Random Continued fractions: L\'evy constant and Chernoff-type estimate

0.23 MB·

by  Lulu Fang

RANDOM CONTINUED FRACTIONS: LE´VY CONSTANT AND
CHERNOFF-TYPE ESTIMATE
6
1 LULUFANG1 MINWU1 NARN-RUEIHSHIEH2 ANDBINGLI1,∗
0
2
n Abstract. Given a stochastic process {An,n ≥ 1} taking values in natural
a numbers, the random continued fractions is defined as [A1,A2,···,An,···]
J analogue to the continued fraction expansion of real numbers. Assume that
0 {An,n ≥ 1} is ergodic and the expectation E(logA1)< ∞, we give a L´evy-
typemetrictheoremwhichcoversthatofrealcasepresentedbyL´evyin1929.
1
Moreover, acorrespondingChernoff-type estimateisobtained under thecon-

] ditions{An,n≥1}isψ-mixingandforeach0<t<1,E(At1)<∞.
T
N
.
h
1. Introduction
t
a
m Let T :[0,1)−→[0,1) be the Gauss transformation defined by
[ 1 − 1 if x∈(0,1)
Tx= x x

1 (0 (cid:4) (cid:5) if x=0,
v
5 where ⌊x⌋ denotes the greatest integer not exceeding x. Then every x ∈ [0,1) can
0 be written as the following recursive form
2
1
2 (1) x= ,
1
0 a (x)+
1
1. a (x)+...+ 1
0 2 an(x)+Tnx
6
where a (x) = ⌊1⌋ and a (x) = a (Tn−1x) for all n ≥ 2 are called the partial
1 1 x n 1
: quotients of x . If there exists n∈N such that Tnx=0, we say that the form (1)
v
is a finite continued fraction expansion of x denoted by [a (x),a (x),··· ,a (x)].
i 1 2 n
X Otherwise,theform(1)issaidtobeaninfinitecontinuedfractionexpansionofxde-
r noted by [a1(x),a2(x),··· ,an(x),···]. For more details about continued fractions,
a we refer the reader to [5],[7],[12],[13] and the references therein.
It is well-known that for every real number x ∈ [0,1), there corresponds a con-
tinued fractionexpansion. Moreover,the expansionis finite if x is rationalandthe
expansion is infinite if x is irrational (see [13], Theorem 14).
For any x∈[0,1) and n≥1, we define the truncated continued fraction
p (x)
n
:=[a (x),a (x),··· ,a (x)],
1 2 n
q (x)
n
where p (x) and q (x) are relatively prime integers. We say p (x)/q (x) the n-th
n n n n
convergentof the continued fraction expansion of x. Clearly these convergents are
Key words and phrases. Randomcontinuedfractions,L´evyconstant, Chernoff-typeestimate.
*Correspondingauthor.
2010AMSSubjectClassifications: 11K50,37A50,60G10.
1
2 LULUFANG1 MINWU1 NARN-RUEIHSHIEH2 ANDBINGLI1,∗
rationalnumbers and p (x)/q (x)→x as n→∞ for all x∈[0,1). More precisely,
n n
for any x∈[0,1) and n≥1,
1 p (x) 1
n
≤ x− ≤ .
2q2 (x) q (x) q2(x)
n+1 (cid:12) n (cid:12) n
(cid:12) (cid:12)
Thisistosaythespeedofpn(x)/qn(x(cid:12)(cid:12))approxim(cid:12)(cid:12)atingtoxisdominatedbyqn−2(x).
Sothedenominatorofthen-thconvergentq (x)playsanimportantroleintheprob-
n
lem of Diophantine approximation. Concerning pointwise asymptotic behaviour of
the sequence {q (x),n≥1}, L´evy [18] obtained the following theorem.
n
Theorem 1 ([18]). For L-almost every x∈[0,1),
1 π2
(2) lim logq (x)= ,
n
n→∞n 12log2
where L denotes the Lebesgue measure.
