A generalized limit probability measure associated with a random system with complete connections for a generalized Gauss-Kuzmin operator, only for a special case, is defined, and its behaviour is investigated. As a consequence a specific version of Gauss-Kuzmin-type problem for the above generalized operator is obtained.

1. Introduction

Let Y=C([0,1]) be the Banach space of complex-valued continuous functions on [0,1] under the supremum norm, and let N*={1,2,…}, N={0,1,2,…}. Then for every f∈Y and for every α≥1 the function Gαf introduced by Fluch [1] and defined by(Gαf)(w)=∑x∈N*α2(αx+w)⋅(αx+α-1+w)⋅f(ααx+α-1+w),
for all w∈[0,1], is called a generalized Gauss-Kuzmin operator.

The present paper arises as an attempt to determine a generalized limit probability measure, only for a special case, associated with a random system with complete connections for the above generalized Gauss-Kuzmin operator obtained in Ganatsiou [2], for every α>2. This will give us the possibility to obtain a specific variant of Gauss-Kuzmin-type problem for the above operator.

Our approach is given in the context of the theory of dependence with complete connections (see Iosifescu and Grigorescu [3]). For a more detailed study of the theory and applications of dependence with complete connections to the metrical problems and other interesting aspects of number theory we refer the reader to [4–9] and others.

The paper is organized as follows. In Section 2, we present all the necessary results regarding the ergodic behaviour of a random system with complete connections associated with the generalized Gauss-Kuzmin operator Gα obtained in [2], in order to make more comprehensible the presentation of the paper. In Section 3, we introduce the determination of a limit probability measure associated with the above random system with complete connections, only for a special case, for every α>2, which will give us the possibility to study in Section 4 a specific version of the associated Gauss-Kuzmin type problem.

2. Auxiliary Results

For every α≥1, we consider the function ρα defined byρα(w)=αα+w,w∈[0,1],
and setgn=Gαnfρα,n∈N,
where Gαn+1f=Gα(Gαnf), for every n∈N and for every f∈Y.

Then we obtain the following statement which gives a relation deriving from an analogous of the Gauss- Kuzmin type equation.

Proposition 2.1.

The function gn satisfies
gn+1(w)=∑x∈N*α⋅(α+w)(αx+w)⋅(αx+α+w)⋅gn(ααx+α-1+w),
for any n∈N and w∈[0,1].

Furthermore we obtain the following.

Proposition 2.2.

For every α≥1, the function
Pα(w,x)=α⋅(α+w)(αx+w)⋅(αx+α+w),w∈[0,1],x∈N*,
defines a transition probability function from ([0,1],B[0,1]) to (X,P(X)), where X=N* andP(X) the power set of X.

Equation (2.3) and Proposition 2.2 lead to the consideration of a family of random systems with complete connections (RSCCs){(W,W)(X,X),uα,Pα},α≥1,
whereW=[0,1],W=B[0,1],X=N*,X=P(X),uα(w,x)=ααx+α-1+w,Pα(w,x)=α⋅(α+w)(αx+w)⋅(αx+α+w),w∈W,x∈X.
In the next, we consider the transition probability function Qα, α≥1, of the Markov chain associated with the family of the RSCCs (2.5) and the corresponding Markov operator Uα, α≥1, defined by
Uαf(w)=∑x∈N*α⋅(α+w)(αx+w)⋅(αx+α+w)⋅f(ααx+α-1+w),
for all complex-valued measurable bounded functions f on [0,1].

This gives us the possibility of obtain the following.

Proposition 2.3.

The family of RSCCs (2.5) is with contraction. Moreover, its associated Markov operator Uα given by (2.7) is regular with respect to L([0,1]), the Banach space of all real-valued bounded Lipschitz functions on [0,1].

