R E S E A R C H
Open Access
Fixed point theorems and explicit
estimates for convergence rates of
continuous time Markov chains
Zhenhai Yan
1,2, Guojun Yan
3and Ikudol Miyamoto
4**Correspondence:
4Graduate School of Mathematics,
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan
Full list of author information is available at the end of the article
Abstract
In this paper we give Banach fixed point theorems and explicit estimates on the rates of convergence of the transition function to the stationary distribution for a class of exponential ergodic Markov chains. Our results are different from earlier estimates using coupling theory, and from estimates using stochastically monotone one. Our estimates show a noticeable improvement on existing results if Markov chains contain instantaneous states or nonconservative states. The proof uses existing results of discrete time Markov chains together withh-skeleton. At last, we apply this result, Ray-Knight compactification and Itô excursion theory to two examples: a class of singular Markov chains and Kolmogorov matrix.
Keywords: exponential ergodicity; Markov chain; fixed point theorem; Poisson point process
1 Introduction
Throughout this paper, unless otherwise specified, let{Xt;t∈[,∞)}be a time homo-geneous, continuous time Markov chain with an honest and standard transition function
pij(t) on a state spaceE={, , , . . .}, and its density matrix isQ= (qij),qi= –qii. LetPxand
Exdenote the probability law and expectation of the Markov chain respectively under the initial condition ofX=x, wherex∈E. LetX= (Ω,F,Ft,Xt,θt,Px) be the right process associated withpij(t).
In this paper we consider the Markov chain which is an exponential ergodicity, that means, there is a unique stationary distribution π= (πj) (j∈E), constantsRi<∞and α> such that
j
pij(t) –πi≤Rie–αt
for alli,j∈E. Our goal is to find out the computable bounds of the constantsRiandα, especiallyα.
There has been considerable recent work on the problem of computable bounds for convergence rates of Markov chains. Recently, the authors (see [–]) gave the bounds of convergence rates for Markov chains. Their main methods are based on renewal theory
©2015 Yan et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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and coupling theory. And in [–], the authors gave the convergence rates of stochasti-cally monotone Markov chains. Their results and methods have the advantages of being applicable to some Markov chains or processes.
However, their methods are not fitted for the general continuous time Markov chains, especially when the symmetric condition, coupling condition or stochastically monotone one is not satisfied. For example, the bounds of Markov chains with instantaneous states such as Kolmogorov matrix, or the regular birth and death process. In this paper, we dis-cuss this problem.
Leti∈Eand suppose thatX=i, define
T=
inf{t> |Xt=i} if this set is not empty,
+∞, otherwise
to be the sojourn time in statei. Define
τj+=
inf{t>T|Xt=j} if this set is not empty,
+∞, otherwise.
Our central result is the following theorem.
Theorem Suppose that pij(t)is an irreducible and ergodic transition function with
sta-tionary distribution{πj,j∈E}and m∈E is a stable state.If there is a positive constantλ
such thatλ<infm∈Eqmand Ei{eλτ
+
m}<∞for all i∈E,then we know that pij(t)is exponen-tially ergodic.Moreover,if
α< λ
λ+ (qm–λ)(Em{eλτ
+
m}– ),
then there exists Ri<∞for some(and then for all)i such that
j
pij(t) –πj≤Rie–αt.
In this paper we shall first develop the methods in [] to the continuous time situation, which leads to considerable improvements of convergence rates. And this result shall be in a wider range of application than existing results in [–]. Next we shall give some fundamental lemmas and the proof of the main theorem in this paper. Finally, we shall apply our result and the Itô excursion theorem to compute two examples in Section , which will show the advantages of our result.
2 Proof of Theorem 1
2.1 Definitions and some fundamental lemmas
Let{Yn}∞n=be a time homogeneous Markov chain with one-step transition matrixΠ= (Πij) on the state spaceE. Suppose that{Yn}∞n=is an aperiodic, irreducible ergodic Markov chain with a transition functionΠijand stationary distributionπj(j∈E). LetΠ= (Πij(n)) be ann-step transition matrix andη+i =inf{n|n≥,Yn=i}for alli∈E.