If lim 1 logq (x) exists, such limit is called the L´evy constant of x, which has
n→∞n n
been studied by many mathematicians, see [8], [10], [36] and [37]. Some other
limit theorems for {q (x),n≥1} in continued fraction expansions of real numbers
n
have been extensively investigated. For instance, the central limit theorem for
{q (x),n ≥ 1} was given in [27] and the corresponding rate of convergence was
n
studied by G. Miseviˇcius [23], the law of the iterated logarithm for {q (x),n ≥ 1}
n
was provided by [11], [33].
Notice that 1 logq (x) converges to π2 in probability, a natural question is
n n 12log2
whattherateofconvergencein(2)is. Weshowthatsuchrateisatmostexponential
by the following theorem which is a Chernoff-type estimate for 1 logq (x).
n n
Theorem 2. For any δ > 0, there exist N > 0, B > 0, α > 0 such that for all
n≥N,
1 π2
L logq − ≥δ ≤Bexp(−αn).
n
n 12log2
(cid:18)(cid:12) (cid:12) (cid:19)
(cid:12) (cid:12)
Furthermore,wewil(cid:12)lproveamoregener(cid:12)alversionofTheorem2underthesetting
(cid:12) (cid:12)
of so-called random continued fractions (see Theorem 4 below). Now we are ready
to state this setting. Let (Ω,F,P) be a probability space and {A ,n ≥ 1} be
n
a stochastic process defined on (Ω,F,P), taking values in the measurable space
({1,2,··· ,n,···},C),where C is the powersetof{1,2,··· ,n,···}. For anyω ∈Ω,
we define
(3)
1
X(ω):=[A (ω),A (ω),··· ,A (ω),···]= .
1 2 n 1
A (ω)+
1
A (ω)+...+ 1
2
A (ω)+...
n
Justasforinfiniteseries,thequestionnaturallyarisesastowhethertheright-hand
sideof(3)formallydefinedisconvergent. Fortunately,theconvergenceoftheright-
hand side of (3) is assured by Theorem 10 in [13] and Proposition 3 in Section 2
guarantees that X is a random variable taking values in the unit interval (0,1).
We saythatX is arandomcontinuedfractionsgeneratedby the stochasticprocess
{A ,n≥1}. Different models of randomcontinued fractions have been considered
n
in literature for instance [17], [19], [20],[32] and the references therein.
LE´VY CONSTANT AND CHERNOFF-TYPE ESTIMATE 3
For any ω ∈Ω and n∈N , the two quantities P and Q are defined as
n n
P (ω) 1
n
:=[A (ω),A (ω),···A (ω)]= ,
Q (ω) 1 2 n 1
n A (ω)+
1
A (ω)+...+ 1
2
A (ω)
n
where P (ω) and Q (ω) are relatively prime integers. We say that P /Q is the
n n n n
n-thconvergentofX. Moreover,if the limit lim 1 logQ exists,suchlimit is said
n→∞n n
to be the L´evy constant of X.
In this paper, we study a limit theorem for {Q ,n ≥ 1} and a corresponding
n
Chernoff-type estimate. We use the notation E(ξ) to denote the expectation of a
randomvariable ξ. The definitions of ergodic and ψ-mixing stochastic process and
somerelatedonesaswellwillbe giveninSection2. Firstlyweobtainthe following
L´evy-typemetrictheoremwhichstatesthattheL´evyconstantofrandomcontinued
fractions exists and equals a constant almost surely (a.s.).
Theorem 3. Let {A ,n ≥ 1} be a stochastic process taking values in natural
n
numbers. If {A ,n≥1} is ergodic and E(logA )<∞, then
n 1
1
(4) lim logQ =− logXdP, P −a.s.
n
n→∞n
ZΩ
Theconvergencein(4)impliesthat{1 logQ ,n≥1}convergesto− logXdP
n n Ω
in probability, that is, for any δ >0,
R
1
(5) lim P logQ + logXdP ≥δ =0.
n
n→∞ (cid:18)(cid:12)n ZΩ (cid:12) (cid:19)
(cid:12) (cid:12)
A natural question is what (cid:12)the rate of convergence(cid:12)in (5) is. We prove that such
(cid:12) (cid:12)
rate is at most exponential by the following theorem which is a type of Chernoff
estimate on the convergence of {1 logQ ,n≥1}.