On the contrary the RSCC associated with a concrete piecewise fractional linear map (see Ganatsiou [10]) is not an RSCC with contraction since r1=1 and its associated Markov chain is not compact and regular with respect to the set L([0,1]), even though there exists a point y*∈(0,1), such that limn→∞|∑n(y)-y*|=0,
for all y∈(0,1). This corrects the escape of [10] gives an RSCC associated with a concrete piecewise fractional linear map which is not uniformly ergodic (a special case of [4]).

By virtue of Proposition 2.3, it follows from [3, Theorem 3.4.5] that the family of RSCCs (2.5) is uniformly ergodic. Furthermore, Theorem 3.1.24 in [3] implies that, for every α≥1, there exists a unique probability measure γα on B[0,1], which is stationary for the kernel Qα, such thatlimn→∞Uαnf=∫01fdγα,f∈L([0,1]).
This means thatγα(B)=∫01Qα(w,B)γα(dw),
whereQα(w,B)=∑X∈BwPα(w,x),
withBw={x∈N*∣uα(w,x)∈B},foreveryB∈W,w∈[0,1].
Moreover, for some c>0and0<θ<1, we have‖Uαnf-∫01fdγα‖≤c⋅θn⋅‖f‖L,
for all n∈N* and f∈L([0,1]), where ∥·∥L denotes the usual norm in L([0,1]), whereUα∞f=∫01f(w)γα(dw).
In general the form of the limit probability measure associated with the family of random systems with complete connections (2.5) cannot be determined but this is possible only for a special case as we prove in the following section.

For the proofs of the above results we refer the reader to Ganatsiou [2].

3. A Limit Probability Measure Associated with the Family of RSCCs

Now, we are able to determine a limit probability measure associated with the family of RSCCs (2.5) as is shown in the following.

Proposition 3.1.

The probability measure γα has the density
ρα(w)=αα+w,for everyw∈[0,1],
with constant 1/α·log(1+α-1) only for the special case a·u-1+1-a[u-1+a-1]<1, for every a>2, 0<u≤1.

Proof.

By virtue of uniqueness of γα we have to show that it satisfies relation (2.10). Since the intervals [0,u), 0<u≤1 generate B[0,1] it is sufficient to verify (2.10) only for B=[0,u), 0<u≤1.

Suppose that B=[0,u). Then, for every w∈[0,1], we haveBw={x∈N*∣uα(w,x)∈[0,u)}={x∈N*∣α(αx+α-1+w)<u}={x∈N*∣x≥[u-1-w⋅α-1+α-1]}.
Hence by (2.11), we have that
Qα(w,[0,u))=α+wα[u-1-w⋅α-1+α-1]+w,
where
[u-1-w⋅α-1+α-1]={[u-1+α-1],if0≤w<α⋅u-1+1-α⋅[u-1+α-1],[u-1+α-1]-1,ifα⋅u-1+1-α⋅[u-1+α-1]<w≤1.

We consider the case αu-1+1-α·[u-1+α-1]<1 or u-1<[u-1+α-1], for every α>2,0<u≤1. Consequently, we obtain that∫01Qα(w,[0,u))⋅ρα(w)dw=1log(1+α-1)⋅∫01dwα⋅[u-1-w⋅α-1+α-1]+w=1log(1+α-1)⋅[log(α⋅u-1+1)-log(α⋅u-1+1-α)+log(α⋅[u-1+α-1]-α+1)-log(α⋅[u-1+α-1])].
In the next we put
I=log(α⋅[u-1+α-1]-α+1)-log(α⋅[u-1+α-1])=log(1-1[u-1+α-1]+1α⋅[u-1+α-1]),II=log(α⋅u-1+1)-log(α⋅u-1+1-α)=log(1+uα)-log(1+uα-u)
By taking the limit of
III=I-log(1+uα-u)=log(1-1[u-1+α-1]+1α⋅[u-1+α-1])-log(1+uα-u)
when u→1 we have that
limu→1log(1+uα-u)=log(1α),limu→1log(1-1[u-1+α-1]+1α⋅[u-1+α-1])=log(1α),foreveryα>2.
So part III tends to 0 when u→1. This means that
limu→1[∫01Qα(w,[0,u))⋅ρα(w)dw]=1log(1+α-1)⋅limu→1log(1+uα)
which is equal to
limu→1∫0uρα(w)dw=limu→1∫0u1α⋅log(1+α-1)⋅αα+wdw=1log(1+α-1)limu→1[log(α+u)-logα]=1log(1+α-1)limu→1log(α+uα)=1log(1+α-1)limu→1log(1+uα)
and the proof is complete.

