Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 503689, 13 pages http://dx.doi.org/10.1155/2013/503689 Research Article On the Dependence of the Limit Functions on the Random Parameters in Random Ergodic Theorems Takeshi Yoshimoto Department of Mathematics, Toyo University, Kawagoe, Saitama 350-8585, Japan Correspondence should be addressed to Takeshi Yoshimoto; yoshimoto t@toyonet.toyo.ac.jp 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 π0 -shift transformation with π0 ∈ {1, 2, . . . , π}, the major finding is that the (random) ergodic limit functions determined in random ergodic theorems depend essentially only on the π—π0 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 πΏ 1 and WosΜ (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 [1] and Ulam-von Neumann’s random ergodic theorem [2] concerning a finite number of measurepreserving transformations and Kakutani’s random ergodic theorem [3] concerning an infinite number of measurepreserving 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 [4]) 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 [4] improved Kakutani’s theorem to the case of random measurepreserving transformations which are not necessarily one-toone and proved that the limit function is essentially independent of the random parameter. Then, later, Ryll-Nardzewski’s theorem was generalized by GΕadysz [5] to the case of a finite number of random parameters. The Ryll-Nardzewski theorem was extended by Cairoli [6] to the case of positive linear contractions on πΏ 1 with an additional condition. Yoshimoto [7] extended both Gladysz’s theorem and Cairoli’s theorem to the case of positive linear contractions on πΏ 1 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 [7]. In what follows, we suppose that there are a given π-finite measure space (π, π½, π) and a probability space (Ω, F, π). Let πΏ π (π) = πΏ π (π, π½, π), 1 ≤ π ≤ ∞ be the usual Banach spaces of eqivalence classes of π½-measurable functions defined on π. From now on, we shall write ππ (π ) for π(π , π) if we wish to regard π(π , π) as a function of π defined on π for π arbitrarily fixed in Ω. 2 Abstract and Applied Analysis It seems to be worthwhile to include the first random ergodic theorems which may be stated, respectively, as follows. exist π-a.e. or in πΏ 2 (π)-mean with probability 1? Revesz made the first step toward the study of this problem (see [8, 9]). Theorem 1 (see [1, 2]). Let π(π ), π(π ) be two given measurepreserving transformations of π into itself, which generate all the combinations of the transformations: π, π, π(π), π(π), π(π), π(π), π(π(π)), π(π(π)), . . .. The ergodic limit exists then for almost every point π of π and almost every choice of the infinite sequence obtained by applying π and π in turn at random, for example, π(π ), π(π(π )), π(π(π(π ))), . . .. The most general formulation of random ergodic theorems is the following Chacon’s type theorem given by Jacobs [10]. Exactly speaking, the first step to the theory of random ergodic theorems was taken by Pitt [1]. The above theorem was stated by Ulam and von Neumann [2] (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 πΏ π (π) (π ≥ 1, π(π) < ∞) 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. PittUlam-von Neumann’s random ergodic theorem concerning a finite number of measure-preserving transformations was extended by Kakutani [3] to the case of an infinite number of measure-preserving transformations as follows. Theorem 2 (Kakutani (1948–1950) [3]). Let Φ = {ππ : π ∈ Ω} be a π½ ⊗ F-measurable family of measure-preserving transformations ππ defined on π, where π(π) = π(π) = 1. Let (Ω∗± , F∗± , π±∗ ) be the two-sided infinite direct product measure space of (Ω, F, π). Then, for any π ∈ πΏ π (π) (π ≥ 1), there exists a (F∗± , π±∗ )-null set π∗ such that for any π∗ ∈ Ω∗± − π∗ , Μ ∗ (π ) ∈ πΏ (π) such that there exists a function π π π 1 π−1 Μ ∗ (π ) ∑ π (πℵπ (π∗ ) ⋅ ⋅ ⋅ πℵ0 (π∗ ) π ) = π π π→∞π π=0 lim π-π.π., (1) where ℵπ (π∗ ) denotes the πth coordinate of π∗ , and this holds also in the norm of πΏ π (π). 