∈[xk−2, ])
#{ ≤n:X −1∈[xk−2, ]}
1An
=On−1/3.
Since
kn

k=2Fxk−2(z)−Fxk(z)=
kn

k=2Fxk−2(z)−Fxk−1(z)+
kn

k=2Fxk−1(z)−Fxk(z)
=Fx0(z)−Fxkn−1(z)+Fx1(z)−Fxkn(z)≤2.
we conclude ha
kn

k=2
EIk
Fx(z)−Fx(z)+dP
X(x)1An=On−1/3.
Fu he mo e, he ough es ima e
EI1
Fx(z)−Fx(z)+dP
X(x)1An≤PX(I1)≤n−1/3
123
Es ima ion and boo s ap o s ochas ically mono one… 37
is ob iously ue, which leads o
ED
Fx(z)−Fx(z)+dP
X(x)1An=On−1/3.(2.4)
We can p o e
ED
Fx(z)−Fx(z)−dP
X(x)1An=On−1/3.(2.5)
in comple e analogy o (2.4). The esul s a ed in Theo em 2.1 ollows om (2.3) o
(2.5). In he gene al case we ha e o ake in o accoun ha he dis ibu ion unc ion FX
is no necessa ily con inuous. This leads o a echnically mo e in ol ed p oo which
is p esen ed in ull de ail in Sec .4.
The ollowing pic u es gi e an imp ession o how he unc ions x→ Fx(z)a e
app oxima ed by 
Fx(z) o di e en alues o z. We simula ed a Poisson-INARCH
p ocess o o de 1, whe e X |X −1,X −2,... ∼Poisson (X −1)and (x)=
min α0+α1x,β. The pa ame e s α0and α1a e chosen as 2.0 and 0.5, espec-
i ely, and he unca ion cons an βis se o 6.0. Fo a sample size n=1000 and
z=0,1,...,11, he ollowing pic u es show Fx(z)( ed lines) and a co esponding
es ima e 
Fx(z)(blue lines). These esul s a e qui e encou aging excep o la ge alues
o x. We conjec u e ha his de iciency is caused by da a spa si y in his egion.
123
38 M. H. Neumann
3 A new boo s ap me hod o Ma ko chains
Ou es ima o 
Fx(z)can be used o boo s apping Ma ko p ocesses, and i is pa ic-
ula ly sui able in case o Ma ko chains wi h a ini e o coun ably in ini e s a e space.
In wha ollows we assume ha (X ) ∈N0is a s a iona y Ma ko chain which has a
123
Es ima ion and boo s ap o s ochas ically mono one… 45
which implies ha

E
Sn−
S∗
n2
=1
n
n

s, =1
co 1(
Xs−1,
Xs)−1(
X∗
s−1,
X∗
s), 1(
X −1,
X )−1(
X∗
−1,
X∗
)
=O
Pφ
X,
X∗(|s− |−1)n−1/3(log n)2.
This implies ha
S∗
n
d
−→ Yin p obabili y.
I in addi ion σ2
∞>0, hen we ob ain by Lemma 2.11 o an de Vaa (1998) ha
sup
xPSn≤x−PS∗
n≤xX0,...,XnP
−→ 0.
Hence, we can use boo s ap quan iles o cons uc con idence in e als o θsuch ha
hei co e age p obabili y con e ges o a p esc ibed le el. Simila implica ions o
o he ypes o s a is ics a e discussed in Leuch and Neumann (2013) and Neumann
(2021).
Rema k 1 In a simila con ex , Papa odi is and Poli is (2002, Theo em 3.3) p o ed
almos su e con e gence o he boo s ap s a iona y dis ibu ion o he s a iona y dis-
ibu ion o he o iginal p ocess. Thei me hod o p oo is comple ely di e en om
ou s and employs classical ools om he heo y o weak con e gence such as Helly’s
heo em and he “uniqueness ick” which uses he ac ha each subsequence con-
ains a u he subsequence con e ging o he same p obabili y measu e. We use a
mo e di ec app oach based on a coupling o he o iginal and he boo s ap p ocess.
The addi ional bene i is ha we ob ain a a e o con e gence a he han consis ency
only.
The ollowing pic u es gi e an imp ession o he e ec o ou coupling. As done
o he pic u es displayed in he p e ious sec ion, we simula ed a Poisson-INARCH
p ocess o o de 1, whe e X |X −1,X −2,... ∼Poisson (X −1)and (x)=
min α0+α1x,β. The pa ame e s α0and α1a e chosen as 2.0 and 0.5, espec i ely,
and he unca ion cons an βis se o 6.0. Fo espec i e sample sizes o n=200 and
n=1000, Figs.1and 2show one ealiza ion o independen and coupled e sions o
X1,...,X50 and X∗
1,...,X∗
50. While he pic u es on he le o Figs. 1and 2le us
a bes hope o a simila beha io o he boo s ap and he o iginal p ocess, hose on
he igh p o ide some e idence ha he boo s ap p ocess success ully mimics he
beha io o he o iginal p ocess.
123