4. A Version of the Gauss-Kuzmin-Type Problem

Let μ be a nonatomic measure on the σ-algebra B[0,1]. Then we may defineVo(w)=μ([0,w]),Vn(w)=Vn(w,μ)=∫0wGαnf(t)dt,n∈N*,w∈[0,1].
Suppose that V0′ exists and it is bounded (μ has bounded density). Then by induction we have that Vn′ exists and it is bounded for any n∈N* withVn′(w)≡Gαnf(w)=Gα[(Gαn-1f)(w)],f∈L([0,1]),n∈N*.
So
∫0wVn′(t)dt=∫0wGαnf(t)dt,Vn(w)=∫0wGαnf(t)dt
while
gn(w)=Gαnf(w)pα(w)≡Vn′(w)pα(w),n∈N.
Now, we are able to determine the limit limn→∞Vn(1/w) and to give the rate of this convergence, that is, a specific version of the associated Gauss-Kuzmin type problem.

Proposition 4.1.

(i) If the density V0′ of μ is a Riemann integrable function, then
limn→∞Vn(1w)=1log(1+α-1)⋅log(αw+1αw),w≥1,α>2,n∈N*.(ii) If the density V0′ of μ is an element of L([0,1)], then there exist two positive constants c and θ<1 such that
limn→∞Vn(1w)=(1+qθn)⋅1log(1+α-1)⋅log(αw+1αw),
for all w≥1, a>2, n∈N*, where q=q(μ,n,w)with|q|≤c.

Proof.

Let V0′∈L([0,1]). Then go∈L([0,1]), and by using relation (2.14) we have
Uα∞g0≡limn→∞Uαng0=∫01g0(w)γα(dw)=∫01V0′(w)dw=1.
According to relation (2.13), there exist two positive constants c and θ<1 such that
Uαng0=Uα∞g0+Tαng0,n∈N*,with‖Tαng0‖≤c⋅θn.
If we consider the Banach space C([0,1]) of all real continuous functions defined on [0,1] with the supremum norm, then since L([0,1]) is a dense subset of C([0,1]) we have
limn→∞|Tαng0|=0,foreverygo∈C([0,1]).
This means that it is valid for any measurable function go which is γα-almost surely continuous, that is, for any Riemann integrable function go. Consequently we obtain
limn→∞Vn(1w)=limn→∞∫01/wUαng0(t)ρα(t)dt=∫01/wρα(t)dt=∫01/w1log(1+α-1)⋅αα+tdt=1log(1+α-1)⋅log(αw+1αw),
that is the solution of the associated Gauss-Kuzmin type problem.

Remarks 4.

(1) It is notable that for α=1 the RSCC associated with the generalized Gauss-Kuzmin operator is identical to that associated with the ordinary continued fraction expansion (see Iosifescu and Grogorescu [3]). Moreover the corresponding limit probability measure associated with the family of RSCCs (2.5) for α=1 is identical to the limit probability measure associated with the above random system with complete connections for the ordinary continued fraction expansion, that is, identical to the Gauss’s measure γ on B[0,1] defined by
γ(A)=1log2∫Adtt+1,A∈B[0,1].

(2) It is an open problem the determination of an analogous limit probability measure for the case αu-1+1-α·[u-1+α-1]>1.

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