ON A CONTRACTIBILITY CONDITION FOR ITERATED RANDOM FUNCTIONS

Radu Theodorescu (12 April 1933 – 14 August 2007), collaborator and friend

ON A CONTRACTIBILITY CONDITION FOR ITERATED RANDOM FUNCTIONS

ULRICH HERKENRATH and MARIUS IOSIFESCU

Wu and Shao [7] considered a geometric moment contractibility condition for ite- rated random functions. We show that this condition is not fulfilled for simple examples of such random functions and propose instead a more general one, able also to incorporate a local contractibility condition considered by Steinsaltz [6].

Moreover, we show that the functional central limit theorem as well as the conver- gence of empirical processes to Gaussian processes in Skorokhod space proved in Wu and Shao’s paper, still hold under our condition, with appropriate alterations of the hypotheses, which reduce to those assumed by these authors under their condition.

AMS 2000 Subject Classification: 60J05, 60F17.

Key words: iterated random function, contractibility, functional central limit theorem.

1. PRELIMINARIES

We adopt the notation from Wu and Shao [7] with just slight modifica- tions intended to better precise a few details (for, e.g., probability distributions and mean value operators).

Let (X, ρ) be a Polish (i.e. complete and separable) metric space with metric ρ, endowed with the σ-algebra of its Borel sets. As usual, define an iterated random function as X_n = F_θn(X_n−1), n ∈ N₊ = {1,2, . . .}, where (i) the θn, n ∈ N+, take values in a second measurable space Θ and are independently distributed with identical marginal probability distributionH on theσ-algebra of measurable sets in Θ; (ii)Fθ(·) is theθ-section of a jointly measurable functionF :X ×Θ→ X, and (iii) X₀ is either a fixed point inX or an X-valued random variable independent of the sequence (θ_n)_n∈N+. On account of (iii), the sequence (X_n)_n∈N, n ∈ N = {0} ∪N₊, is supported by a probability space, say, (Ω,K, P_H) or (Ω,K, P_h,H) according as X₀ is a fixed point x∈ X or anX-valued random variable with probability distribution h on the Borel subsets ofX.

REV. ROUMAINE MATH. PURES APPL.,52(2007),5, 563–571

Set Xn(x) = F_θn ◦ · · · ◦F_θ1(x) and introduce the backward iteration process Z_n(x) = F_θ1 ◦ · · · ◦F_θn(x), x ∈ X, n ∈ N₊. It is obvious that for all x ∈ X and n∈ N₊ the probability distributions of X_n(x) and Z_n(x) are identical.

It has been known for long that (X_n(x))_n∈N+ is a Markov process while the sequence (Z_n(x))_n∈N+ is a very different sort of object (actually, a chain of infinite order). Also (cf. Letac [4]), if F_θ(·) is continuous for any fixed θ ∈ Θ and if Z_∞ := lim_n→∞Z_n(x) exists P_H-a.s. and does not depend on x∈ X, then the probability distribution of Z_∞, say, π is the only stationary distribution of (Xn(x))n∈N+.

2. A GENERAL CONTRACTIBILITY CONDITION

Wu and Shao [7] proved the P_H-a.s. convergence of Z_n(x) as n → ∞ under Conditions 1 and 2 below. In what follows, E_H(E_h,_H) stands for the mean-value operator with respect toP_H(P_h,_H).

Condition 1. There exist x₀ ∈ X and α∈(0,1] such that EH(ρ^α(x₀, Fθ₁(x₀))) =

Θρ^α(x₀, Fθ(x₀))H(dθ)<∞.

Condition 2. Withx₀ ∈ X andα ∈(0,1] as in Condition 1, there exist r(α) :=r∈(0,1) andC(α) :=C >0 such that

E_H(ρ^α(X_n(x), X_n(x₀)))≤Crⁿρ^α(x, x₀) for any x∈ X and n∈N.

In a somewhat more restrictive context, where X is a convex subset of R^m for somem∈N+ withρ the Euclidean distance, Steinsaltz [6] introduced a local contractibility condition amounting to the existence of a drift function ϕ:X →(0,∞) and of some r∈(0,1) such that

(1) EH

lim sup

y→x

ρ(Xn(y), Xn(x)) ρ(y, x)

≤ϕ(x)rⁿ

for any x∈ X and n∈N. It was proved by Steinsaltz [6, p. 1959] that under (1) in conjunction with a suitable assumption onϕ, which is a generalization of Condition 1, a more general condition than Condition 2 does hold, but not Condition 2 itself.

Our point is that it is possible to introduce a more general condition able to fully integrate both Steinsaltz’s and Wu and Shao’s ones.

To be precise, we introduce Condition C below.