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Definition We say that{Yn}∞n=isρ-geometrically ergodic (for short, geometrically er-godic) if there exits a numberρwith <ρ< such that
Πij(n) –πj<Cijρn ()
for anyn∈Nandi,j∈E, whereρis called ergodic index.
Lemma SupposeΠijandπjare defined as above,m∈E is a fixed state,a< ,b> ,there
is a function V(x)≥on E such that
ij
ΠijV(j) =aV(i) +bI{m}(i) ()
(called drift inequality).IfΠmm>δ> ,then we have
j
Πij(n) –πj≤ ρ
ρ– ( –M–)V(i)ρ
n ()
for >ρ> –M–,where
M= ( –a)
–a+b+b+ – δ
δ
b
–a
( –a)b+b . ()
Proof From () together with Theorems . and . in [], we can get the proof of
Lemma .
Definition Given a numberh> , the discrete time Markov chain{Xnh}∞n=having a one-step transition functionpij(h) (and therefore ann-step transition functionpij(nh)) is called theh-skeleton of{Xt,t≥}.
Lemma Suppose that pij(t)is an irreducible and ergodic transition function,m∈E is a
fixed state;for a constantλ( <λ<qm),we have Em{eλτ
+
m}<∞.Let
ηhm=inf{nh|n≥,Xnh=m}
for all h> .If( –e(λ–qm)h)Em{eλτm+}< ,then we know that Eieληhm≤ e
(λ–qm)hEi{eλτm+}
– ( –e(λ–qm)h)Em{eλτm+} (i=m), () Emeληhm≤ e
(λ–qm)h
– ( –e(λ–qm)h)Em{eλτm+}. ()
Proof It is obvious thatmis not an absorbing state, otherwisepij(t) is reducible. Suppose thatEi{eλτm+}<∞. Let
τ=inf{t|Xt=m}, γ=inf{t|t>τ,Xt=m},
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τk+=inf{t|t>γk,Xt=m} and
γk+=inf{t|t>τk+,Xt=m},
wherek= , , . . . andmis recurrent. Then the stopping times mentioned above are almost surely finite and
τ<γ<τ<γ<· · ·.
From the strong Markov property ofX, it is easily known thatγ–τ,γ–τ,γ–τ, . . . are independent identically distributed exponential random variables with meanqm. So we havePi{γ
k–τk≤h,∀k}= .
We can easily getηih≤τ+hon{γ–τ>h}andηmh ≤τk++hon
{γk+–τk+>h,γn–τn≤h,∀n≤k}. Ifi=m, then we have
Eieληhm=Eieληhm;γ
–τ>h
+ ∞
k=
Eieληhm;γ
k+–τk+>h,γn–τn≤h,∀n≤k
≤eλhEieτm+;T
◦θτm+ >h
+eλh
∞
k=
Eieλτk+;γ
k+–τk+>h,γn–τn≤h,∀n≤k . () Ifi=mandτ= , then we have
Eieληhm=Eieληhm;γ
–τ>h
+ ∞
k=
Eieληhm;γ
k+–τk+>h,γn–τn≤h,∀n≤k
≤eλhPi{T >h} +eλh
∞
k=
Eieλτk+;γ
k+–τk+>h,γn–τn≤h,∀n≤k . () Ifi=m, then we have, for eachk≥,
Eieλτk+;γ
k+–τk+>h,γn–τn≤h,∀n≤k =EiEieλτk+;γ
k+–τk+>h,γn–τn≤h,∀n≤k|Fτk+
=e–qmhEieλτk+;γ
n–τn≤h,∀n≤k =e–qmhEiEieλτk+;γ
n–τn≤h,∀n≤k|Fτk
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=e–qmhEiEieλ(τm+◦θτk+τk);T
◦θτk≤h, . . .|Fτk
=e–qmhEieλτkEmeλτm+;T
≤h ;γn–τn≤h,∀n≤k–
=e–qmhEmeλτm+
;T≤hEi
eλτk;γ
n–τn≤h,∀n≤k–
=· · ·
=e–qmhEmeλτm+;T
≤h kEi
eλτ
=e–qmhEmeλτm+;T
≤h k
Eieλτm+
and
Emeλτm+;T
>h =Em
Emeλ(τi+◦θh+h);T
>h|Fh =EmeλhEmeλτi+;T>h
=e(λ–qm)hEieλτi+.