46 M. H. Neumann
Fig. 1 Independen and coupled p ocesses, n = 200
Fig. 2 Independen and coupled p ocesses, n = 1000
4 P oo s
4.1 P oo s o he main esul s
P oo o Theo em 2.1 Ou s a egy o p o e his esul is al eady ske ched in Sec .2,in
he special case whe e he dis ibu ion unc ion FXassocia ed o PXis con inuous. In
he gene al case wi h a possibly discon inuous unc ion FX, we ha e o ake g ea ca e
since we canno spli he domain Din o in e als Iksuch ha PX(Ik)=1/kn, whe e
kn=n1/3. I could be he case ha PXhas masses conside ably la ge han 1/kna
single poin s which equi es a modi ica ion o ou p e ious app oach.
To ob ain an app op ia e collec ion o in e als Ik, we de ine again sui able
g id poin s x0,xn,...,xKn. Fo echnical easons we choose hem as a dec easing
sequence. We se x0:= ∞ and de ine ecu si ely xk:= in {x:PX((x,xk−1)) ≤
n−1/3} o k≥1. This p ocedu e will e mina e when xKn=0 and D=Ro when
xKn=−∞, o someKn. In bo h cases we ha e ha D=[x1,x0)∪···∪[xKn,xKn−1).
123
Es ima ion and boo s ap o s ochas ically mono one… 47
Fo k=1,...,Kn−1, i.e. wi h a possible excep ion o k=Kn,weha e
PX(xk,xk−1)≤n−1/3≤lim
m→∞ PX(xk−1/m,xk−1)=PX[xk,xk−1),
whe e he la e equali y ollows since he p obabili y measu e PXis con inuous om
abo e. In he ollowing we show ha
ED
Fx(z)−Fx(z)+dP
X(x)=On−1/3.(4.1)
To his end, we conside he con ibu ions by E[xk,xk−1)(
Fx(z)−Fx(z))+dP
X(x)
sepa a ely. We dis inguish be ween h ee possible cases.
Case 1 I PX[xk,xk−1)≤2n−1/3and k<Kn, hen we use o all x∈[xk,xk−1)
in case o Nn,k:=  ≤n:X −1∈[xk,xk−1)= 0 he es ima e

Fx(z)−Fx(z)+≤
Fxk(z)−Fxk−1(z)+
≤max
: ≥xkn
=1[1(X ≤z)−FX −1(z)]1(X −1∈[xk+1, ])
#{ ≤n:X −1∈[xk+1, ]}∨1
+Fxk+1(z)−Fxk−1(z),
which leads o
E[xk,xk−1)
Fx(z)−Fx(z)+dP
X(x)1{Nn,k=0}
≤PX[xk,xk−1)
Emax
: ≥xkn
=1[1(X ≤z)−FX −1(z)]1(X −1∈[xk+1, ])
#{ ≤n:X −1∈[xk+1, ]}∨1
+Fxk+1(z)−Fxk−1(z)
=OPX[xk,xk−1)n−1/3+(Fxk+1(z)−Fxk−1(z)).(4.2a)
Case 2 I PX[xk,xk−1)>2n−1/3 hen PXhas a xka poin mass g ea e han
n−1/3and we a gue di e en ly. In his case, we use o all x∈(xk,xk−1)in case o
Nn,k:=  ≤n:X −1=xk= 0 he es ima e