Condition C.(i) There existx₀ ∈ X, α∈(0,1], r∈(0,1), and measur- able functionsϕ_n :X →[0, ∞), n∈N₊, such that bothR:= 1/lim sup

n→∞ δn^1/n

andRx:= 1/lim sup

n→∞ ϕ¹n^/n(x) strictly exceed r for any x∈ X, where δ_n:=E_H(ϕ_n(X₁(x₀))ρ^α(x₀, X₁(x₀))), n∈N₊. (ii) For anyx∈ X and n∈N+ one has

EH(ρ^α(Xn(x), Xn(x₀)))≤rⁿϕn(x)ρ^α(x, x₀).

Clearly, R and R_x, x∈ X, are the convergence radii of the power series

n∈N+δ_ntⁿ and

n∈N+ϕ_n(x)tⁿ, t∈R, respectively.

Wu and Shao’s Conditions 1 and 2 correspond to the case where ϕ_n is a constant independent of n∈N+. Steinsaltz’s conditions imply that Condi- tion C holds with an arbitrary x₀ ∈ X whileϕn is independent of n∈N+, it having the form

ϕ_n(x) = sup

0≤t≤1ϕ(x₀+t(x−x₀)), x∈ X, whereϕ:X →[0,∞) is the function occurring in (1).

3. A SIMPLE EXAMPLE

Doubtlessly, Condition 2 is restrictive enough, as simple examples show.

Such an example is given by

F_θ(x) =ax²+bθ

withx∈ X = [0,∞), θ∈[0,1] anda, b∈(0,1).Let (θn)n∈N+ be a sequence of i.i.d. Bernoulli variables with success probability ¹₂, so that the corresponding iterated random function is defined recursively by the equation

(2) X_n(x) =aX_n²₋₁(x) +bθ_n, n∈N₊,

withX₀(x) =x. It is easy to check that X_n(x) is an even 2ⁿth degree poly- nomial with positive coefficients, part of which are random as functions of θ_i, 1≤i≤n, n∈N₊. Hence the ratio

|X_n(x)−X_n(x₀)|/|x−x₀|

exceeds|Xn(x)−Xn(0)|/x, so that the choice x₀ = 0 in Condition 2 appears to be the natural one. Next, as X₀(·) is an increasing function, one sees by induction thatXn(·) is an increasing function for any n∈N+. Then, letting

q_n(x) = X_n(x)−X_n(0)

x , n∈N₊, 0=x∈ X,

so thatq₁(x) =ax,equation (2) yields

qn(x) =a(Xn−1(x) +Xn−1(0))qn−1(x)≥2aXn−1(0)qn−1(x), n∈N₊, whence

(3) q_n(x)≥(2a)ⁿ⁻¹ax

n−1 j=1

X_j(0), n≥2.

Now, note thatXj(0)≥bθj, j ∈N+, so that (4)

n−1 j=1

X_j(0)≥bⁿ⁻¹θ₁· · ·θ_n−1, n≥2.

Since theθ_i are independent andE_H(θ_i) = ¹₂, i∈N₊, it follows from (3) and (4) that

E_H(q_n(x))≥(ab)ⁿ⁻¹ax, n∈N₊, 0=x∈ X,

that is, withρ denoting the Euclidean distance on the real line, we have (5) E_H(ρ(X_n(x), X_n(0)))≥(ab)ⁿ⁻¹axρ(x,0)

for any n∈N₊ and x∈ X.

Clearly, (5) shows thatCondition 2cannot hold for our example asX is unbounded.

On the other hand, it is easy to see that E_H(ρ(X₂(x), X₂(0))) =a²x

ax²+b

ρ(x,0), E_H(ρ(X₃(x), X₃(0))) =a³x

ax²+b a³x⁴+a²bx²+2ab²+b

ρ(x,0). These equations actually exhibit ϕ₂(x) and ϕ₃(x). It then appears that the mean valueE_H(ρ(X_n(x), X_n(0))) has the form

aⁿx

ax²+b f_n

ax²+b

ρ(x,0)

for some functionf_n: [0,∞)→R, n∈N₊, showing that our example should enter the class of iterated function systems that obey Condition C under suit- able assumptions on the coefficientsaandb.We shall not go any further into this matter as our goal was a different one.

4. MAIN RESULT Our main result is stated below.

Theorem 1. Assume Condition Cholds.

(j)There exists an X-valued σ(θ₁, θ₂, . . .)-measurable random variable Z_∞to whichZ_n(x)convergesP_H-a.s. asn→ ∞for anyx∈ X and, moreover,

EH(ρ^α(Zn(x), Z_∞))≤rⁿϕn(x)ρ^α(x, x₀) +

m≥n

r^mδm.