So
Emeλτm+;T
≤h =Em
eλτi+ –Emeλτm+;T
>h =
–e(λ–qm)hEmeλτm+. ()
By () and the equations above we have
Eieλτk+;γ
k+–τk+>h,γn–τn≤h,∀n≤k =e–qmh –e(λ–qm)hkEmeλτm+ kEieλτm+
by (). If
–e(λ–qm)hEmeλτm+< ,
then we have
Eieληhm≤ e
(λ–qm)hEi{eλτm+}
– ( –e(λ–qm)h)Em{eλτm+}.
And by () we have
Emeληhm≤ e
(λ–qm)h
– ( –e(λ–qm)h)Em{eλτm+}.
So Lemma is proved.
Remark From () we get that whenh↓ andi=m,
Eieληhm= + (q
m–λ)
Emeλτm+– h+Oh Eieλτm+. ()
From () we see that whenh↓,
Emeληhm– = (q
m–λ)
Emeλτm+– h+Oh. ()
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Proposition (see [], p.) Let{Gk,k= , , . . .}be at most countable collection of
un-bounded open subsets of(,∞).Then,in any nonempty open subinterval I of(,∞),there exits a number h with the property that for each k,nh∈Gkfor infinitely many integers n.
2.2 Proof of Theorem 1
() For eachh> such that ( –e(λ–qm)h)Em{eλτm+}< , we writeph
ij(n) for the transition function ofh-skeleton{Xnh}∞n=. Consider
Vh(i) =
Ei{eληhm} ifi=m,
ifi=m
and
ah=e–λh, bh=e–λh
Emeληhm– .
For anyi=m, by () we have
eλh
j∈E
phijVh(j) =eλh
j=m
phijEjeληhm+eλhph
im
=Eieλ(ηhm◦θh+h);X
h=m +Ei
eλh;Xh=m =Eieληhm=V
h(i).
Similarly we can get
eλh
j∈E
phmjVh(j) =eλh
j=m
pmjh Ejeληmh+eλhph
mm
=Emeλ(ηhm◦θh+h);X
h=m +Em
eλh;Xh=m =Emeληhm= +Emeληhm– .
By () we have
j∈E
phijVh(j) =ahVh(i) +bhI{m}(i)
for anyi∈E. By () we know thatph
ij(n) satisfies the drift inequality. Letδh=e–qmh, obviously we havephmm>δh. Let
Mh= ( –ah)
–ah+bh+bh+
– δ h δh
bh –ah
( –ah)bh+bh .
By () and () we obtain
j∈E
pij(nh) –πj≤ ρ
ρ– ( –Mh–)Vh(i)ρ
n ()
for any >ρ> –M–h from Lemma .
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() From () we have
Mh= –ah
+(qm–λ)(E
m{eλτm+}– )
λ +O(h)
,
which gives
M–h = λ
λ+ (qm–λ)(Em{eλτ
+
m}– )h+O
h.
Hence, for any
α< λ
λ+ (qm–λ)(Em{eλτ
+
m}– ),
there existε> and <h<εsuch thate–αh> –M–
h . By () we get
j∈E
pij(nh) –πj≤
e–αh
e–αh– ( –M– h )
Vh(i)e–αnh. ()
() For eachi∈E, let βi=inf
βeβt j∈E
pij(t) –πjis bounded on (,∞)
.
We have
fil(t) =e(βi+l
–)t
j∈E
pij(t) –πj
for anyl> , which is a continuous and unbounded function on (,∞). Then we know that
Gil=
t|fil(t) >
,
fori∈Eandl> , which is a class of nonempty and unbounded open sets on (,∞). From Proposition , for everyGil, there exists <h<εsuch that there are infinitely manynh belonging toGil, wheren= , , . . . .
Ifnh∈Gil, then by () we have
fil(nh) =e(βi+l
–)nh
j∈E
pij(nh) –πj≤
e–αh
e–αh– ( –M– h )
Vh(i)e(β+l
––α)nh
,
which givesβi+l––α≥.