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 (πℵπ (π∗ ) )π≥0 of measure-preserving transformations on π 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 (ππ )π≥1 at random, not necessarily with the same distribution but independently from a given set Φ of measure-preserving transformations on π, under what condition does the limit 1 π ∑ π (ππ ⋅ ⋅ ⋅ π1 π ) π→∞π π=1 lim (2) Jacobs’ General Random Ergodic Theorem [10]. Let π be an endomorphism of (Ω, F, π) and let {ππ : π ∈ Ω} be a strongly F-measurable family of random linear contractions on πΏ 1 (π). Let {ππ }π≥0 be a sequence of π½ ⊗ F-measurable functions defined on π × Ω which is admissible for {ππ : π ∈ Ω}. This means that if β ∈ πΏ 1 (π ⊗ π) and |β(π , π)| ≤ ππ (π , π) π ⊗ π-a.e., then |ππ βππ (π )| ≤ ππ+1 (π , π) π ⊗ π-a.e. Then, for any function π ∈ πΏ 1 (π), there exists a π-null set π ∈ F such that for any π ∈ Ω − π, ∑ππ=0 ππ πππ ⋅ ⋅ ⋅ πππ π π (π ) Μ (π , π)) (= π π→∞ ∑ππ=0 ππ (π , π) lim (3) exists and is finite π-a.e. on the set {π : ∑∞ π=0 ππ (π , π) > 0} (cf. [11] which includes a further weighted generalization of Jacobs’theorem). If all ππ are positive, then Jacobs’ theorem yields the following Chacon-Ornstein’s type random ergodic theorem (cf. [10, 12]); for π ∈ πΏ 1 (π) and π ∈ πΏ+1 (π), there exists a π-null set π such that for any π ∈ Ω − π, ∑ππ=0 ππ πππ ⋅ ⋅ ⋅ πππ π π (π ) Μ (π , π)) (= π π π → ∞ ∑π π π ⋅ ⋅ ⋅ π π π (π ) π π π=0 π ππ lim (4) exists and is finite π-a.e. on the set {π : ∑∞ π=0 ππ πππ ⋅ ⋅ ⋅ πππ π π(π ) > 0}. Moreover, if the family {ππ , π ∈ Ω} has a strictly positive invariant function π ∈ πΏ+1 (π), then for every π ∈ πΏ 1 (π), there exists a π-null set π such that for any π∈Ω−π 1 π ∑ ππ πππ ⋅ ⋅ ⋅ πππ π π (π ) (= π∗ (π , π)) π→∞π + 1 π=0 lim (5) exists and is finite π-a.e. Unfortunately, this CesaΜro-type result does not hold in general without assuming the existence of a strictly positive invariant function. However, if the family {ππ : π ∈ Ω} satisfies the norm conditions βππ βπΏ 1 (π) ≤ 1 and βππ βπΏ ∞ (π) ≤ 1 for all π ∈ Ω, then the above CesaΜrotype random ergodic theorem holds even without assuming the existence of a strictly positive invariant function in πΏ 1 (π). Μ π) = [π] Μ (π ), π Μ (π , π) = The above limit functions π(π , π π ∗ ∗ Μ ] (π ), and π (π , π) = [π ] (π ) depend generally on [π π π π the random parameter π. Our particular interest is in the relationship between the limit functions and the random parameters in the case that π is the shift transformation of the random parameter space (Ω∗ , F∗ , π∗ ) being the one-sided infinite product of the same probability space (Ω, F, π). Abstract and Applied Analysis 3 2. Random Ergodic Theorems Throughout all that follows, let (Ω∗ , F∗ , π∗ ) be the one-sided infinite product measure space of (Ω, F, π): ∞ ∞ ∞ (Ω∗ , F∗ , π∗ ) = (∏Ωπ , β¨ Fπ , β¨ ππ ) , π=1 π=1 π=1 π π π π=1 π=1 π=1 (Ω∗π , F∗π , ππ∗ ) = (∏Ωπ , β¨ Fπ , β¨ ππ ) , (6) (π: a positive integer) , (Ωπ , Fπ , ππ ) = (Ω, F, π) , ∗ ℵπ (ππ ) = ℵπ+1 (π ) , π = 1, 2, . . . . ∗ 1 π π Μ (π , π∗ ) ∑ π π (π , π∗ ) = π π→∞π π=1 lim (7) ∗ Then, π is clearly a F -measurable and π -measurepreserving transformation defined on Ω∗ . Let π be a fixed positive integer. For simplicity, we let [π∗ ]π = (π1 , . . . , ππ ) for any π∗ = (π1 , . . . , ππ , . . .) ∈ Ω∗ . Fix an π0 ∈ {1, 2, . . . , π}. Suppose that to each [π∗ ]π ∈ Ω∗π , there corresponds a linear contraction operator π[π∗ ] on πΏ 1 (π). The family {π[π∗ ]π : π∗ ∈ Ω∗ } is said to be strongly F∗π -measurable if for any β ∈ πΏ 1 (π) the function π[π∗ ]π β is strongly F∗π -measurable as an πΏ 1 (π)-valued function defined on Ω∗π , namely, for the mapping Ψβ : [π∗ ]π → π[π∗ ]π β of F∗π into πΏ 1 (π), ππ∗ β Ψβ−1 has a separable support (cf. [13]). Our main result is stated as follows. Theorem 4. Let π0 be any fixed integer with 1 ≤ π0 ≤ π. Let {π[π∗ ]π : π∗ ∈ Ω∗ } be a strongly F∗π -measurable family of positive linear contractions on πΏ 1 (π). Suppose that there exists a strictly positive πΏ 1 (π)-function invariant under {π[π∗ ]π : π∗ ∈ Ω∗ }. Then, for every π ∈ πΏ 1 (π), there exists a π∗ -null set π∗ ∈ F∗ such that for any π∗ ∈ Ω∗ − π∗ , there exists a Μ ∗ function π [π ]π−π ∈ πΏ 1 (π) such that Μ = π. Μ ππ lim Μ ∗ = π [π ]π−π (π ) 0 (8) π-π.