(jj) Let π denote the probability distribution of Z_∞ and let X₀, X₀ be X-valued random variables independent of (θ_n)_n∈N+ with common probability distribution π. Then

(6) E_π,H

ρ^α

X_n(X₀), X_n(X₀)

≤2

m≥n

r^mδ_m, n∈N₊.

Proof. (j) As in the proof of Theorem 2 in Wu and Shao [7, p. 427], we deduce that

EH(ρ^α(Zn(x₀), Zn+1(x₀)))≤rⁿEH ϕn

Fθ_n+1(x₀) ρ^α

x₀, Fθ_n+1(x₀)

=rⁿδn

for anyn∈N+.Fixr₀ ∈(0,1) with _R^r < r₀ <1. Then, by Markov’s inequality, PH(ρ^α(Zn(x₀), Zn+1(x₀))≥rⁿ₀)≤

r r₀

n

δn

for any n∈N₊. As by Condition C(i) the series with general term (r/r₀)ⁿδ_n is convergent, the Borel-Cantelli lemma implies that

PH(ρ^α(Zn(x₀), Zn+1(x₀))≥rⁿ₀ infinitely often) = 0,

henceZn(x₀)→Z_∞, say,PH-a.s. asn→ ∞by the completeness ofX. Clearly, Z_∞ isσ(θ₁, θ₂, . . .)-measurable. Next, by the triangle inequality,

E_H(ρ^α(Z_n(x₀), Z_∞))≤E_H

j∈N

ρ^α(Z_n+j(x₀), Z_n+j+1(x₀))

≤

≤

j∈N

EH(ρ^α(Zn+j(x₀), Zn+j+1(x₀)))≤δ_n :=

m≥n

r^mδm

for any n∈N₊. Finally, by Condition C(ii), for anyx∈ X,

E_H(ρ^α(Z_n(x), Z_∞))≤E_H(ρ^α(Z_n(x), Z_n(x₀))) +E_H(ρ^α(Z_n(x₀), Z_∞))≤

≤rⁿϕ_n(x)ρ^α(x, x₀) +δ_n,

whence we conclude that Zn(x) → Z_∞ PH-a.s. as n → ∞, by invoking the argument already used to show that Z_n(x₀) → Z_∞ P_H-a.s. as n → ∞. We should now chooser₀ ∈(0,1) withr/min (R, R_x)< r₀<1 and note then that both series

n∈N+ϕ_n(x) (r/r₀)ⁿ and

n∈N+m≥n

(r/r₀)^mδm=

n∈N+

n(r/r₀)ⁿδn

are convergent by Condition C(i). The proof of (j) is complete.

(jj) This follows as in Wu and Shao [7, p. 428] with the corresponding new upper bound as indicated.

Remark 1. Theorem 1 allows one to estimate the rate of convergence of then-step transition probability functionPⁿof the Markov chain (X_n(x))_n∈N+ to its stationary probability distributionπ. Recall [cf., e.g., Hofmann-Jorgensen [1, pp. 80–82]] that whateverα∈(0,1] a metricρ_L,α which metrizes the topol- ogy of weak convergence on the set of probability measures on the Borel sets ofX is defined by

ρ_L,α(µ, ν) =

= sup

Xfd (µ−ν)

: 0≤f≤1, |f(x)−f(x)| ≤ρ^α(x, x), x, x∈X

for two such probability measuresµandν, and note thatρL,α ≥ρL,1[cf., e.g., Iosifescu [2, p. 73]]. It then follows from Theorem 1 that

ρ_L,α(Pⁿ(x,·), π)≤rⁿϕ_n(x)ρ^α(x, x₀) +

m≥n

r^mδ_m for any n∈N₊ and x∈ X since

Xfd (Pⁿ(x,·)−π) =E_H(f(X_n(x))−f(Z_∞)) =E_H(f(Z_n(x))−f(Z_∞)) for any continuous bounded functionf :X →R.

Remark 2. As in Wu and Shao [7, p. 428], Theorem 1 above allows to defineX-valued random variables X_i, i≤0, in such a way that the equation X_i = F_θi(X_i−1) still holds. Let (θ_i)_i∈Z be a doubly infinite sequence of i.i.d.

Θ-valued random variables with common distributionH. Then, as in the proof of Theorem 1(j), the limit

mlim→∞F_θi◦F_θi−1◦ · · · ◦F_θi−m(x) :=X_i(x), x∈ X,

exists P_H-a.s. for any i ≤ 0, is a σ(. . . , θ_i−1, θ_i)-measurable function which does not actually depend onx, has probability distributionπ, and satisfies the equationX_i=F_θi(X_i−1) for any i≤0.Notice that withX_i, i≤0,so defined, the doubly infinite sequence (X_i)_i∈Zconstructed by using the latter recurrence equation forany i∈Z,is a strictly stationary sequence, as well as a Markov process, on a suitable probability space, to be still denoted (Ω,K, Pπ,H).