n n
Theorem 4. Let {A ,n ≥ 1} be a stochastic process taking values in natural
n
numbers. If {A ,n ≥1} is ψ-mixing and E(At)<∞ for each 0<t <1, then for
n 1
any δ >0, there exist N >0, B >0, α>0 such that for all n≥N, we have
1
P logQ + logXdP ≥δ ≤Bexp(−αn).
n
n
(cid:18)(cid:12) ZΩ (cid:12) (cid:19)
(cid:12) (cid:12)
Now we turn to s(cid:12)howing the continued fr(cid:12)action expansion of real numbers can
(cid:12) (cid:12)
be regarded as a special case of random continued fractions. Let I = [0,1)∩Qc
andB be the Borelσ-algebraonI, where Qc denotes the set ofirrationalnumbers.
If ν is a probability measure on the measurable space (I,B), then the sequence
of partial quotients {a ,n ≥ 1} with respect to (w.r.t.) the probability measure ν
n
formsastochasticprocesstakingvaluesinnaturalnumbers. Furthermore,ifν isan
invariant measure w.r.t. the Gauss transformation T, then the stochastic process
{a ,n ≥ 1} is stationary. In 1800, Gauss found such an invariant measure called
n
the Gauss measure, which is given by
1 1
µ(A)= dx
log2 1+x
ZA
foranyBorelsetA⊆[0,1)andisequivalenttotheLebesguemeasureL. Moreover,
the dynamic systems (I,B,µ,T) is an ergodic system (see [5], Theorem 3.5.1) and
the partial quotients sequence {a ,n ≥ 1} is ψ-mixing with exponential rate (see
n
4 LULUFANG1 MINWU1 NARN-RUEIHSHIEH2 ANDBINGLI1,∗
[29], Lemma 2.1). If we choose (Ω,F,P) = (I,B,µ), then a , p and q play the
n n n
same roles as A , P and Q respectively in the definition of random continued
n n n
fractions. In other words, the continued fraction expansion of real numbers is a
special model of random continued fractions.
It is worth pointing out that the sequence of partial quotients {a ,n≥1} is an
n
ergodicandψ-mixingstochasticprocess(see[29],Lemma2.1)anditisnotdifficult
to check that
∞
1 1
E(loga )= loga (x)dµ(x) = logk·log 1+ <∞.
1 1
log2 k(k+2)
Z[0,1) k=1 (cid:18) (cid:19)
X
Therefore, {a ,n≥1} satisfies the conditions of Theorem 3. Thus the application
n
of Theorem 3 to {a ,n≥1} yields that for µ-almost every x∈[0,1),
n
1 π2
lim logq (x)=− logxdµ= ,
n
n→∞n Z[0,1) 12log2
whichis the resultof Theorem1 givenby L´evy[18] in 1929. Furthermore,for each
0<t<1, the expectation
∞
1 1
E(at)= at(x)dµ(x) = ktlog 1+ <∞.
1 1 log2 k(k+2)
Z[0,1) k=1 (cid:18) (cid:19)
X
ApplyingTheorem4tothecontinuedfractionexpansionofrealnumbers,weobtain
a Chernoff-type estimate for 1 logq given in Theorem 2.
n n
Remark 1. (i) Given a distribution of A , it induces a measure ν by
1
ν(B)=P(X ∈B),
where B ∈ B. The stationary of {A ,n ≥ 1} guarantees that ν is T-invariant,
n
where T is Gauss transformation on [0,1). Conversely any T-invariant measure
gives a distribution of {a ,n ≥ 1} which is a special case of random continued
n
fractions. Therefore, random continued fractions of the form (3) can completely
characterize the set of the invariant measures with respect to T.
(ii) Theorem 3 is parallel to Theorem 1 and Theorem 4 is parallel to Theorem 2;
while Theorem 2 is even new, to our knowledge.