π. Proof. We need the following lemmas. F∗π -measurability Lemma 5 (see [10, 14]). The strong of the operator family {π[π∗ ]π : π∗ ∈ Ω∗ } guarantees that for any π ∈ πΏ 1 (π ⊗ π∗ ), there exists a uniquely determined π½ ⊗ F∗ measurable version [π[π∗ ]π πππ0 π∗ ](π ) of π[π∗ ]π πππ0 π∗ (π ) such that excepting a π∗ -null set, [π[π∗ ]π πππ0 π∗ ] (π ) = π[π∗ ]π πππ0 π∗ (π ) π-a.e. (9) Using the measurable version appearing in Lemma 5, we define ππ (π , π∗ ) = [π[π∗ ]π πππ0 π∗ ] (π ) . (11) (10) (12) ∗ One can easily verify that excepting a suitable π -null set π1∗ ∈ F∗ , ππ π (π , π∗ ) = π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) π-a.e., (13) Μ π∗ ) does for all π = 1, 2, . . .. Next we wish to show that π(π , depend essentially only on (π , [π∗ ]π−π0 ). To do this, we define sub-π-fields ℘π , π = 1, 2, . . ., of π½ ⊗ F∗ by ℘π = π½ ⊗ F1 ⊗ ⋅ ⋅ ⋅ ⊗ π½π ⊗ Fπ ⊗ {0, Ωπ+1 } ⊗ ⋅ ⋅ ⋅ , π = 1, 2, . . . , (14) (℘0 = π½ ⊗ {0, Ω1 } ⊗ {0, Ω2 } ⊗ ⋅ ⋅ ⋅) . It is clear that if π < π, then ℘π ⊂ ℘π and that if π ∈ πΏ 1 (π ⊗ π∗ ), ππ = ℘π π, then ℘π ππ = ππ whenever π ≤ π. Therefore, the system {ππ , ℘π : π = 1, 2, . . .} forms a martingale. For each π, let πΈ(⋅ | ℘π ) denote the conditional expectation operator with respect to the sub-π-field ℘π . Let β(π , π∗ ) be of the form β β (π , π∗ ) = π (π ) ⋅ ∏ππ (ππ ) , (15) π=1 0 1 π ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) π→∞π π=1 π ⊗ π-a.e. Moreover, as is easily checked, we find that π = 1, 2, . . . . Let π be the (one-sided) shift transformation defined on Ω∗ which means that using the coordinate functions ℵπ (⋅), ∗ From the norm conditions of {π[π∗ ]π : π∗ ∈ Ω∗ } in Theorem 4, it turns out that π is a linear operator on πΏ 1 (π ⊗ π∗ ) with βπβπΏ 1 (π⊗π∗ ) ≤ 1. Moreover, it is easy to check that there exists a strictly positive πΏ 1 (π ⊗ π∗ )-function invariant under π. Thus, for any π ∈ πΏ 1 (π ⊗ π∗ ), we can apply ChaconOrnstein’s ergodic theorem [12] (cf. Hopf ’s ergodic theorem Μ ∈ πΏ (π ⊗ π∗ ) [15]) to ensure the existence of a function π 1 such that where π ∈ πΏ 1 (π) ∩ πΏ ∞ (π), ππ ∈ πΏ ∞ (Ωπ ), π = 1, 2, . . . , β. Then, the linear combinations of functions of the form (15) are everywhere dense in πΏ 1 (π ⊗ π∗ ). Thus, for the question Μ |℘ )= confronting us, it suffices to prove the relation πΈ(π π−π0 Μ Μ π only for the case when π, is of the form (15). Lemma 6. It holds that πΈ(ππ ⋅ | ℘π−π0 ) = ππ πΈ(⋅ | ℘(π−1)π0 +π ), π = 1, 2, . . .. Proof. It follows that for a sufficiently large β and π < β, πΈ(β | ℘π ) is such that π πΈ (β | ℘π ) (π , π∗ ) = π (π ) ⋅ ∏ππ (ππ ) π=1 β ⋅ ∏ ∫ ππ (π) ππ (π) π=π+1 Ω π ⊗ π∗ -a.e. (16) 4 Abstract and Applied Analysis Thus, excepting a π∗ -null set π2∗ , we get We return to the proof of the theorem. By Lemma 6 and the martingale convergence theorem (cf. [16, 17]), we have π ∗ (π∗ πΈ (β | ℘π ))π∗ (π ) = [π[π ∗ ] π] (π ) ∏ππ (ππ+π ) 0 π π=1 β × ∏ ∫ ππ (π) ππ (π) π=π+1 Ω π-a.e. (17) On the other hand, since excepting a π∗ -null set β ∗ (π∗ β)π∗ (π ) = [π[π ∗ ] π] (π ) ∏ππ (ππ+π ) 0 π σ΅©σ΅© Μ Μ σ΅©σ΅©σ΅©σ΅© σ΅©σ΅©πΈ (π | ℘π−π0 ) − π σ΅©πΏ 1 (π⊗π∗ ) σ΅© σ΅© Μ | ℘ ) − ππ π Μ σ΅©σ΅©σ΅©σ΅© = σ΅©σ΅©σ΅©σ΅©πΈ (ππ π π−π0 σ΅©πΏ 1 (π⊗π∗ ) σ΅© π Μσ΅© σ΅© Μ |℘ ≤ σ΅©σ΅©σ΅©σ΅©ππ πΈ (π σ΅©σ΅©πΏ 1 (π⊗π∗ ) (π−1)π0 +π ) − π πσ΅© σ΅© Μ Μ σ΅©σ΅©σ΅©σ΅© | ℘(π−1)π0 +π ) − π ≤ σ΅©σ΅©σ΅©σ΅©πΈ (π σ΅©πΏ 1 (π⊗π∗ ) σ³¨→ 0 (25) as π → ∞, and thus π-a.e., Μ Μ | ℘ ) = π, πΈ (π π−π0 (18) π=1 (26) Μ π∗ ) depends essentially only on which implies that π(π , ∗ (π , [π ]π−π0 ). Hence, the theorem follows from (11), (26), and Fubini’s theorem. The proof of Theorem 4 has hereby been completed. we have that if π ≥ π − π0 , then π (πΈ (π∗ β | ℘π+π0 ))π∗ (π ) = [π[π∗ ]π π] (π ) ∏ππ (ππ+π0 ) π=1 β × ∏ ππ (π) ππ (π) π-a.e. π=π+1 (19) Consequently, π∗ πΈ (β | ℘π ) = πΈ (π∗ β | ℘π+π0 ) π ⊗ π∗ -a.e., (20) and by approximation, ∗ If we take π0 = π in Theorem 4, we have Cairoli’s Μ π∗ ) does not depend essentially on theorem in which π(π , ∗ π . The π random parameter generalization (see [7]) of Cairoli’s theorem is obtained by taking π0 = 1 in Theorem 4. Adapting the (almost) same argument as used in the proof of Theorem 4, we have the following. Theorem 