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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 503689, 13 pages
On the Dependence of the Limit Functions on the Random Parameters in Random Ergodic Theorems
Department of Mathematics, Toyo University, Kawagoe, Saitama 350-8585, Japan
Received 16 October 2012; Accepted 28 December 2012
Academic Editor: Haydar Akca
Copyright © 2013 Takeshi Yoshimoto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the structure of the ergodic limit functions determined in random ergodic theorems. When the r random parameters are shifted by the -shift transformation with , the major finding is that the (random) ergodic limit functions determined in random ergodic theorems depend essentially only on the random parameters. Some of the results obtained here improve the earlier random ergodic theorems of Ryll-Nardzewski (1954), Gladysz (1956), Cairoli (1964), and Yoshimoto (1977) for positive linear contractions on and Woś (1982) for sub-Markovian operators. Moreover, applications of these results to nonlinear random ergodic theorems for affine operators are also included. Some examples are given for illustrating the relationship between the ergodic limit functions and the random parameters in random ergodic theorems.
1. A General Argument
The present paper is concerned with the relations between the limit functions in random ergodic theorems and the random parameters concomitant to the limit functions. The first results of the random ergodic theory include Pitt’s random ergodic theorem  and Ulam-von Neumann’s random ergodic theorem  concerning a finite number of measure-preserving transformations and Kakutani’s random ergodic theorem  concerning an infinite number of measure-preserving transformations. Furthermore, Kakutani dealt with the relationship between the random ergodic theorem and the theory of Markov processes with a stable distribution. The random ergodic theorem is usually obtained by using the so-called skew product method as natural extensions of ergodic theorems and has received a great deal of attention from the wider point of view including operator-theoretical treatment. In fact, interesting extensions have been made by many authors.
It was pointed out by Marczewski (see ) that the proof of Kakutani’s theorem should be found which would not use the hypothesis that the transformations in question are one-to-one. Answering this question, Ryll-Nardzewski  improved Kakutani’s theorem to the case of random measure-preserving transformations which are not necessarily one-to-one and proved that the limit function is essentially independent of the random parameter. Then, later, Ryll-Nardzewski’s theorem was generalized by Gładysz  to the case of a finite number of random parameters. The Ryll-Nardzewski theorem was extended by Cairoli  to the case of positive linear contractions on with an additional condition. Yoshimoto  extended both Gladysz’s theorem and Cairoli’s theorem to the case of positive linear contractions on with a finite number of random parameters. In this paper we inquire further into the problem of the dependence of the limit functions upon the random parameters in random ergodic theorems, and we have an intention of improving the previous random ergodic theorem of Yoshimoto .
In what follows, we suppose that there are a given -finite measure space and a probability space . Let , be the usual Banach spaces of eqivalence classes of β-measurable functions defined on . From now on, we shall write for if we wish to regard as a function of defined on for arbitrarily fixed in .
It seems to be worthwhile to include the first random ergodic theorems which may be stated, respectively, as follows.
Theorem 1 (see [1, 2]). Let , be two given measure-preserving transformations of into itself, which generate all the combinations of the transformations: , , , , , , , . The ergodic limit exists then for almost every point of and almost every choice of the infinite sequence obtained by applying and in turn at random, for example, , , .
Exactly speaking, the first step to the theory of random ergodic theorems was taken by Pitt . The above theorem was stated by Ulam and von Neumann  (independently of Pitt), but the essence of the contents is the same as the theorem of Pitt who proved both the pointwise convergence and the , mean convergence of random averages in question. Ulam and von Neumann announced the pointwise convergence of random averages in an abstract form, but the proof has never been published. Pitt-Ulam-von Neumann’s random ergodic theorem concerning a finite number of measure-preserving transformations was extended by Kakutani  to the case of an infinite number of measure-preserving transformations as follows.
Theorem 2 (Kakutani (1948–1950) ). Let be a -measurable family of measure-preserving transformations defined on , where . Let be the two-sided infinite direct product measure space of . Then, for any , there exists a -null set such that for any , there exists a function such that where denotes the th coordinate of , and this holds also in the norm of .
Kakutani’s paper on the random ergodic theorem was published in 1950, but the random ergodic theorem had already been dealt with by Kawada (“Random ergodic theorems”, Suritokeikenkyu (Japanese) 2, 1948). Later, Kawada reminisced about the circumstances of an affair of his paper. It is the rights of matter that Kawada’s result is due to Kakutani’s kind suggestion.
Remark 3. In Kakutani’s random ergodic theorem (as well as in Pitt-Ulam-von Neumann’ theorem), the sequence of measure-preserving transformations on is chosen at random with the same distribution and independently. In connection with this question, an interesting problem is the following: if we choose a sequence at random, not necessarily with the same distribution but independently from a given set of measure-preserving transformations on , under what condition does the limit exist . or in -mean with probability ? Revesz made the first step toward the study of this problem (see [8, 9]).