Fx(z)−Fx(z)+≤max
: ≥xkn
=1[1(X ≤z)−FX −1(z)]1(X −1∈[xk, ])
#{ ≤n:X −1∈[xk, ]}∨1
+Fxk(z)−Fxk−1(z),
123
48 M. H. Neumann
which implies
E(xk,xk−1)
Fx(z)−Fx(z)+dP
X(x)1{Nn,k=0}
≤PX(xk,xk−1)
Emax
: ≥xkn
=1[1(X ≤z)−FX −1(z)]1(X −1∈[xk, ])
#{ ≤n:X −1∈[xk, ]}∨1
+Fxk(z)−Fxk−1(z)
=OPX(xk,xk−1)n−1/3+(Fxk(z)−Fxk−1(z)).(4.2b)
Fo x=xk, we use he simple es ima e

Fx(z)−Fx(z)+≤max
: >xkn
=1[1(X ≤z)−FX −1(z)]1(X −1∈[xk, ])
#{ ≤n:X −1∈[xk, ]}∨1,
and we ob ain
E{xk}
Fx(z)−Fx(z)+dP
X(x)1{Nn,k=0}
≤PX{xk}Emax
: >xkn
=1[1(X ≤z)−FX −1(z)]1(X −1∈[xk, ])
#{ ≤n:X −1∈[xk, ]}∨11An
=OPX{xk}n−1/3.(4.2c)
Case 3 I PX([xKn,xKn−1)) ≤2n−1/3, hen we can simply use he es ima e
E[xKn,xKn−1)
Fx(z)−Fx(z)+dP
X(x)≤2n−1/3.(4.2d)
Finally, i ollows om Lemma 4.2 ha Pk{ω:Nn,k(ω) =0}=O(n−1/3),
which implies ha
ED
Fx(z)−Fx(z)+dP
X(x)1k{Nn,k=0}
≤P !
k{ω:Nn,k(ω) =0}"=On−1/3.(4.2e)
F om (4.2a) o(4.2e) we ob ain (4.1).
The e m D(
Fx(z)−Fx(z))−dP
X(x)can be analogously es ima ed which com-
ple es he p oo o he heo em. 
123
Es ima ion and boo s ap o s ochas ically mono one… 49
P oo o Theo em 3.1 (i) We cons uc a coupling o he o iginal p ocess and i s
boo s ap coun e pa , whe e we use π(x,x∗), (y,y∗)de ined by (3.5a) and
(3.5b) as ansi ion p obabili ies and 
Pas ansi ion ke nel. The ini ial alues
a e chosen such ha 
X0=
X∗
0∼PX. Then, o each ∈N0, condi ioned on
(
X ,
X∗
), he nex pai (
X +1,
X∗
+1)is gene a ed acco ding o 
P. I ollows om
(3.4) and (3.6) in pa icula ha

P
X +1= 
X∗
+1,
X =
X∗

=
x∈N0
P
X +1= 
X∗
+1|
X =
X∗
=x
P
X =
X∗
=x
=
x∈N0
δx,x
P
X =
X∗
=x
≤1
2
x
yπ(x,y)−π∗(x,y)PX({x})
=O
Pn−1/3log n.
This implies i s