7. Let π0 be any fixed integer with 1 ≤ π0 ≤ π. Let {π[π∗ ]π : π∗ ∈ Ω∗ } be a strongly F∗π -measurable family of linear contractions on πΏ 1 (π) with βπ[π∗ ]π βπΏ (π) ≤ 1 for all ∞ π∗ ∈ Ω∗ . Then, for every π ∈ πΏ π (π ⊗ π∗ ) with 1 ≤ π < ∞, there exists a π∗ -null set π∗ ∈ F∗ such that for any π∗ ∈ Μ ∗ Ω∗ − π∗ , there exists a function π [π ]π−π ∈ πΏ π (π) such that 0 ∗ π πΈ (⋅ | ℘π ) = πΈ (π ⋅ | ℘π+π0 ) , (21) 1 π ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) π→∞π π=1 lim Μ ∗ = π [π ]π−π (π ) and so by iteration, 0 π∗π πΈ (⋅ | ℘π ) = πΈ (π∗π ⋅ | ℘π+ππ0 ) , π = 0, 1, 2, . . . , π ≥ π − π0 . (22) π = 1, 2, . . . . (23) ∗ Since π is the adjoint operator of π , we have thus πΈ (ππ ⋅ | ℘π−π0 ) = ππ πΈ (⋅ | ℘(π−1)π0 +π ) , π-π.π., and that if 1 < π < ∞, then In particular, π∗π πΈ (⋅ | ℘π−π0 ) = πΈ (π∗π ⋅ | ℘(π−1)π0 +π ) , (27) π = 1, 2, . . . . (24) σ΅©σ΅© π σ΅©σ΅© 1 lim σ΅©σ΅©σ΅© ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (⋅) π→∞ σ΅© σ΅©σ΅© π π=1 σ΅©σ΅©σ΅© σ΅©σ΅© Μ ∗ −π = 0, (⋅) σ΅©σ΅© [π ]π−π0 σ΅©σ΅© σ΅©πΏ π (π) (28) and that if π(π) < ∞ and π ∈ πΏ(π × Ω∗ )log+ πΏ(π × Ω∗ ), then σ΅©σ΅© π σ΅©σ΅© 1 lim σ΅©σ΅©σ΅© ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (⋅) π→∞ σ΅© σ΅©σ΅© π π=1 σ΅©σ΅©σ΅© σ΅©σ΅© Μ ∗ −π = 0. (⋅) σ΅©σ΅© [π ]π−π0 σ΅©σ΅© σ΅©πΏ 1 (π) (29) Abstract and Applied Analysis 5 Proof. As before, we define ππ(π , π∗ ) = [π[π∗ ]π πππ0 π∗ ](π ) for π ∈ πΏ 1 (π ⊗ π∗ ). From the norm conditions of {π[π∗ ]π : π∗ ∈ Ω∗ } in Theorem 7, it turns out that π is a linear operator on πΏ 1 (π ⊗ π∗ ) with βπβπΏ 1 (π⊗π∗ ) ≤ 1 and βπβπΏ ∞ (π⊗π∗ ) ≤ 1. Then, It follows from the Riesz convexity theorem that βπβπΏ π (π⊗π∗ ) ≤ 1 for all π with 1 < π < ∞. Thus, for any π ∈ πΏ π (π ⊗ π∗ ) (1 ≤ π < ∞), we can apply Dunford and Schwartz’s ergodic theorem [18] to ensure the existence of a Μ ∈ πΏ (π ⊗ π∗ ) such that function π π 1 π Μ (π , π∗ ) lim ∑ ππ π (π , π∗ ) = π π→∞π π=1 π ⊗ π-a.e., (30) Μ = π, Μ ππ σ΅©σ΅© π σ΅©σ΅© σ΅©σ΅© 1 σ΅© Μ σ΅©σ΅©σ΅© lim σ΅©σ΅©σ΅© ∑ ππ π − π = 0, σ΅©σ΅© π → ∞σ΅© π σ΅©σ΅© π=1 σ΅©σ΅©πΏ (π⊗π∗ ) π + ∗ (31) ∗ and that if π(π) < ∞ and π ∈ πΏ(π × Ω )log πΏ(π × Ω ), then σ΅©σ΅© π σ΅©σ΅© σ΅©σ΅© 1 σ΅©σ΅© π Μ σ΅© = 0. lim σ΅© ∑ π π − πσ΅©σ΅©σ΅© π → ∞σ΅© σ΅©σ΅© π π=1 σ΅©σ΅© σ΅© σ΅©πΏ 1 (π⊗π∗ ) (32) Now, adapting the (almost) same argument as used in the proof of Theorem 4, we can find that ∗ π ⊗ π -a.e. (or, ∈ πΏ 1 (π ⊗ π∗ )) , (34) where |π| denotes the linear modulus of π (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 [4] by taking π0 = π = 1 and to the π random parameter result of GΕadysz [5] by taking π0 = 1. (π) : Theorem 8. Let π1 , . . . , ππ be π positive integers. Let {π[π ∗] ππ π∗ ∈ Ω∗ }, π = 1, 2, . . . , π, be strongly F(π) ππ (π = 1, 2, . . . , π)measurable families of linear contractions on πΏ 1 (π) with (π) βπ[π ≤ 1. For each π = 1, 2, . . . , π, we set ∗] β ππ π 1 π πΏ ∞ (π) π Μ ∗ (π ) π-a.e. as π → ∞, . . . , π → ∞ converge to π 1 π [π ]π−1 Μ independently. Here, π ∗ (⋅) means that the number of [π ]π−1 random parameters is at most π − 1. (π) ππ β (π , π∗ ) = [π[π ∗ ] βπππ π∗ ] (π ) , π π = 1, 2, . . . , π. π (π) (π) (π) ππ (π, [π∗ ]ππ ) = π[π , ∗ ] π[ππ∗ ] ⋅ ⋅ ⋅ π π−1 ∗ [π π ] π π π ππ π = 1, 2, ⋅ ⋅ ⋅ , π π 1 π π 1 1 π ∑ ⋅ ⋅ ⋅ ∑ π1 1 ⋅ ⋅ ⋅ ππππ π = πΈ1 ⋅ ⋅ ⋅ πΈπ π π1 ⋅ ⋅ ⋅ ππ π =1 π =1 (38) π ⊗ π∗ -a.e. converge to πΈ1 ⋅ ⋅ ⋅ πΈπ π almost everywhere on π × Ω∗ as π1 → ∞, . . . , ππ → ∞ independently, where each πΈπ is a projection of πΏ π (π⊗π∗ ) onto the manifold π(πΌ−ππ ) = {π ∈ πΏ π (π⊗π∗ ) : ππ π = π} with σ΅©σ΅© ππ σ΅©σ΅© σ΅©σ΅© 1 σ΅©σ΅© ππ σ΅© lim σ΅©σ΅©σ΅© ∑ ππ π − πΈπ πσ΅©σ΅©σ΅©σ΅© = 0, ππ → ∞σ΅© π σ΅©σ΅© σ΅©σ΅© π ππ =1 σ΅©πΏ π (π⊗π∗ ) π = 1, 2, . . . , π. (39) Note here that πΈπ π ∈ π (πΌ − ππ ) , πΈπ−1 πΈπ π ∈ π (πΌ − ππ−1 ) , . . . , πΈ1 ⋅ ⋅ ⋅ πΈπ π ∈ π (πΌ − π1 ) . (40) Thus, by Lemma 6 and the (πΏ π ) martingale convergence theorem, we find that excepting a π∗ -null set πΈπ π (π , π∗ ) = (πΈπ π)[π∗ ] ππ −1 π-a.e., (π ) πΈπ−1 πΈπ π (π , π∗ ) = (πΈπ−1 πΈπ π)[π∗ ] πΈ1 ⋅ ⋅ ⋅ πΈπ π (π , π∗ ) = (πΈ1 ⋅ ⋅ ⋅ πΈπ π)[π∗ ] (π ) π-a.e., max(π1 ,...,ππ )−1 (ππ (0, [π∗ ]ππ ) = the identity operator) . (37) Each ππ turns out to be a linear contraction on