The most general formulation of random ergodic theorems is the following Chacon’s type theorem given by Jacobs .
Jacobs’ General Random Ergodic Theorem . Let be an endomorphism of and let be a strongly -measurable family of random linear contractions on . Let be a sequence of -measurable functions defined on which is admissible for . This means that if and ., then . Then, for any function , there exists a -null set such that for any , exists and is finite . on the set (cf.  which includes a further weighted generalization of Jacobs’theorem).
If all are positive, then Jacobs’ theorem yields the following Chacon-Ornstein’s type random ergodic theorem (cf. [10, 12]); for and , there exists a -null set such that for any , exists and is finite . on the set . Moreover, if the family has a strictly positive invariant function , then for every , there exists a -null set such that for any exists and is finite . Unfortunately, this Cesàro-type result does not hold in general without assuming the existence of a strictly positive invariant function. However, if the family satisfies the norm conditions and for all , then the above Cesàro-type random ergodic theorem holds even without assuming the existence of a strictly positive invariant function in . The above limit functions , , and depend generally on the random parameter . Our particular interest is in the relationship between the limit functions and the random parameters in the case that is the shift transformation of the random parameter space being the one-sided infinite product of the same probability space .
2. Random Ergodic Theorems
Throughout all that follows, let be the one-sided infinite product measure space of :
Let be the (one-sided) shift transformation defined on which means that using the coordinate functions ,
Then, is clearly a -measurable and -measure-preserving transformation defined on . Let be a fixed positive integer. For simplicity, we let for any . Fix an . Suppose that to each , there corresponds a linear contraction operator on . The family is said to be strongly -measurable if for any the function is strongly -measurable as an -valued function defined on , namely, for the mapping of into , has a separable support (cf. ).
Our main result is stated as follows.
Theorem 4. Let be any fixed integer with . Let be a strongly -measurable family of positive linear contractions on . Suppose that there exists a strictly positive -function invariant under . Then, for every , there exists a -null set such that for any , there exists a function such that
Proof. We need the following lemmas.
Lemma 5 (see [10, 14]). The strong -measurability of the operator family guarantees that for any , there exists a uniquely determined -measurable version of such that excepting a -null set,
Using the measurable version appearing in Lemma 5, we define
From the norm conditions of in Theorem 4, it turns out that is a linear operator on with . Moreover, it is easy to check that there exists a strictly positive -function invariant under . Thus, for any , we can apply Chacon-Ornstein’s ergodic theorem  (cf. Hopf’s ergodic theorem ) to ensure the existence of a function such that
Moreover, as is easily checked, we find that
One can easily verify that excepting a suitable -null set , for all . Next we wish to show that does depend essentially only on . To do this, we define sub--fields , , of by
It is clear that if , then and that if , , then whenever . Therefore, the system forms a martingale. For each , let denote the conditional expectation operator with respect to the sub--field . Let be of the form where , , . Then, the linear combinations of functions of the form (15) are everywhere dense in . Thus, for the question confronting us, it suffices to prove the relation only for the case when , is of the form (15).
Lemma 6. It holds that .
Proof. It follows that for a sufficiently large and , is such that
Thus, excepting a -null set , we get On the other hand, since excepting a -null set we have that if , then Consequently, and by approximation, and so by iteration, In particular, Since is the adjoint operator of , we have thus
We return to the proof of the theorem. By Lemma 6 and the martingale convergence theorem (cf. [16, 17]), we have as , and thus which implies that depends essentially only on . Hence, the theorem follows from (11), (26), and Fubini’s theorem. The proof of Theorem 4 has hereby been completed.
If we take in Theorem 4, we have Cairoli’s theorem in which does not depend essentially on . The random parameter generalization (see ) of Cairoli’s theorem is obtained by taking in Theorem 4. Adapting the (almost) same argument as used in the proof of Theorem 4, we have the following.
Theorem 7. Let be any fixed integer with . Let be a strongly -measurable family of linear contractions on with for all . Then, for every with , there exists a -null set such that for any , there exists a function such that and that if , then and that if and , then
Proof. As before, we define for . From the norm conditions of in Theorem 7, it turns out that is a linear operator on with and . Then, It follows from the Riesz convexity theorem that for all with . Thus, for any , we can apply Dunford and Schwartz’s ergodic theorem  to ensure the existence of a function such that
and that if , then
and that if and , then
Now, adapting the (almost) same argument as used in the proof of Theorem 4, we can find that Note here that if , , (or , then where denotes the linear modulus of (see [18, 19]). Therefore, (27) follows from (30), (33), and Fubini’s theorem. Equations (28) and (29) follow from (30), (31), (32), (33), (34), and Fubini’s theorem.