P
X1= 
X∗
1=O
Pn−1/3log n,
hen

P
X2= 
X∗
2≤
P
X2= 
X∗
2,
X1=
X∗
1+
P
X1= 
X∗
1
=O
Pn−1/3log n,
and a e Knsuch s eps
dTVPX,
P
X∗
Kn≤
P
XKn= 
X∗
Kn=O
Pn−1/3log nK
n.
On he o he hand, (X∗
) ∈N0, and he e o e (
X∗
) ∈N0as well, a e geome ically
e godic. Hence, o Kn=Klog nand Ksu icien ly la ge,
dTV 
P
X∗
Kn,P∗
X∗=O
Pn−1/3,
which leads o
dTVPX,P∗
X∗≤dTV PX,
P
X∗
Kn+dTV 
P
X∗
Kn,P∗
X∗
=O
Pn−1/3(log n)2.
123
50 M. H. Neumann
(ii) We couple he o iginal and he boo s ap p ocess acco ding o (3.5a) and (3.5b)
and show i s ha

P(
X ,
X∗
)∈S×S
X −1=x,
X∗
−1=x∗
≥PX ∈SX −1=x·P∗X∗
∈SX∗
−1=x∗(4.3)
holds o all x,x∗∈N0.Le x,x∗∈N0be a bi a y. To simpli y no a ion we se ,
o a gene ic se B⊆N0,π(B)=y∈Bπ(x,y),π∗(B)=y∈Bπ∗(x∗,y),
and π∧π∗(B)=y∈Bπ(x,y)∧π∗(x∗,y).
I π∧π∗(S)≥π(S)·π∗(S), hen (4.3) ollows immedia ely. Suppose now he
opposi e, π∧π∗(S)<π(S)·π∗(S). Then δx,x∗>0, and i ollows om (3.5a)
and (3.5b)

P(
X ,
X∗
)∈S×S
X −1=x,
X∗
−1=x∗
=
y∈S
π(x,y)∧π∗(x∗,y),
+
y,y∗∈Sπ(x,y)−π(x,y)∧π∗(x∗,y)π∗(x∗,y∗)−π(x,y∗)∧π∗(x∗,y∗)
δx,x∗
=π∧π∗(S)+1
δx,x∗π(S)−π∧π∗(S)π∗(S)−π∧π∗(S).
=π(S)·π∗(S)+1
δx,x∗#δx,x∗π∧π∗(S)−π(S)π∗(S)
+π(S)−π∧π∗(S)π∗(S)−π∧π∗(S)$.
Since δx,x∗=1−π∧π∗(N0)=π(S)−π∧π∗(S)+π(Sc)−π∧π∗(Sc)
he e m in cu ly b aces is equal o
π(S)−π∧π∗(S)π∧π∗(S)−π(S)π∗(S)
+π(Sc)−π∧π∗(Sc)π∧π∗(S)−π(S)π∗(S)
+π(S)−π∧π∗(S)π∗(S)−π∧π∗(S)
=π(S)−π∧π∗(S)π∗(S)π(Sc)
+π(Sc)−π∧π∗(Sc)π∧π∗(S)−π(S)π∗(S)
=π(Sc)π∧π∗(S)−π∧π∗(S)π∗(S)
+π∧π∗(Sc)π(S)π∗(S)−π∧π∗(S),
and is he e o e non-nega i e. This p o es (4.3).
I ollows om (3.4) ha , o y,y∗∈Ssuch ha PX({y∗})>0,

P
X +1=
X∗
+1=z
X =y,
X∗
=y∗=π(y,z)∧π∗(y∗,z)
≥κQ{z}+OPn−1/3log n.(4.4)
123

Es ima ion and boo s ap o s ochas ically mono one… 51
We ob ain om (4.3) and (4.4) he e exis some κ∗>0 such ha

P(
X +1,
X∗
+1)=(z,z)
X −1=x,
X∗
−1=x∗
≥
y,y∗∈S
P(
X +1,
X∗
+1)=(z,z)
X =y,
X∗
=y∗