πΏ 1 (π ⊗ π∗ ) with βππ βπΏ ∞ (π⊗π∗ ) ≤ 1. So, it follows from the Riesz convexity theorem that βππ βπΏ π (π⊗π∗ ) ≤ 1 for 1 < π < ∞. Hence, from Dunford-Schwartz’s ergodic theorem [18], we have that for every π ∈ πΏ π (π ⊗ π∗ ) the multiple averages max(ππ−1 ,ππ )−1 π (36) × (ππ , [ππ1 +⋅⋅⋅+ππ−1 π∗ ]π ) π (π ) (33) Note here that if π ∈ πΏ π (π ⊗ π∗ ), 1 < π < ∞, (or π ∈ πΏ(π × Ω∗ )log+ πΏ(π × Ω∗ )), then 1 π σ΅¨ σ΅¨ sup ∑ |π|π σ΅¨σ΅¨σ΅¨πσ΅¨σ΅¨σ΅¨ ∈ πΏ π (π ⊗ π∗ ) π≥1 π π=1 π π 1 1 ∑ ⋅ ⋅ ⋅ ∑ π1 (π1 , [π∗ ]π1 ) ⋅ ⋅ ⋅ ππ π1 ⋅ ⋅ ⋅ ππ π =1 π =1 Proof. As already seen above, we can define the operators π1 , . . . , ππ on πΏ 1 (π ⊗ π∗ ) as follows: for β ∈ πΏ 1 (π ⊗ π∗ ) and that if 1 < π < ∞, then Μ (π , π∗ ) = π Μ (π , [π∗ ] ) π π−π0 Then, for every π ∈ πΏ 1 (π), there exists a π∗ -null set π∗ such Μ ∗ ∈ that for every π∗ ∈ Ω∗ − π∗ , there exists a function π [π ]π−1 πΏ 1 (π) with π = max(π1 , . . . , ππ ) such that the multiple averages (π ) π-a.e. (35) (41) 6 Abstract and Applied Analysis Finally, observe that excepting a π∗ -null set, we get (π) (π) (π) πππ π (π , π∗ ) = π[π πππ π∗ (π ) ∗ ] π[ππ∗ ] ⋅ ⋅ ⋅ π π−1 ∗ [π π ] π π π ππ π = ππ (π, [π∗ ]ππ ) ππ∗ (π ) π = 1, 2, . . . , π-a.e., π = 1, 2, . . . , π, (42) ππ (ππ , [ππ1 +⋅⋅⋅+ππ−1 π∗ ]π ) (43) π π-a.e. π π (1) (1) 2 π1 1 π2 2 π (π , π∗ ) = π[π ⋅ ⋅ ⋅ π[π ∗] π1 −1 π∗ ] [π2 π] π1 ∗ (π ) π π π (1) (1) (2) = π[π ⋅ ⋅ ⋅ π[π ⋅⋅⋅ ∗] π1 −1 π∗ ] π[ππ1 π∗ ] π π1 1 π2 (2) π[π π1 +π2 −1 π∗ ] πππ1 +π2 π∗ (π ) (44) π2 = π1 (π1 , [π∗ ]π1 ) π2 (π2 , [ππ1 π∗ ]π ) π-a.e. To complete the proof of the above equality, assume that (43) has already been established for the π−1 operators π2 , . . . , ππ . Then, it is easily verified by the induction hypothesis that (43) holds for the π operators π1 , . . . , ππ . Hence, taking π(π , π∗ ) = π(π )π(π∗ ) (π(⋅) = 1), the theorem follows from the above arguments. In particular, if the operators in question are commutative then we have the following. (π) : Theorem 9. Let π1 , . . . , ππ be π positive integers. Let {π[π ∗] ππ π∗ ∈ Ω∗ }, π = 1, 2, . . . , π, be a strongly F(π) ππ (π = 1, 2, . . . , π)measurable commuting family of linear contractions on πΏ 1 (π) (π) with βπ[π ≤ 1. Then, for every π ∈ πΏ 1 (π), there exists ∗] β ππ πΏ ∞ (π) a π∗ -null set π∗ such that for every π∗ ∈ Ω∗ − π∗ , there exists Μ ∗ ∈ πΏ (π) with π = max(π , . . . , π ) such that a function π 1 1 π [π ]π−1 lim 1 π → ∞ ππ π−1 π−1 π1 =0 ππ =0 ∑ ⋅ ⋅ ⋅ ∑ π1 (π1 , [π∗ ]π1 ) ⋅ ⋅ ⋅ ππ (ππ , [ππ1 +⋅⋅⋅+ππ−1 π∗ ]π ) π (π ) (45) π Μ ∗ (π ) =π [π ]π−1 πΏ ∞ (π) π π 1 π π1 +⋅⋅⋅+ππ−1 ππ (ππ , [π (46) ∗ π ]π ) π (π ) π Μ ∗ (π ) π-a.e. as π’(π) (= (π , . . . , π ) ≥ (1, . . . , converge to π 1 π [π ]π−1 1)) → ∞ in a sector of Zπ+ . 2 × π (π , π∗ ) ππ π 1 1 ∑ ⋅ ⋅ ⋅ ∑ π1 (π1 , [π∗ ]π1 ) ⋅ ⋅ ⋅ π1 ⋅ ⋅ ⋅ ππ π =1 π =1 π1 1 π∗ ∈ Ω∗ }, π = 1, 2, . . . , π, be strongly F(π) ππ (π = 1, 2, . . . , π)measurable commuting family of linear contractions on πΏ 1 (π) (π) with βπ[π ≤ 1. Then, for every π ∈ πΏ 1 (π), there exists ∗] β a π∗ -null set π∗ such that for every π∗ ∈ Ω∗ − π∗ , there exists Μ ∗ ∈ πΏ (π) with π = max(π , . . . , π ) such that a function π 1 1 π [π ]π−1 the multiple averages In fact, we have for two operators π1 and π2 π (π) : Theorem 10. Let π1 , . . . , ππ be π positive integers. Let {π[π ∗] ππ π π1 1 ⋅ ⋅ ⋅ ππππ π (π , π∗ ) = π1 (π1 , [π∗ ]π1 ) ⋅ ⋅ ⋅ × ππ∗ (π ) πΆ0 > 0 such that the ratios ππ /ππ are bounded by πΆ0 for 1 ≤ π, π ≤ π and all π’(π) = (π1 , . . . , ππ ) (see [20, page 203]). Appealing to Brunel-Dunford-Schwartz’ theorem (see [20, Theorem 3.5, page 215]), we find that the multiple averages (38) converge π-a.e. in a sector Zπ+ . In addition, in this case, Theorem 8 implies that the limit function depends essentially only on the π − 1 random parameters. Hence, we have the following. π-a.e. In passing, we make mention of the a.e. convergence for sectorial restricted random averages. We say that a sequence π’(π) ⊂ Zπ+ remains in a sector of Zπ+ if there is a constant A sub-Markovian operator π∗ on πΏ ∞ (π) means that π∗ is a positive linear contraction on πΏ ∞ (π) with 1 subinvariant under π∗ (i.e., π∗ 1 ≤ 1) such that limπ → ∞ π∗ βπ = 0 π-a.e. for any (decreasing) sequence (βπ )π≥1 ⊂ πΏ ∞ (π) with limπ → ∞ βπ = 0 π-a.e. Let π denote the positive linear contraction on πΏ 1 (π) with π∗ as the adjoint operator of π. When π = ππ with a π-subinvariant function π ≥ 0 (π =ΜΈ 0), π is called a π-subinvariant measure. Here, the subinvariant function is not necessarily integrable. ∗ : π∗ ∈ Ω∗ } be a strongly F∗π Now, let Π∗ = {π[π ∗] π measurable family of sub-Markovian operators on πΏ ∞ (π). Let Π = {π[π∗ ]π : π∗ ∈ Ω∗ } be a strongly F∗π -measurable family of positive linear contractions on πΏ 1 (π), where each π[π∗ ]π is the adjoint operator of the corresponding operator ∗ π[π ∗ ] . Then, there exists a positive linear contraction π on π πΏ 1 (π ⊗ π∗ ) such that for any π ∈ πΏ 1 (π ⊗ π∗ ), ππ (π , π∗ ) = [π[π∗ ]π πππ0 π∗ ] (π ) π × π∗ -a.e (47) In addition, if βπ[π∗ ]π βπΏ (π) ≤ 1 for all π∗ ∈ Ω∗ , π is ∞ also a contraction on πΏ ∞ (π ⊗ π∗ ) (cf. [10, 14]). The operator π extends uniquely to a positive linear transformation on the class of nonnegative π½⊗F∗ -measurable functions defined on π × Ω∗ (see [20, page 51]). Lemma 11. Let the measure π be finite. Assume that π is πΈ(π⋅ | ℘0 )-subinvariant. Then, π turns out to be a positive Dunford-Schwartz operator on πΏ 1 (π ⊗ π∗ ). Abstract and Applied Analysis 7 Proof. Since π is assumed to be πΈ(π⋅ | ℘0 )-subinvariant, the function 1 (∈ πΏ 1 (π ⊗ π∗ )) is πΈ(π⋅ | ℘0 )-subinvariant. According to Lemma 1.5 of WosΜ [21], we get π1 = πΈ (π1 | ℘0 ) , (48) so that 1 is also π-subinvariant. Thus, for π ∈ πΏ 1 (π ⊗ π∗ ) ∩ πΏ ∞ (π ⊗ π∗ ), σ΅©σ΅© σ΅©σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© σ΅© (49) σ΅©σ΅©ππσ΅©σ΅©∞ ≤ πσ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©∞ = σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©∞ π1 ≤ σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©∞ . Since π is a positive contraction on πΏ 1 (π ⊗ π∗ ), and π is a positive Dunford-Schwartz operator on πΏ 1 (π ⊗ π∗ ). 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 WosΜ [21] (In proving the key Lemma 1.5 of [21], WosΜ showed that if π ∈ πΏ 1 (π ⊗ π∗ ) does not depend essentially on parameter π∗ , then ππ = πΈ(ππ | ℘0 ). Using this equality, he deduced π1 ≤ 1 from πΈ(π1 | ℘0 ) ≤ 1 in order to prove Theorems 2.8 and 3.4 of [21]. But obviously, 1 does not necessarily belong to πΏ 1 (π ⊗ π∗ ) in case where π is π-finite. His arguments are of course correct in the case that π is finite (see, e.g., conditions (i) and (ii) in Theorem 12)). Theorem 12. Let Π = {π[π∗ ]π : π∗ ∈ Ω∗ } be a strongly F∗π measurable family of positive linear contractions on πΏ 1 (π), where each π[π∗ ]π is the adjoint operator of the corresponding ∗ sub-Markovian operator π[π ∗ ] . Assume either of the following π conditions: (i) π is finite and πΈ(π⋅ | ℘0 )-subinvariant, (ii) π is π-finite and βπ[π∗ ]π βπΏ (π) ≤ 1 for all π∗ ∈ Ω∗ . ∞ Then, for any π ∈ πΏ π (π ⊗ π∗ ), 1 ≤ π < ∞, there exists a π∗ -null set π∗ such that for every π∗ ∈ Ω∗ − π∗ , there exists Μ ∗ a function π [π ]π−π ∈ πΏ π (π) such that 0 π 1 lim ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) π→∞π π=1 Μ ∗ = π [π ]π−π (π ) 0 3. (πΆ, πΌ)-Type Random Ergodic Theorems In this section, we establish a (πΆ, πΌ)-type random ergodic theorem for measure-preserving transformations on π. In this section, we assume that (π, π½, π) is a probability space. For real πΌ > −1 and π(= 0, 1, 2, . . .), let π΄πΌπ be the (πΆ, πΌ) coefficient of order πΌ, which is defined by the generating function ∞ 1 = π΄πΌπ π‘π , ∑ (1 − π‘)πΌ+1 π=0 0 < π‘ < 1. (53) Then, we can easily check that π΄πΌπ is decreasing in π for −1 < πΌ < 0 and increasing in π for πΌ > 0. For πΌ > −1, we have π΄πΌπ > 0, π΄πΌ0 = 1, π΄0π = 1, π΄πΌ−1 = π΄πΌπ − π΄πΌπ−1 , π π π+πΌ π΄πΌπ = ∑ π΄πΌ−1 π−π = ( π ) , π=0 π΄πΌπ ∼ ππΌ Γ (πΌ + 1) (π σ³¨→ ∞) , (54) and moreover, ππΌ (π + 1)πΌ ≤ π΄πΌπ ≤ Γ (πΌ + 1) Γ (πΌ + 1) π΄πΌ−1 π ππΌ−1 ≤ Γ (πΌ) for 0 < πΌ ≤ 1, (55) for π > 0. In general, using Hille’s theorem (see [22, Theorem 8]), we have the following. (50) Theorem 13. Let π0 be any fixed integer with 1 ≤ π0 ≤ π. Let {π[π∗ ]π : π∗ ∈ Ω∗ } be a strongly F∗π -measurable family of linear contractions on πΏ π (π) (1 ≤ π < ∞). Let πΌ > 0 be