By the arbitrariness oflandα, we get
βi≥
λ λ+ (qm–λ)(Em{eλτ
+
m}– ).
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From the definition ofβi, it is easy to know that for anyα> ,
α< λ
λ+ (qm–λ)(Em{eλτ
+
m}– )
andi∈E, there existsRi> such that
j∈E
pij(t) –πj≤Rie–αt.
Definition Let
α∗=supα|for alli,j∈E,∃Rij> s.t.pij(t) –πj≤Rijeαt
.
The constantα∗is called the maximal exponentially ergodic constant of a transition func-tionpij(t).
Remark Ifpij(t) is irreducible,mis a stable state andλ> which makeEm{eλτ
+
m}<∞,
then we know thatpij(t) is still ergodic and the result in Theorem remains valid from the proof of Theorem .
Remark From Theorem and Definition we know
α∗≥ λ
λ+ (qm–λ)(Em{eλτ
+
m}– ). 3 Two examples
In this section we compute the maximal exponentially ergodic constants for two types of chains: a kind of singular Markov chain in which all states are not conservative and Kolmogorov matrix in which state is an instantaneous state.
3.1 A kind of singular Markov chain
SupposeE={, , . . .}and
Q=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
–q · · · –q · · · –q · · · –q · · ·
..
. ... ... ... · · ·
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, ()
where <q<q<· · ·<qn<· · ·<∞andinfiq–i = fori= , , . . . . In this case the tran-sition function withQ-matrix above is not unique (see [, ]), but the honest transition functionpij(t) withQ-matrix is unique and its resolvent is
Rij(λ) =Rminij (λ) +Ei
e–λσ
k∈EakRminkj (λ) λk∈Em∈EakRminkm(λ)
, ()
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whereak(k∈E) are sequences of nonnegative real numbers such that
k∈E
ak=∞ and
k∈E
m∈E
akRminkm() = ,
where Rmin
ij (λ) is the resolvent of the minimal transition functionpminij (t). From [] it is known that this chain is not symmetric, so we cannot discuss its ergodicity with coupling theory. We also cannot adapt existing results to this chain. The following are our main methods and result.
.. Ray-Knight compactification
Theorem For the Markov chain with Q-matrix(),the Ray-Knight compactification of the state space E is E=E∪ {∞},X= (Ω,F,Ft,Xt,θt,Px)is the right processes with the
transition function pij(t).
Proof Consider
σ=inft|t> ,∀ε> , there are infinite jumps ofXin (t–ε,t+ε). Then we have
Rminij (λ) =Ei
∞
e–λtI {j}(Xt)dt
=δijEi
T
e–λtdt
= δij λ+qi
and
Eie–λσ=Eie–λT
= qi λ+qi
.
Then by () we have
Rij(λ) = δij
λ+qi + qi
λ+qi aj
λ+qj
λ∞k= ak
λ+qk
,
which gives
lim
i→∞Rij(λ) = aj
λ+qj
λ∞k= ak
λ+qk
.
So we can get that the Ray-Knight compactification based onpij(t) isE=E∪ {∞}and
R∞j(λ) = aj
λ+qj
λ∞k= ak
λ+qk
.
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After the Ray-Knight compactification, the Markov chainX= (Ω,F,Ft,Xt,θt,Px) is the
right process with the transition functionpij(t).
Remark This chain also holds the strong Markov property.
.. Excursion leaving from state∞
For eachi∈E, let σi=inf{t≥|Xt=i}
be the hitting time of statei. ByT,σandσidefined above, we havePi[σ<∞] = (i∈E). Consider excursion leaving from state∞ofX, and letϕ(x) =Ex[e–σ∞] forx∈E∪ {∞}; it is easily verified thatϕ(·) is a -excessive function ofX.
And then we have the following result.
Theorem There exists a continuous additive function Ltof X such that
() Ex{∞
e–sdLs}=ϕ(x)for anyx∈E∪ {∞};
() suppdL={t|Xt=∞};
() L∞=∞. Let
U=w(·)|w(·)∈DE¯[,∞),∃s> ,w(s)∈E,w(·)≡ ∞on
η∞(w),∞,
whereη∞(w) =inf{t|t> ,w(t) =∞}. We writeU for Boolean algebra onU,{Wt}t≥for coordinate process,{Ut}t>for natural filtration andθt for shift operator. (U,U) is called the excursion space.