At the end, we compare our work with some literature on the large deviation
principlefordynamicalsystems. OreyandPelikan[25]providedanexplicitdescrip-
tion of the rate function of the large deviations for the convergence of trajectory
averages of Anosov diffeomorphisms. Young [38] dealt with the problem of large
deviations for continuous maps in compact metric spaces. Kifer [14] exhibited a
unified approachto large deviations ofdynamicalsystems andstochastic processes
basing on the existence of a pressure function and the uniqueness of equilibrium
states for certain dense sets of functions. Melbourne and Nicol [21] studied a class
of nonuniformly hyperbolic systems modelled by a Young tower and with a return
timefunctiontothebasewithaexponentialorpolynomialdecay. Formorenewre-
sults,see[3],[4],[16],[22],[30]andthereferencestherein. Weemphasizethatthose
papers mentioned above do not cover the case of the Gauss transformation since
the Gauss transformation is a piecewise map and is not continuous on [0,1). On
theotherhand,someauthorsconsideredthelargedeviationprincipleforstationary
processundervariousdependencestructures. OreyandPelikan[24]establishedthe
large deviations for certain classes of stationary processes satisfying a ratio-mixing
LE´VY CONSTANT AND CHERNOFF-TYPE ESTIMATE 5
condition. Bryc [1] proved the large deviation principle for the empirical field of a
stationary Zd-indexed random field under strong mixing dependence assumptions.
Bryc [2] also gave the large deviation principle for the arithmetic means of a se-
quence which has either fast enough ϕ-mixing rate or is ψ-mixing. Although we
assume that the stochastic process {A ,n ≥ 1} is ψ-mixing, those results about
n
thelargedeviationestimatesforstationaryprocesscannotbeappliedtoobtainour
Theorem 4 since the mixing property of {A ,n≥1} do not transfer to that of the
n
process {X ,n≥1} (see Section 2).
n
2. Definitions and properties of random continued fractions
Let B(R)be the Borel σ-algebra on R, for any B ∈B(R), 1≤i≤n, n∈N, the
i
set
C(B ×B ×···×B )={x=(x ,x ,···):x ∈B , for all i=1,2,··· ,n}
1 2 n 1 2 i i
is called a cylinder.
Denote
RN :=R×R×···={x=(x ,x ,···):x ∈R, for all i=1,2,···}
1 2 i
and
B(RN):=B(R)×B(R)×··· is the σ-algebra generated by all cylinders.
The measurable space (RN,B(RN)) is called the direct product of the measurable
space (R,B(R)). For more details about the direct product space, we refer to the
reader to Shiryaev’s book [31] in Section 2.2.
Let{ξ ,n≥1}beastochasticprocessdefinedontheprobabilityspace(Ω,F,P),
n
taking values in the measurable space (R,B(R)).
Definition 1. A stochastic process {ξ ,n≥1} is stationary if for all n∈N,
n
P((ξ ,ξ ,···)∈B)=P((ξ ,ξ ,···)∈B) for all B ∈B(RN).
1 2 n+1 n+2
Remark 2. (i)Asequenceofindependentandidenticallydistributed(i.i.d.) random
variables is stationary.
(ii) Let (Ω,F,P) = (I,B,µ), where µ is the Gauss measure. The sequence of
partial quotients {a ,n≥1} is stationary (see [5], [18]).
n
Definition 2. A set A ∈ F is invariant with respect to the stochastic process
{ξ ,n≥1} if there is a set B ∈B(RN) such that for all n∈N,
n
A={ω :(ξ (ω),ξ (ω),···)∈B}.
n n+1
Definition 3. A stationary stochastic process {ξ ,,n≥ 1} is ergodic if the prob-
n
ability of every invariant set is either 0 or 1.
Remark 3. (i) A sequence of i.i.d. random variables is ergodic.
(ii) The sequence of partial quotients {a ,n ≥ 1} is ergodic (see[5], [7], [18],
n
[26]).
Foralln∈N,letσ(ξ ,··· ,ξ )bethesmallestσ-algebrathatmakesallξ ,··· ,ξ
1 n 1 n
measurableandσ(ξ ,··· ,ξ ,···)be the smallestσ-algebrathatmakes{ξ ,n≥1}
1 n n
measurable.