P(
X ,
X∗
)=(y,y∗)
X −1=x,
X∗
−1=x∗
≥κ∗(4.5)
holds wi h a p obabili y ending o 1. Hence, wi h a p obabili y ending o 1, he
coupled p ocess is φ-mixing wi h geome ically decaying coe icien s.
(iii) Acco ding o (4.5), he coupled p ocess (
X ,
X∗
) ∈N0sa is ies Doeblin’s con-
di ion which implies in pa icula ha his p ocess has a unique s a iona y
dis ibu ion. Le (
X0
,
X∗,0
) ∈N0be a s a iona y e sion o he coupled p o-
cess. Since (
X ,
X∗
) ∈N0is geome ically e godic we ob ain

P
X0
= 
X∗,0
≤
P
XKn= 
X∗
Kn+dTV
P(
X0
Kn,
X∗,0
Kn),
P(
XKn,
X∗
Kn)
=O
Pn−1/3(log n)2.

4.2 Some auxilia y lemmas
Lemma 4.1 Suppose ha (X ) ∈N0is a Ma ko chain wi h s a e space D ⊆Rsuch
ha (A2) is ul illed. Fo a bi a y I ⊆D, le
η := 1(X ≤z)−P(X ≤z|X −1)1(X −1∈I),
whe e I ⊆D. Then, o a bi a y γ<1,
E⎡
⎣ n

=1
η "4⎤
⎦=O(np
I)2+np
γ
I,
whe e pI:= P(X0∈I).
P oo In iew o En
=1η 4=n
s, ,u, =1E[ηsη ηuη ]we i s conside he
e ms E[ηsη ηuη ]. Le he indices be ch onologically o de ed, i.e. 1 ≤s≤ ≤u≤
≤n. Then i ollows om he Ma ko p ope y ha
Eηsη ηuη =0i u< .
Conside ing he emaining cases o s≤ ≤u= , we make use o he ollowing
equali ies.
123
52 M. H. Neumann
(a) s= =u=
Then E[ηsη ηuη ]=Eη4
s.
(b) s= <u=
Then E[ηsη ηuη ]=co (η2
s,η
2
u)+Eη2
sEη2
u.
(c) s< ≤u=
Then E[ηsη ηuη ]=co (ηs,η
η2
u)=co (ηsη ,η
2
u).
Fo s<u, he e exis 4
2=6 quad upels ( 1, 2, 3, 4)such ha i= j=s o
some i= j, and k= l=u o some k= l.Fo s< <u, he e exis 4 ·3=12
quad upels ( 1, 2, 3, 4)such ha i=s, j= and k= l=u o some i,j,k,l,
k= l. Finally, o s< =u, he e exis 4 quad upels ( 1, 2, 3, 4)such ha i=s
o some iand j=u o j= i. The e o e we ob ain
E⎡
⎣ n

=1
η "4⎤
⎦≤
n

=1
Eη4
+6
1≤s<u≤n
Eη2
sEη2
u
+12
n−1

=1
(s, ,u)∈T(1)
n, co (ηs,η
η2
u)
+12
n−1

=1
(s, ,u)∈T(2)
n, co (ηsη ,η
2
u),(4.6)
whe e
T(1)
n, =(s, ,u):1≤s< ≤u≤n, := −s≥u− 
T(2)
n, =(s, ,u):1≤s≤ <u≤n, := u− > −s.
To es ima e he las wo e ms on he igh -hand side o (4.6)weuseawell-known
co a iance inequali y o α-mixing andom a iables,
co (X,Y)≤4α(σ(X), σ (Y))1−1/α−1/β XαYβ,
whe e α, β ∈(1,∞)a e such ha 1/α +1/β < 1 and Xα<∞,Yβ<∞;
see e.g. B adley (2007a, Co olla y 10.16). Choosing α=β=2/γ and aking in o
accoun ha |ηs|≤1 and E|ηs|≤pIwe ob ain ha
co (ηs,η
η2
u)≤αX( −s−1)1−γ/2−γ/2ηs2/γ η η2
u2/γ
≤αX( −s−1)1−γpγ
I
as well as
co (ηsη ,η
2
u)≤αX(u− −1)1−γpγ
I.
123
Es ima ion and boo s ap o s ochas ically mono one… 53
Using #T(1)
n, ≤n( +1)and #T(2)
n, ≤n we ob ain om (4.6)
E⎡
⎣ n