fixed. If for all π ∈ πΏ π (π ⊗ π∗ ) there exists a π∗ -null set π∗ such that ∗ for any π∗ ∈ Ω∗ − π∗ there exists a function π[π ∈ πΏ π (π) ∗] π−π0 such that (51) (i) limπ → ∞ (1/π΄πΌπ ) ∑ππ=0 π΄πΌ−1 π−π π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ Μ ∗ (π ) for π-almost all π[πππ0 π∗ ]π ππ(π+1)π0 π∗ (π ) = π [π ]π−π0 π ∈ π, π-a.e., and that if 1 < π < ∞, then σ΅©σ΅© π σ΅©σ΅© 1 lim σ΅©σ΅©σ΅© ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (⋅) π→∞ σ΅©π σ΅©σ΅© π=1 σ΅©σ΅© σ΅©σ΅© Μ ∗ σ΅©σ΅© −π =0 (⋅) [π ]π−π0 σ΅©σ΅© σ΅©σ΅©πΏ (π) π Proof of Theorem 1. In view of Lemma 11, either condition (i) or condition (ii) guarantees that π is a positive DunfordSchwartz operator on πΏ 1 (π ⊗ π∗ ). Hence, we can apply Theorem 7 to conclude that Theorem 12 follows. and that if π is finite and π ∈ πΏ(π × Ω∗ )log+ πΏ(π × Ω∗ ), then σ΅©σ΅© π σ΅©σ΅© 1 lim σ΅©σ΅©σ΅© ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (⋅) π→∞ σ΅© σ΅©σ΅© π π=1 (52) σ΅©σ΅© σ΅©σ΅© Μ −π[π∗ ]π−π (⋅) σ΅©σ΅©σ΅© = 0. σ΅©σ΅© 0 σ΅©πΏ 1 (π) (π+1) π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ (ii) limπ → 1+0 (π − 1) ∑∞ π=0 π Μ ∗ (π ), π[πππ0 π∗ ]π ππ(π+1)π0 π∗ (π ) = π [π ]π−π 0 πΌ (iii) limπ → ∞ (1/π )π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[πππ0 π∗ ]π ππ(π+1)π0 π∗ (π ) = 0, for π-almost all π ∈ π. Conversely, if (ii) holds but (iii) be replaced by the condition that there exists a π½⊗F∗ -measurable 8 Abstract and Applied Analysis ∗ = (π ⋅ π) (π , π∗ ) function π(π , π∗ ) finite almost everywhere such that excepting a π∗ -null set σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ 1 π π½−1 σ΅¨σ΅¨ ∑ π΄ π[π∗ ] π[ππ0 π∗ ] ⋅ ⋅ ⋅ π[πππ0 π∗ ] ππ(π+1)π0 π∗ (π )σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ π½ π−π π π π σ΅¨σ΅¨ σ΅¨σ΅¨ π΄ π π=0 ≤ ππ∗ (π ) π ⊗ π∗ -a.e., σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ 1 π ∗ σ΅¨σ΅¨σ΅¨ sup σ΅¨σ΅¨σ΅¨ πΌ ∑ π΄πΌ−1 (π ⋅ π) (π , π ) σ΅¨σ΅¨ π−π σ΅¨σ΅¨ π≥0 σ΅¨σ΅¨σ΅¨ π΄ π π=0 σ΅¨ ∈ πΏ π (π ⊗ π∗ ) (⊂ πΏ 1 (π ⊗ π∗ )) , (56) π-a.e., 1 π πΌ−1 ∑ π΄ π−π (π ⋅ π) (Φ∗π (π , π∗ )) π → ∞ π΄πΌ π π=0 lim for all π ≥ 0, where π½ ≥ 0, then (i) holds for all πΌ > π½. ∗ = (π ⋅ π) (π , π∗ ) Note here that in Theorem 13, (ii) and (iii) do not necessarily imply (i) in general. Theorem 14. Let π0 be any fixed integer with 1 ≤ π0 ≤ π and {π[π∗ ]π : π∗ ∈ Ω∗ } be a π½ ⊗ F∗π -measurable family of measurepreserving transformations on π. Let 0 < πΌ < 1, πΌπ > 1, and π ∈ πΏ π (π). Then, there exists a π∗ -null set π∗ such that for ∗ every π∗ ∈ Ω∗ −π∗ , there exists a function π[π (⋅) ∈ πΏ π (π) ∗] π−π0 such that (59) Furthermore, applying Lemma 6 to the operator π∗ induced by Φ∗ , it follows that ∗ = lim (π − 1) ∑ π−(π+1) π (π[πππ0 π∗ ]π ⋅ ⋅ ⋅ π[π∗ ]π π ) = π=0 ∗ π[π ∗] π−π0 (π ) π-π.π., σ΅¨σ΅¨ σ΅¨σ΅¨ 1 π lim {∫ σ΅¨σ΅¨σ΅¨ πΌ ∑ π΄πΌ−1 π−π π (π[πππ0 π∗ ]π ⋅ ⋅ ⋅ π[π∗ ]π π ) π→∞ π σ΅¨σ΅¨σ΅¨ π΄ π π=0 ∗ −π[π ∗] π−π0 (60) so that to complete the proof of the theorem, we may take ∗ ∗ π[π (π ) = (π ⋅ π)[π∗ ]π−π (π ) . ∗] π−π 0 ∞ π → 1+0 ∗ (π ⋅ π) (π , π∗ ) = (π ⋅ π) (π , [π∗ ]π−π0 ) , π 1 lim ∑ π΄πΌ−1 π−π π (π[πππ0 π∗ ]π ⋅ ⋅ ⋅ π[π∗ ]π π ) π → ∞ π΄πΌ π π=0 π ⊗ π∗ -a.e. 0 (61) Remark 15. For example, if π0 = π = 1 in Theorem 14, then ∗ ∗ π[π ∗ ]π−π (π ) = π (π ). It is worthwhile to note that if 0 < πΌ < 1 0 and πΌπ = 1 (so, π = πΌ−1 > 1), then the pointwise (πΆ, πΌ)convergence for Φ∗ does not hold in general (see [24]). For the case of a positive linear contraction on πΏ 1/πΌ (π ⊗ π∗ ), see Irmisch [25]. In particular, applying Irmisch’s theorem to subMarkovian operators, we have the following. 1/π σ΅¨σ΅¨π σ΅¨σ΅¨ σ΅¨ (π ) σ΅¨σ΅¨ ππ} = 0. σ΅¨σ΅¨ σ΅¨ (57) Proof. Define the skew product Φ∗ of π (the one-sided shift transformation of Ω∗ ) and {π[π∗ ]π } as follows: Theorem 16. Let Π = {π[π∗ ]π : π∗ ∈ Ω∗ } be a strongly F∗π measurable family of positive linear contractions on πΏ 1 (π), where each π[π∗ ]π is the adjoint operator of the corresponding ∗ sub-Markovian operator π[π ∗ ] . Assume that π is πΈ(π⋅ | ℘0 )π subinvariant. Let 0 < πΌ < 1, πΌπ > 1 and π ∈ πΏ π (π). Then, there exists a π∗ -null set π∗ such that for every π∗ ∈ Ω∗ − π∗ , ∗ (⋅) ∈ πΏ π (π) such that there exists a function π[π ∗] π−π 0 ∗ ∗ π0 ∗ Φ (π , π ) = (π[π∗ ]π π , π π ) , ∗ ∗ (π , π ) ∈ π × Ω . (58) Let (π ⋅ π)(π , π∗ ) = π(π ) ⋅ π(π∗ ), where π(π∗ ) = 1 for all π∗ ∈ Ω∗ . Then, (π ⋅ π) ∈ πΏ π (π ⊗ π∗ ). Since πΏ π (π ⊗ π∗ ) is reflexive, we see from Yosida-Kakutani’s mean ergodic theorem [23] and DeΜniel’s theorem [24] that there exists a function (π ⋅ π)∗ ∈ πΏ π (π ⊗ π∗ ) such that 1 π πΌ−1 ∑ π΄ π−π π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[πππ0 π∗ ]π π → ∞ π΄πΌ π π=0 lim ∞ = lim (π − 1) ∑ π−(π+1) π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[πππ0 π∗ ]π π → 1+0 π=0 ∗ = π[π (π ) ∗] π−π 0 π-π.