6 LULUFANG1 MINWU1 NARN-RUEIHSHIEH2 ANDBINGLI1,∗
Definition 4. A stationary stochastic process {ξ ,,n ≥ 1} is ψ-mixing if for all
n
m,n∈Nandforanyintegrablefunctionsf ∈σ(ξ ,··· ,ξ ),g ∈σ(ξ ,ξ ,···),
1 m m+n m+n+1
we have
|E(fg)−E(f)E(g)|≤ψ(n)E(|f|)E(|g|),
where f ∈ σ(ξ ,··· ,ξ ) means that f is a σ(ξ ,··· ,ξ )-measurable function, ψ is
1 n 1 n
non-negative with ψ(n)→0 as n→∞.
Remark 4. (i) The ψ-mixing stochastic process is ergodic.
(ii) The sequence of partial quotients {a ,n ≥ 1} is ψ-mixing (see [29], Lemma
n
2.1).
Inthefollowing,wewillgivesomepropertiesofrandomcontinuedfractions. For
all n∈N and any ω ∈Ω, let
1
X (ω)= ,
n 1
A (ω)+
n
A (ω)+...
n+1
it is clear that for any ω ∈Ω, X (ω)=X(ω) by form (3).
1
Proposition 1. With the convention, P ≡ 1, Q ≡ 0, P ≡ 0, Q ≡ 1. The
−1 −1 0 0
following properties hold for all n∈N and any ω ∈Ω.
(i) P =A P +P , Q =A Q +Q .
n n n−1 n−2 n n n−1 n−2
n−1
Moreover, Qn ≥Qn−1+Qn−2 and Qn ≥2 2 .
(ii) P Q −Q P =(−1)n−1.
n n−1 n n−1
(iii) P Q −Q P =(−1)nA .
n n−2 n n−2 n
X Q −P P +X P
1 n n n n+1 n−1
(iv) X =− . That is, X = .
n+1 1
X Q −P Q +X Q
1 n−1 n−1 n n+1 n−1
1 P 1
n−1
(v) ≤ X − ≤ .
1
2Q Q Q Q Q
n n−1 (cid:12) n−1(cid:12) n n−1
(vi) X ·X ·····(cid:12)X =|X Q(cid:12) −P |.
1 2 (cid:12) n 1 (cid:12)n−1 n−1
(cid:12) (cid:12)
Remark 5. All proofs of the above properties are similar to that of the continued
fraction expansion of real numbers (see [5], [7], [12], [13]).
Proposition 2. For all n≥1 and any ω ∈Ω, we have
n
1 1 1
logQ + logX ≤ log2.
n k
(cid:12)n n (cid:12) n
(cid:12) Xk=1 (cid:12)
(cid:12) (cid:12)
Proof. By Proposition 1(cid:12) (v) and (vi), we have (cid:12)
(cid:12) (cid:12)
1 1
≤X ·X ·····X ≤ .
1 2 n
2Q Q
n n
Taking the logarithm on both sides of the above inequalities and divided by n,
we obtain that
n
1 1 1 1
− logQ − log2≤ logX ≤− logQ .
n k n
n n n n
k=1
X
LE´VY CONSTANT AND CHERNOFF-TYPE ESTIMATE 7
Therefore,
n
1 1 1
logQ + logX ≤ log2.
n k
(cid:12)n n (cid:12) n
(cid:12) Xk=1 (cid:12)
(cid:12) (cid:12) (cid:3)
(cid:12) (cid:12)
(cid:12) (cid:12)
Remark 6. By Proposition2, we know that for any ε>0, there exists N >0 such
that for all ω ∈Ω and all n≥N, we have
n
1 1
logQ (ω)+ logX (ω) <ε.
n k
(cid:12)n n (cid:12)
(cid:12) kX=1 (cid:12)
(cid:12) (cid:12)
This means that lim (cid:12)(cid:12) 1 logQ + 1 n logX =(cid:12)(cid:12)0 uniformly.
n→∞ n n n k=1 k
(cid:18) (cid:19)
P
Next we investigate which properties of X can be inherited from A.
Proposition 3. Let h:({1,2,··· ,n,···}N,S))−→((0,1),B((0,1))) defined by
1
h(x ,x ,··· ,x ,···)= ,
1 2 n 1
x +
1 x +...+ 1
2 x +...
n
where the measurable space ({1,2,··· ,n,···}N,S) is the direct product of measur-
able space ({1,2,··· ,n,···},C). Then h is a measurable function.