In the setting of measure-preserving transformations, Theorem 7 is reduced to the random one-parameter result of Ryll-Nardzewski  by taking and to the random parameter result of Gładysz  by taking .
Theorem 8. Let be positive integers. Let , , be strongly -measurable families of linear contractions on with . For each , we set Then, for every , there exists a -null set such that for every , there exists a function with such that the multiple averages converge to . as independently. Here, means that the number of random parameters is at most .
Proof. As already seen above, we can define the operators on as follows for Each turns out to be a linear contraction on with . So, it follows from the Riesz convexity theorem that for . Hence, from Dunford-Schwartz’s ergodic theorem , we have that for every the multiple averages converge to almost everywhere on as independently, where each is a projection of onto the manifold with Note here that Thus, by Lemma 6 and the martingale convergence theorem, we find that excepting a -null set Finally, observe that excepting a -null set, we get In fact, we have for two operators and To complete the proof of the above equality, assume that (43) has already been established for the operators . Then, it is easily verified by the induction hypothesis that (43) holds for the operators . Hence, taking , the theorem follows from the above arguments.
In particular, if the operators in question are commutative then we have the following.
Theorem 9. Let be positive integers. Let , , be a strongly -measurable commuting family of linear contractions on with . Then, for every , there exists a -null set such that for every , there exists a function with such that
In passing, we make mention of the . convergence for sectorial restricted random averages. We say that a sequence remains in a sector of if there is a constant such that the ratios are bounded by for and all (see [20, page ]). Appealing to Brunel-Dunford-Schwartz’ theorem (see [20, Theorem 3.5, page ]), we find that the multiple averages (38) converge . in a sector . In addition, in this case, Theorem 8 implies that the limit function depends essentially only on the random parameters. Hence, we have the following.
Theorem 10. Let be positive integers. Let , , be strongly -measurable commuting family of linear contractions on with . Then, for every , there exists a -null set such that for every , there exists a function with such that the multiple averages converge to . as in a sector of .
A sub-Markovian operator on means that is a positive linear contraction on with subinvariant under (i.e., ) such that . for any (decreasing) sequence with . Let denote the positive linear contraction on with as the adjoint operator of . When with a -subinvariant function (), is called a -subinvariant measure. Here, the subinvariant function is not necessarily integrable.
Now, let be a strongly -measurable family of sub-Markovian operators on . Let be a strongly -measurable family of positive linear contractions on , where each is the adjoint operator of the corresponding operator . Then, there exists a positive linear contraction on such that for any ,
In addition, if for all , is also a contraction on (cf. [10, 14]). The operator extends uniquely to a positive linear transformation on the class of nonnegative -measurable functions defined on (see [20, page ]).
Lemma 11. Let the measure be finite. Assume that is -subinvariant. Then, turns out to be a positive Dunford-Schwartz operator on .
Proof. Since is assumed to be -subinvariant, the function is -subinvariant. According to Lemma 1.5 of Woś , we get so that is also -subinvariant. Thus, for , Since is a positive contraction on , and is a positive Dunford-Schwartz operator on .
Now, the above general theorems can also be applied to sub-Markovian operators. For example, Theorem 7 yields the following theorem which extends both Theorems 3.4 and 3.6 of Woś  (In proving the key Lemma 1.5 of , Woś showed that if does not depend essentially on parameter , then . Using this equality, he deduced from in order to prove Theorems 2.8 and 3.4 of . But obviously, 1 does not necessarily belong to in case where is -finite. His arguments are of course correct in the case that is finite (see, e.g., conditions (i) and (ii) in Theorem 12)).
Theorem 12. Let be a strongly -measurable family of positive linear contractions on , where each is the adjoint operator of the corresponding sub-Markovian operator . Assume either of the following conditions:(i)is finite and -subinvariant,(ii) is -finite and for all .
Then, for any , , there exists a -null set such that for every , there exists a function such that and that if , then and that if is finite and , then
Proof of Theorem 12. In view of Lemma 11, either condition (i) or condition (ii) guarantees that is a positive Dunford-Schwartz operator on . Hence, we can apply Theorem 7 to conclude that Theorem 12 follows.
3. -Type Random Ergodic Theorems
In this section, we establish a -type random ergodic theorem for measure-preserving transformations on . In this section, we assume that is a probability space. For real and , let be the coefficient of order , which is defined by the generating function Then, we can easily check that is decreasing in for and increasing in for . For , we have and moreover, In general, using Hille’s theorem (see [22, Theorem 8]), we have the following.