5. LIMIT THEOREMS

Theorem 3 (a central limit theorem) in Wu and Shao’s paper can be restated in terms of our Condition C and its implication (6). Let (X_i)i∈Z be the doubly infinite sequence defined in Remark 2 above. For a fixed ∈ N₊ consider a measurable functiong:X →Rand defineY_i =g(X_i−+1, . . . , X_i), i∈Z, S_n, =

n

i=1Y_i, n ∈N₊, S_0, = 0.For a real-valued random variable Z on (Ω,K, P_π,H) denote by Z itsL₂-norm and define ∆g(δ) = sup{||[g(W)− g(W)]1₍ρ(W,W)≤δ)||:W and W areX-valued random variables distributed asY₁},where

ρ w, w

=

i=1

ρ² wi, w_i

12

forw= (w₁, . . . , w) andw = (w₁, . . . , w), so that ρ^α

w, w

≤

i=1

ρ^α w_i, w_i forα∈(0,1]. Now, we can state

Theorem 2. Assume (6)holds,E_π,H(g(Y₁)) = 0, E_π,H(|g^p(Y₁)|)<∞ for some p >2, that

(7)

₁

0

∆_g(t)

t dt <∞, and that

n∈N+

δ_n +· · ·+δ_n+−1

r₁^−αn₍p−2)/2p <∞ for some r₁ ∈(0,1).

Then there exists σ_g ≥0such that

S_nu,/√n, 0≤u≤1

converges weakly toσ_gB.Here Bstands for a standard Brownian motion and for integer part.

Also, for π-almost all x∈X,the same conclusion holds for

1≤i≤nu

g(Xi(x), . . . , X_i+−1(x))/√

n, 0≤u≤1

,

that is, for the Markov process started at x.

The proof is mutatis mutandis identical to that of Theorem 3 on page 430 in Wu and Shao’s paper on account of (6), by using at appropriate places our assumptions.

Remark 3. It is an open problem whether the proviso ‘π-almost allx∈ X’ can be removed by using recent work of Peligrad and Utev [5], as this has been done in Lager˚as and Stenflo [3] in another context.

Corollary 1 and Theorem 4 in Wu and Shao’s paper concerning em- pirical processes, can be also restated. Let X = R with ρ the Euclidean distance. LetG(x) =Pπ,H(X₁ ≤x), Gn(x) =_n

i=11_(Xi≤x)/n and Pn(x) =

√n(G_n(x)−G(x)), x∈R, n∈N₊. It is then easy to check that defining g_y(u) =1₍u<y)−G(y), y, u∈R,

one has ∆²_g(δ) ≤ 2 min (G(y+δ)−G(y), G(y)−G(y−δ)) := 2τ(y;δ). So that, condition (7) in our Theorem 2 holds if one assumes that

(8)

₁

0

τ(x:t)

t dt <∞.

The result corresponding to Wu and Shao’s Corollary 1 can now be stated as follows.

Corollary 1. Assume the conditions in Theorem 1 with (7) replaced by (8). Then for any x ∈ X there exists some σ(x) <∞ such that P_n(x) ⇒ N

0, σ²(x)

(convergence in distribution).

As to Wu and Shao’s Theorem 4, the corresponding result can be stated as follows.

Theorem 3. Assume that

supx∈R|G(x+δ)−G(x)| ≤Clog^−k δ⁻¹ for any δ∈

0,¹₂

with some k >5/2andC >0,and that δ_n =O

r^n/₂ ²n^−k as n→ ∞ for some 0< r₂ < r²₁^α.Then, under the assumptions of Corollary 1, {P_n(y), y∈R} converges in D(R) to a Gaussian process (Γt)t∈R with mean zero and covariance function

E_π,H(ΓxΓy) =

k∈Z

cov(1₍X₀≤x) 1₍X_k≤y))), x, y∈R.

Acknowledgement. Marius Iosifescu gratefully acknowledges support from the Deutsche Forschungsgemeinschaft under Grant 936 RUM 113/21/0-1 and from Con- tract 2-CEx-06-11-97 of the Romanian Authority for Research.

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Received 28 April 2006 Universit¨at Duisburg-Essen, Campus Duisburg Institut f¨ur Mathematik

D-47048 Duisburg, Germany [email protected]

and Romanian Academy Institute of Mathematical Statistics

and Applied Mathematics Calea 13 Septembrie nr.13 050711 Bucharest 5, Romania

[email protected]