Proof. Let
A={I(a ,a ,··· ,a ):a ∈{1,2,···}, for all n∈N, i=1,··· ,n},
1 2 n i
where I(a ,a ,··· ,a ) = {x ∈ (0,1) : a (x) = a ,a (x) = a ,··· ,a (x) = a },
1 2 n 1 1 2 2 n n
a ∈{1,2,···} for each 1≤i≤n, n∈N.
i
It is well-known that A is a semi-algebra and can generate the Borel σ-algebra
B((0,1)). So it suffices to show that for every I(a ,a ,··· ,a )∈A, we have
1 2 n
h−1(I(a ,a ,··· ,a ))∈S.
1 2 n
In fact,
h−1(I(a ,a ,··· ,a ))={(x ,··· ,x ,···):h(x ,··· ,x ,···)∈I(a ,··· ,a )}
1 2 n 1 n 1 n 1 n
={(x ,··· ,x ,···):x =a ,··· ,x =a }
1 n 1 1 n n
={a }×···×{a }×N×··· .
1 n
Therefore,
h−1(I(a ,a ,··· ,a ))∈S.
1 2 n
(cid:3)
Proposition 4 ([6], Theorem 6.1.1 and Theorem 6.1.3). Let {ξ ,n ≥ 1} be a
n
stochasticprocess, F :RN −→Rbeameasurablefunctionandη =F(ξ ,ξ ,···)
n n n+1
foralln≥1. Thenif{ξ ,n≥1}isstationary, sois {η ,n≥1};andif{ξ ,n≥1}
n n n
is ergodic, so is {η ,n≥1}.
n
ByProposition3,Proposition4andX =h(A ,A ,···),wecanimmediately
n n n+1
obtain the following corollary.
8 LULUFANG1 MINWU1 NARN-RUEIHSHIEH2 ANDBINGLI1,∗
Corollary 1. If the stochastic process {A ,n≥1} is stationary and ergodic, then
n
{X ,n≥1} is also stationary and ergodic respectively.
n
3. Proofs of Theorem 3 and Theorem 4
The following ergodic theorem(see [31], Section 5.3)will be needed in the proof
of Theorem 3.
Theorem 5 (Ergodic theorem [31]). Let {ξ ,n≥1} be an ergodic stochastic pro-
n
cess. Then for every real-valued measurable function f with E(|f(ξ )|) < ∞, we
1
have
n
1
lim f(ξ (ω))=E(f(ξ )), P −a.s.
k 1
n→∞n
k=1
X
Now, we are ready to prove Theorem 3.
Proof of Theorem 3. Since {A ,n ≥ 1} is an ergodic stochastic process, by Corol-
n
lary 1, we obtain that {X ,n≥1} is ergodic.
n
Note that (A + 1)−1 < X ≤ A−1 and E(logA ) < ∞, so E(|logX |) =
1 1 1 1 1
−E(logX ) ≤ E(log(A +1)) ≤ E(log2A ) = log2+E(logA ) < ∞. By Theo-
1 1 1 1
rem 3, we have
n
1
lim logX = logXdP, P −a.s.
k
n→∞n
k=1 ZΩ
X
On the other hand, by Proposition 2, for any ε > 0, there exists N > 0 such
that for all ω ∈Ω and all n≥N, we have
n
1 1
logQ (ω)+ logX (ω) <ε.
n k
(cid:12)n n (cid:12)
(cid:12) kX=1 (cid:12)
(cid:12) (cid:12)
Therefore, (cid:12) (cid:12)
(cid:12) (cid:12)
1
lim logQ =− logXdP, P −a.s.
n
n→∞n ZΩ
(cid:3)
The followingLemma1,Lemma2andLemma3arethekeylemmasforproving
Theorem 4.