For anyt≥, defineβt=inf{s|Ls>t},{βt}is the right reverse of local timeLt. LetDp(ω) =
{t|βt–<βt}. We have known that between excursion leaving from state∞ofXandDp(ω) is one-to-one correspondence (see [, ]).
Theorem For any t∈Dp,let
Yt(ω) =
Xβt–+s(ω), if≤s<βt–βt–,
∞, if s≥βt–βt–,
then{Yt;t∈Dp}is the Poisson point process on the excursion space(U,U) (see[]),and
the characteristic measurePˆ(·)satisfies
ˆ
P{Wt=i, . . . ,Wtn=in}
=
k∈E
akpminki(t)· · ·p
min
in–in(tn–tn–).
Remark This means that the characteristic measurePˆ(·) has the same distribution as
k∈EakPk{·}.
Remark
() The state space of the right processXisE∪ {∞}, where∞is the branching point;
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() The local timeLtofXonS∞is continuous;
() The excursion measurePˆ(·)isσ-finite and satisfies
ˆ
P{W∈/E}
= and Pˆ{W=k}
=ak
for allk∈E.
.. Maximal exponentially ergodic constant
The following theorem gives the stationary distribution.
Theorem The transition function pij(t)defined above is ergodic,and its stationary
dis-tribution is
πi=
aiq–i
∞ k=akq–k
, ∀i∈E.
Proof () According toRmin ij (λ) =
δij
λ+qiand
=
k∈E
ak
m∈E
Rminkm() = k∈E
ak +qk
,
we have∞k=akq–k < +∞. () The resolvent ofpij(t) is
Rij(λ) = δij λ+qi
+ qi λ+qi
aj
λ+qj
λ∞k= ak
λ+qk
,
which gives
πi=lim
λ→λRii(λ) =
aiq–i
∞ k=akq–k
<∞.
Then we complete the proof.
In the following we discuss the conditions of exponential ergodicity and convergence rate of exponential ergodicity.
Theorem If
<λ<∞ a
k=ak(qk–q)–∧
q, ()
then for some(and then for all)i∈E,Ei[eλσ] <∞,p
ij(t)is exponentially ergodic.Moreover,
for thisλ,if
α< λ
λ+ (q–λ)(E{eλτ
+ }– ),
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where
Eeλτ+= a
q–λ
a
a–
∞ k=
akλ
qk–λ
,
then there exists Ri<∞for any i∈E such that
j∈E
pij(t) –πj≤Rie–αt.
Proof () For anyi∈E(i= ),
Eieλσ =Eieλ(T+σ◦θT) = a
qi–λ
E∞eλσ .
In the following we computeE∞[eλσ].
Consider the coordinate processW(s) on the excursion space (U,U). For anyi∈E, define
ηi=inf
s|W(s) =i (i∈E), η∞=inf{s|Ws=∞}. Whens>η∞,W(·) equal to∞, soηi>η∞equal toηi=∞. Let
C={W|W= }, C={W|W= }.
ObviouslyC,C∈U andC∪C=U. Letτ=inf{t|βt>σ}(t> ) and
Zt={s|s∈Dp,s≤t,Ys∈C},
whereAdenotes the cardinal ofA, then we have
P∞{τ >t}=P∞Zt= =e–at,
which gives thatτ is exponential random variable with meana. And hence we have
E∞eλσ =E∞
exp
λ
s∈G
I(,σ](s)σ∞I{C}(Ys)
◦θs
=E∞exp[λβτ]
=
∞
E∞eλβta
e–atdt.
From the computation of the Poisson point process, we know that
E∞eλβt=E∞eλzt
=exptPˆeλσ∞– I{C}(Ys)
,
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which gives
ˆ
Peλσ∞– I{C}(Ys)
= ∞
k=
akEk
eλσ– = ∞
k=
akλ
qk–λ .
Hence
E∞eλσ =
∞
aexpt
∞
k=
akλ
qk–λ –a
dt
= a
a–
∞ k=
akλ
qk–λ
.