=1
η "4⎤
⎦≤np
I+6(np
I)2
+12 n
n−1

=1
(2 +1)αX( −1)1−γpγ
I
=O(np
I)2+np
γ
I,
which comple es he p oo . 
Lemma 4.2 Suppose ha (X ) ∈N0is a Ma ko chain wi h s a e space D ⊆Rand
s a iona y dis ibu ion PXsuch ha (A2) is ul illed. Fo a bi a y I ⊆D,le Nn(I):=
#{ ≤n:X −1∈I}. Then, o a bi a y δ>0,κ<∞, and PX(I)≥nδ−1,
P|Nn(I)−nP
X(I)|>nP
X(I)/2=On−κ.
P oo Le q∈2Nand >0. Since
∞

=1
q−2[αX( )]/(q+) <∞
i ollows om an ex ension o Rosen hal’s inequali y (see e.g. Theo em 2 in Sec-
ion 1.4.1 in Doukhan (1994)) ha
E)Nn(I)−nP
X(I)q=E
n

=11(X −1∈I)−PX(I)q
≤Cqnq/2PX(I)q/(2+) +nP
X(I)q/(q+).
(4.7)
Choosing >0 small enough we ha e ha nP
X(I)2−2/(2+) ≥nδ o some δ>0.
The e o e we ob ain om Ma ko ’s inequali y ha
P|Nn(I)−nP
X(I)|>nP
X(I)/2
≤Cq
nq/2PX(I)q/(2+) +nP
X(I)q/(q+)
(n¶X(I)/2)q
=OnP
X(I)1−1/(2+)−q/2+n(nP
X(I))−q=On−κ,
i qis chosen su icien ly la ge. 
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54 M. H. Neumann
Lemma 4.3 Suppose ha (X ) ∈N0is a Ma ko chain wi h s a e space D ⊆Rand
s a iona y dis ibu ion PXsuch ha (A2) is ul illed. Then he e exis some C <∞
such ha , o a bi a y x ≤x wi h PX([x,x])≥n−1/3,
Esup
: ≥xn
=11(X ≤z)−P(X ≤z|X −1)1(X −1∈[x, ])
#{ ≤n:X −1∈[x, ]}∨1≤Cn
−1/3
(4.8a)
and
Esup
u:u≤xn
=11(X ≤z)−P(X ≤z|X −1)1(X −1∈[u,x])
#{ ≤n:X −1∈[u,x]}∨1≤Cn
−1/3.
(4.8b)
P oo We p o e only (4.8a) since he p oo o (4.8b) is comple ely analogous. The
p oo is ca ied ou in wo s eps. Fi s we conside he echnically simple case whe e
he dis ibu ion unc ion FXis con inuous. This allows us o de ine a sui able dyadic
amily o in e als which leads o a eadily comp ehensible p oo . A e wa ds we
ex end he esul o he gene al case.
S ep 1 Suppose ha FXis con inuous. Fi s we p o e ha o a bi a y δ>0 and each
≥x he e exis s some C<∞such ha
Esup
x:x≤x≤ 
n

=11(X ≤z)−P(X ≤z|X −1)1(X −1∈[x,x])
≤C*nP
X([x, ])+nδ.(4.9)
To deal wi h he sup emum we de ine a sui able sys em o dyadic in e als. Le Jn∈N
be such ha nδ−1/2<2−JnPX([x, ])≤nδ−1.Fo j=1,2,...,Jnand k=
1,2,...,2j,wese
xj,k=F−1
XFX(x)+k2−jPX([x, ])(4.10a)
and, o j=1,...,Jn,
Bj,k=[x,xj,1]i k=1,
(xj,k−1,xj,k]i k=2,...,2j.(4.10b)
We ha e ha
PXBj,k=2−jPX([x, ]). (4.11)
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