Lemma 1. For any δ >0, there exists N >0, such that for all n>N , we have
0 0
n
1 1 δ
ω : logQ (ω)+ logXdP ≥δ ⊆ ω : logX (ω)− logXdP ≥ .
n k
P(cid:26)roof(cid:12)(cid:12)(cid:12)(cid:12).nIt suffices to pZrΩove that if(cid:12)(cid:12)(cid:12)(cid:12)for a(cid:27)ny δ(>0,(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)tnheXkr=e1exists N0 >0Z,Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 2)
1
(6) logQ + logXdP ≥δ
n
n
(cid:12) ZΩ (cid:12)
(cid:12) (cid:12)
holds for all n>N0, then (cid:12) (cid:12)
(cid:12) (cid:12)
n
1 δ
logX − logXdP ≥ .
k
(cid:12)(cid:12)nXk=1 ZΩ (cid:12)(cid:12) 2
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
LE´VY CONSTANT AND CHERNOFF-TYPE ESTIMATE 9
Since
1
lim log2=0,
n→∞n
by Proposition 2, we obtain that for any δ > 0, there exists N > 0 such that for
0
all n>N , we have
0
n
1 1 1 δ
(7) logQ + logX ≤ log2< .
n k
(cid:12)n n (cid:12) n 2
(cid:12) Xk=1 (cid:12)
(cid:12) (cid:12)
By the triangle inequ(cid:12)ality, we deduce that for(cid:12)all n>N0,
(cid:12) (cid:12)
n
1
logX − logXdP
k
(cid:12)(cid:12)nkX=1 ZΩ (cid:12)(cid:12)
(cid:12) n (cid:12)
(cid:12)1 1 (cid:12)1
=(cid:12) logX + logQ −(cid:12) logQ − logXdP
k n n
(cid:12)(cid:12)nkX=1 n n ZΩ (cid:12)(cid:12)
(cid:12) n (cid:12)
(cid:12)1 1 1 (cid:12) δ
≥(cid:12) logQ + logXdP − logQ + logX(cid:12) ≥ ,
n n k
where the las(cid:12)(cid:12)(cid:12)(cid:12)tninequality fZoΩllows from(cid:12)(cid:12)(cid:12)(cid:12) (6)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)nand (7). ThniskX=co1mplete(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)s the2proof. (cid:3)
Lemma 2. Let {A ,n ≥ 1} be a ψ-mixing stochastic process and 0 < t < 1.
n
Then for any ε > 0, there exists N > 0 such that for any integrable function
g ∈σ(A ,A ,···), we have
N N+1
E Xt|g| ≤(1+ε)2E Xt E(|g|).
1 1
Proof. Recallthat,since{(cid:0)An,n≥(cid:1) 1}isaψ-mix(cid:0)ing(cid:1)stochasticprocess,forallm,n∈
N and for any integrable functions f ∈σ(A ,··· ,A ), g ∈σ(A ,A ,···),
1 m m+n m+n+1
we have
|E(fg)−E(f)E(g)|≤ψ(n)E(|f|)E(|g|),
with ψ(n)→0 as n→∞ being 0<E(Xt)≤1.
1
For any ε>0, we can choose N >0, such that ψ(N )≤ε and
1 1
1
(8) 2−(N1−1)t ≤ εE Xt
2 1
holds. The first implies (cid:0) (cid:1)
(9) |E(fg)−E(f)E(g)|≤εE(|f|)E(|g|).
For all m∈N, let Y = Pm. Then Y ∈σ(A ,··· ,A ). Proposition 1 (i) and
m Qm m 1 m
(v) imply that
1 1
(10) |X −Y |≤ ≤ ≤2−(m−1).
1 m Q Q Q2
m m+1 m
Note that X >0, Y >0, 0<t<1, we know that Xt−Yt ≤|X −Y |t.
1 N1 1 N1 1 N1
Applying m=N1 in (10) leads to X1t−YNt1 ≤2−(N1−1(cid:12))t, that is,(cid:12)
(cid:12) (cid:12)
(11) −2−(N1−1)t ≤(cid:12) Xt−Yt(cid:12) ≤2−(N1−1)t.
(cid:12) 1 N(cid:12)1
Therefore,
(12) E Xt|g| ≤E (Yt +2−(N1−1)t)|g| =E Yt |g| +2−(N1−1)tE(|g|),
1 N1 N1
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
10 LULUFANG1 MINWU1 NARN-RUEIHSHIEH2 ANDBINGLI1,∗
for any integrable function g ∈σ(A ,A ,···) with N =2N .