Therefore, if () is satisfied, then we haveE∞[eλσ] <∞, which givesEi[eλσ] <∞. From
[], Lemma ., we know thatpij(t) is exponentially ergodic. () Ifi= , thenτ+=σ. Ifλsatisfies (), then we have
Eieλτ+=Eieλσ <∞
for anyi∈Efrom Theorems and .
Ifm= in Theorem , by the method of () above, we have
Eeλτ+=Eeλ(T+σ◦T)
=EeλT E∞eλσ
= a
q–λ
a
a–
∞ k=
akλ
qk–λ
.
We complete the proof of this theorem.
Remark Obviously, the maximal exponentially ergodic constant satisfies
α∗≥ λ
λ+ (q–λ)(E{eλτ
+ }– )
.
3.2 Kolmogorov matrix
This following example contains an instantaneous state.
Suppose thatq,q, . . . are sequences of positive real numbers and considerQ-matrix as follows: Q= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ –∞ · · ·
q –q · · ·
q –q · · ·
q –q · · · .. . ... ... ... · · · ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,
where∞i=qi–<∞. This matrix is called the Kolmogorov matrix. There are infinitely many dishonest processes with thisQ-matrix. The authors (see [, , ]) have shown that the process with the following resolvents is the only honest one.
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Let
R(λ) = λ
+ ∞
k= λ+qk
– ,
Rj(λ) =R(λ)· λ+qj
(j≥),
Ri(λ) =
qi λ+qi·
R(λ) (i≥) and
Rij(λ) =
qi λ+qi
·R(λ)· λ+qj
+ δij λ+qj
(i,j≥), whereλ> and the state space isE={, , , . . .}.
Obviously, the transition functionpij(t) which corresponds with the resolvents above is the only honest one. Though this chain is weakly symmetric, its convergence rate is still unknown because of its instantaneous state.
.. Ray-Knight compactification
Theorem For the Markov chain with Q-matrix above,the Ray-Knight compactification of the state space E is still E,X= (Ω,F,Ft,Xt,θt,Px)is the right process with the transition
function pij(t).
Proof It is obvious that
lim
i→∞Rij(λ) =Rj(λ).
By using the methods in [], we show thatE=E. In the Ray-Knight topology, instanta-neous state is the limit point of sequences{, , . . .}. So we know that the Markov chain
X= (Ω,F,Ft,Xt,θt,Px) is the right process with the transition functionpij(t) (see [,
]).
Remark This chain holds the strong Markov property.
.. Excursion leaving from state
For eachi∈E, the definition ofT,σ,σias above. Then obviously for eachi∈E, we have
Pi[σ<∞] = , which means that instantaneous state is a recurrent state ofX. Consider excursion leaving from state ofX, and let
ϕ(x) =Exe–σ
for allx∈E; it is easily verified thatϕ(·) is a -excessive function ofX. And then we have the following result.
Theorem There exists a continuous additive function Ltof X such that
() Ex{∞
e–sdLs}=ϕ(x)for allx∈E;
RETRACTED
() suppdL={t|Xt= };
() L=∞. Let
U=w(·)|w(·)∈DE[,∞),∃s> ,w(s)∈E,w(·)≡ on
η(w),∞
,
whereη(w) =inf{t|t> ,w(t) = }. We writeUfor Boolean algebra onU,{Wt}t≥for co-ordinate process,{Ut}t>for natural filtration andθtfor shift operator. (U,U) is called the excursion space.
For anyt≥, letβt=inf{s|Ls>t},{βt}be the right reverse of local timeLt. LetDp(ω) =
{t|βt–<βt}. We have known that between excursion leaving from state ofXandDp(ω) is a one-to-one correspondence.
Theorem For any t∈Dp,define
Yt(ω) =
Xβt–+s(ω), if≤s<βt–βt–;
, if s≥βt–βt–,
then{Yt;t∈Dp}is the Poisson point process on the excursion space(U,U),and the
char-acteristic measurePˆ(·)satisfies
ˆ
P{Wt=i, . . . ,Wtn=in}
=
k∈E
pminki(t)· · ·piminn–in(tn–tn–).
Remark Theorem means that the characteristic measurePˆ(·) has the same distribu-tion ask∈EPk{·}.