Theorem 13. Let be any fixed integer with . Let be a strongly -measurable family of linear contractions on . Let be fixed. If for all there exists a -null set such that for any there exists a function such that(i)⋯ for -almost all ,(ii)⋯,(iii)⋯ ,
for -almost all . Conversely, if (ii) holds but (iii) be replaced by the condition that there exists a -measurable function finite almost everywhere such that excepting a -null setfor all , where , then (i) holds for all .
Note here that in Theorem 13, (ii) and (iii) do not necessarily imply (i) in general.
Theorem 14. Let be any fixed integer with and be a -measurable family of measure-preserving transformations on . Let , , and . Then, there exists a -null set such that for every , there exists a function such that
Proof. Define the skew product of (the one-sided shift transformation of ) and as follows:
Let , where for all . Then, . Since is reflexive, we see from Yosida-Kakutani’s mean ergodic theorem  and Déniel’s theorem  that there exists a function such that Furthermore, applying Lemma 6 to the operator induced by , it follows that so that to complete the proof of the theorem, we may take
Remark 15. For example, if in Theorem 14, then . It is worthwhile to note that if and (so, ), then the pointwise -convergence for does not hold in general (see ). For the case of a positive linear contraction on , see Irmisch .
In particular, applying Irmisch’s theorem to sub-Markovian operators, we have the following.
Theorem 16. Let be a strongly -measurable family of positive linear contractions on , where each is the adjoint operator of the corresponding sub-Markovian operator . Assume that is -subinvariant. Let , and . Then, there exists a -null set such that for every , there exists a function such that
Proof. In view of Lemma 11, is a positive linear contraction on as well as on . Thus, it follows from the Riesz convexity theorem that for . Therefore, we reach the assertion of Theorem 16 through Theorems 7 and 13 appealed to Irmisch’s theorem .
Remark 17. The relations between the random ergodic limit functions and the random parameters have been investigated (with satisfactory formulations) only in discrete parameter cases so far. So, it is very interesting to study the continuous analogs of the theorems obtained above. But no continuous results are known from the point of view of the dependence of the limit functions on the random parameters. Here, it is worthwhile to notice that Anzai has obtained a continuous version of Kakutani’s random ergodic theorem for Brownian motion in continuous parameter cases (see ). Let () be a Brownian motion (or Wiener process) on a probability space . This process has independent increments; that is, for arbitrary , the random variables are independent. In fact, since the process is Gaussian with and by definition, it is sufficient to verify only that the increments are uncorrelated. Thus, if , then
Let be an arbitrary ergodic measurable semiflow on a finite measure space . Then, Anzai’s result may be stated as follows: for any and for almost all , there exists a null set such that holds for any . This is an immediate consequence of the ergodicity of the measure-preserving skew product semiflow defined by ,, where is the ergodic semiflow on given by . It is an interesting and important problem to generalize Anzai’s result for Brownian motions to the case of contraction operator quasisemigroups in associated with . We do not discuss it in the present paper.
4. Applications to Nonlinear Random Ergodic Theorems
The random ergodic theorems obtained above can be applied to the nonlinear random ergodic theorems for affine systems (see ). An affine operator on is an operator of the type , where is a linear contraction on , and is a fixed element of . Then, is nonlinear and nonexpansive. The fixed points of are solutions of Poisson’s equation for , which is . When we assume to be mean ergodic, the averages converge if and only if . If , then there exists a unique such that , and the limit of is , where is the limit of . Therefore, iterating will yield almost everywhere convergence of the averages of for any . Now, let be a random affine system on such that is a strongly measurable family of linear contractions on as well as on and such that for some , where for (cf. [11, 27]). For each , we define a sequence of random functions , , in inductively by
Theorem 18. Let be the sequence of random functions associated with which is determined by a random affine system given in with . Then, there exists a -null set such that for any , there exists a function such that
Proof. It follows that there exists a -null set such that for any , Moreover, we have Therefore, by Theorem 7, there exists a -null set such that for every , there exist functions , such that Hence, the theorem follows by putting and
As far as we are concerned with the ergodic behaviors of Cesàro-type processes for nonexpansive operators on , one can only expect weak convergence in general. In fact, the pointwise convergence of the averages of nonlinear and nonexpansive operators on may fail to hold. In addition, these averages do not need converge in the strong operator topology of (see [27, 28]). The so-called nonlinear sums introduced by Wittmann  make it possible to consider the pointwise convergence and the strong convergence under some additional conditions. To make the most of advantageous results in the theory of linear ergodic theorems, it is very rational to consider a class of affine operators as a model case (cf. ). Under the above setting, observe that Then, Theorem 7, together with Theorems 2.1 and 2.2 of , yields the following theorem.
Theorem 19. Let