N N+1 1
Since Y ∈ σ(A ,··· ,A ), g ∈ σ(A ,A ,···) with N = 2N , by (8) and
N1 1 N1 N N+1 1
(9), we deduce that
1
(13) E Yt |g| +2−(N1−1)tE(|g|)≤E Yt E(|g|)(1+ε)+ εE Xt E(|g|).
N1 N1 2 1
The first(cid:0)inequa(cid:1)lity of (11) implies that (cid:0) (cid:1) (cid:0) (cid:1)
1
(14) E Yt ≤E Xt+2−(N1−1)t =E Xt +2−(N1−1)t ≤(1+ ε)E Xt ,
N1 1 1 2 1
where th(cid:0)e last(cid:1)inequ(cid:16)ality follows from(cid:17) (8). (cid:0) (cid:1) (cid:0) (cid:1)
Combining the inequalities (12), (13) and (14), for any ε > 0 and for any inte-
grable function g ∈σ(A ,A ,···), we have
N N+1
1 1
E Xt|g| ≤(1+ ε)(1+ε)E Xt E(|g|)+ εE Xt E(|g|)
1 2 1 2 1
(cid:0) (cid:1)≤(1+ε)2E Xt E(cid:0)(|g|)(cid:1). (cid:0) (cid:1)
1
(cid:3)
(cid:0) (cid:1)
Lemma 3. Suppose that for each 0 < t < 1, E(At) < ∞. Let {A ,n ≥ 1} be a
1 n
ψ-mixing stochastic process. Then for any 0<t<1 and for any ε>0, there exists
N >0 such that for any integrable function g ∈σ(A ,A ,···), we have
N N+1
E X−t|g| ≤(1+ε)2E X−t E(|g|).
1 1
Proof. Since (A +1)−1(cid:0)<X ≤(cid:1)A−1 and E(A(cid:0)t)<(cid:1)∞ for each 0<t<1, we have
1 1 1 1
0<E At ≤E X−t ≤E (A +1)t ≤2tE At <∞.
1 1 1 1
Since {An,n ≥ 1}(cid:0)is a(cid:1)ψ-mix(cid:0)ing s(cid:1)tochas(cid:0)tic process(cid:1), for any(cid:0) ε >(cid:1) 0, we can choose
N >0,suchthatforallm∈Nandforanyintegrablefunctionf ∈σ(A ,··· ,A ),
2 1 m
g ∈σ(A ,A ,···), we obtain that
m+N2 m+N2+1
(15) |E(fg)−E(f)E(g)|≤εE(|f|)E(|g|)
and
1
(16) 2−(N2−2)t ≤ εE X−t .
2 1
Forallm∈N,letY = Pm,soY ∈σ(A ,·(cid:0)·· ,A(cid:1) ). NotethatX−1 =A +X
m Qm m 1 m 1 1 2
and Y−1 =A +Y′ , where Y′ =[A ,A ,··· ,A ].
m 1 m m 2 3 m
Therefore,
(17) X−1−Y−1 = X −Y′ ≤2−(m−2),
1 m 2 m
whNeroetitcheetlhaasttiXne−q1ua>l(cid:12)(cid:12)it0y,fYol−lo1w>s fr0(cid:12)(cid:12)oman(cid:12)(cid:12)(cid:12)d(100).< t <(cid:12)(cid:12)(cid:12) 1, we know that X−t−Y−t ≤
1 m 1 m
isX,1−1−Ym−1 t. Applying m = N2 in (17) leads to X1−t−YN−2t ≤(cid:12)(cid:12)2−(N2−2)t, t(cid:12)(cid:12)hat
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)(18) (cid:12) −2−(N2−2)t ≤X−t−Y−t ≤2(cid:12)−(N2−2)t. (cid:12)
1 N2
Therefore,
(19) E X−t|g| ≤E (Y−t+2−(N2−2)t)|g| =E Y−t|g| +2−(N2−2)tE(|g|),
1 N2 N2
for any in(cid:0)tegrabl(cid:1)e funct(cid:16)ion g ∈σ(A ,A ,(cid:17)···) wi(cid:0)th N =(cid:1)2N .
N N+1 2