Remark
() The state space of the right processXisE, whereis the branching point; () The local timeLtofXonSis continuous;
() The excursion measurePˆ(·)isσ-finite and satisfies
ˆ
P{W= }
= and Pˆ{W=k}
=
for allk∈E\{}.
.. Maximal exponentially ergodic constant
The following theorem will give the stationary distribution.
Theorem The transition function pij(t)defined above is ergodic,and we know that its
stationary distribution is
π=
+∞k=q–k and πj= q–j
+∞k=q–k
for all j∈E\{}.
RETRACTED
Proof () According to Theorem . in [], p., we have
πj=lim
λ→λRij(λ) =tlim→∞pij(t), which gives
π=lim
λ→λRi(λ) =λlim→λR(λ) = +∞k=q–
k <∞ and
πj=lim
λ→λRij(λ) =
q– j +∞k=q–
k <∞ for eachi≥.
Thus the proof is completed.
In the following we discuss the conditions of exponential ergodicity and the convergence rates of exponential ergodicity.
Theorem If
<λ<∞
k=(qk–q)–∧
q, ()
then for some(and then for all)i∈E,Ei[eλσ] <∞,so p
ij(t)is exponentially ergodic.
More-over,for thisλ,if
α< λ
λ+ (q–λ)(E{eλτ
+ }– ),
where
Eeλτ+=
q–λ –∞k= akλ
qk–λ
,
then there exists Ri<∞for any i∈E such that
j∈E
pij(t) –πj≤Rie–αt.
Proof () For anyi∈E(i≥), we know
Eieλσ=Eieλ(T+σ◦θT)=
qi–λ
Eeλσ.
Then we will computeE{eλσ}. Consider the coordinate processW(s) on the excursion
space (U,U). For anyi∈E, define ηi=inf
s|W(s) =i.
RETRACTED
Let
C={W|W= } and C={W|W= }, then we know thatC,C∈U andC∪C=U.
Letτ=inf{t|βt>σ}(t> ) and
Zt={s|s∈Dp,s≤t,Ys∈C}, we have
P{τ >t}=PˆZt = =e–P(Wˆ =)t=e–t,
which givesτis exponential random variable with mean and
Eeλσ=E
exp
λ
s∈G
I(,σ](s)σ∞I{C}(Ys)
◦θs
=Eexp[λβτ]
=
∞
Eeλβte–tdt.
From the computation of the Poisson point process, we have
Eeλβt=Eeλzt
=exptPˆeλσ– I
{C}(Ys)
,
which gives
ˆ
Peλσ– I
{C}(Ys)
= ∞
k=
Ekeλσ– =
∞
k= λ
qk–λ .
So
Eeλσ=
∞
expt ∞
k= λ
qk–λ –
dt
=
–∞k=qλ
k–λ
.
Therefore, if () is satisfied, then we get that E[eλσ] <∞ andEi{eλσ}<∞. From
Lemma . in [], p., we know thatpij(t) is exponentially ergodic. () Ifi≥, then we haveτ+=σ. Ifλsatisfies (), then we have
Eieλτ+=Eieλσ<∞
for anyi∈Efrom Theorem .
RETRACTED
Letm= in Theorem , by the method of () above, we have
Eeλτ+=Eeλ(T+σ◦T)
=EeλT Eeλσ
=
q–λ –∞k=qλ
k–λ
.
So the result is proved from Theorem .
Remark According to the results above, the maximal exponentially ergodic constant of this example satisfies
α∗≥ λ
λ+ (q–λ)(E{eλτ
+ }– )
.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally to this work and read and approved the final manuscript.
Author details
1School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China.2School of Statistics,
Henan University of Economics and Law, Zhengzhou, Henan 450046, China.3School of Mathematics and Statistics,
Zhengzhou University, Zhengzhou, Henan 450001, China.4Graduate School of Mathematics, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan.
Acknowledgements
This paper was written during a short stay of the corresponding author at the Graduate School of Mathematics of Nagoya University as a visiting professor. I would also like to thank the Graduate School of Mathematics and their members for their warm hospitality.
Received: 11 May 2015 Accepted: 14 October 2015
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