π. (62) σ΅¨σ΅¨ { σ΅¨σ΅¨ 1 π lim { β¬ σ΅¨σ΅¨σ΅¨ πΌ ∑ π΄πΌ−1 (π ⋅ π) (Φ∗π (π , π∗ )) π→∞ σ΅¨σ΅¨ π΄ π π=0 π−π {π×Ω∗ σ΅¨ 1/π σ΅¨σ΅¨π σ΅¨σ΅¨ ∗ −(π ⋅ π) (π , π ) σ΅¨σ΅¨σ΅¨ ππ ⊗ π } = 0, σ΅¨σ΅¨ σ΅¨ ∗ ∗ (π ⋅ π) (Φ∗ (π , π∗ )) ∗ Proof. In view of Lemma 11, π is a positive linear contraction on πΏ 1 (π ⊗ π∗ ) as well as on πΏ ∞ (π ⊗ π∗ ). Thus, it follows from the Riesz convexity theorem that βπβπΏ π (π⊗π∗ ) ≤ 1 for 1 < π < ∞. Therefore, we reach the assertion of Theorem 16 through Theorems 7 and 13 appealed to Irmisch’s theorem [25]. Remark 17. The relations between the random ergodic limit functions and the random parameters have been investigated Abstract and Applied Analysis 9 (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 [26]). Let π = (ππ‘ )π‘≥0 (ππ‘ = ππ‘ (π), π0 = 0) be a Brownian motion (or Wiener process) on a probability space (Ω, F, π). This process has independent increments; that is, for arbitrary π‘1 < π‘2 < ⋅ ⋅ ⋅ < π‘π , the random variables ππ‘2 − ππ‘1 , . . . , ππ‘π − ππ‘π−1 are independent. In fact, since the process is Gaussian with πΈ(ππ‘ ) = 0 and πΈ(ππ‘ ππ ) = min(π‘, π ) by definition, it is sufficient to verify only that the increments are uncorrelated. Thus, if π < π‘ < π’ < V, then πΈ ((ππ‘ − ππ ) (πV − ππ’ )) = 1 ((π‘ − π ) + (V − π’) − |π‘ − π + V − π’|) = 0. 2 (63) Let {ππ‘ : π‘ ≥ 0} be an arbitrary ergodic measurable semiflow on a finite measure space (π, π½, π). Then, Anzai’s result may be stated as follows: for any π ∈ πΏ 1 (π) and for almost all π ∈ Ω, there exists a null set π = ππ (π) ∈ F such that 1 πΌ ∫ π (πππ‘ (π) π ) ππ‘ = ∫ π (π ) ππ πΌ→∞πΌ 0 π lim (64) holds for any π ∈ π − π. This is an immediate consequence of the ergodicity of the measure-preserving skew product semiflow {Ψπ‘ π‘ ≥ 0} defined by Ψπ‘ (π, π ) = (ππ‘ π,πππ‘ (π) π ), where {ππ‘ : π‘ ≥ 0} is the ergodic semiflow on Ω given by ππ (ππ‘ π) = ππ+π‘ (π). It is an interesting and important problem to generalize Anzai’s result for Brownian motions to the case of contraction operator quasisemigroups in πΏ 1 (π) associated with {ππ‘ }. We do not discuss it in the present paper. 4. Applications to Nonlinear Random Ergodic Theorems π[π∗ ]π = π[π∗ ]π + [(πΌ − π) π]π∗ , π∗ ∈ Ω∗ , (66) where ππ (π , π∗ ) = [π[π∗ ]π πππ0 π∗ ] (π ) , (67) for π ∈ πΏ 1 (π ⊗ π∗ ) (cf. [11, 27]). For each π ∈ πΏ 1 (π ⊗ π∗ ), we define a sequence of random functions {ππ (π, π∗ ) : π∗ ∈ Ω∗ }, π = 0, 1, . . ., in πΏ 1 (π) inductively by ππ (0, π∗ ) = ππ∗ , ππ (1, π∗ ) = ππ∗ + π[π∗ ]π ππ (0, ππ0 π∗ ) , (68) .. . ππ (π + 1, π∗ ) = ππ∗ + π[π∗ ]π ππ (π, ππ0 π∗ ) . Theorem 18. Let {ππ (π, π∗ ) : π∗ ∈ Ω∗ } be the sequence of random functions associated with π ∈ πΏ 1 (π ⊗ π∗ ) which is determined by a random affine system {(π[π∗ ]π , π[π∗ ]π , [(πΌ − π)π]π∗ , ππ0 ) : π∗ ∈ Ω∗ } given in πΏ 1 (π) with βπ[π∗ ]π βπΏ (π) ≤ ∞ 1. Then, there exists a π∗ -null set π∗ such that for any π∗ ∈ Μ ∗ Ω∗ − π∗ , there exists a function π [π ]π−π ∈ πΏ 1 (π) such that 0 lim π→∞ The random ergodic theorems obtained above can be applied to the nonlinear random ergodic theorems for affine systems (see [27]). An affine operator π on πΏ 1 (π) is an operator of the type ππ = ππ + β, where π is a linear contraction on πΏ 1 (π), and β is a fixed element of πΏ 1 (π). Then, π is nonlinear and nonexpansive. The fixed points of π are solutions of Poisson’s equation for π, which is (πΌ − π)π = β. When we assume π to be mean ergodic, the averages 1 π 1 π π−1 1 π π ∑ π π = ∑ ππ π + ∑ ∑ ππ β π π=1 π π=1 π π=1 π=0 π)π will yield almost everywhere convergence of the averages of ππ π for any π ∈ πΏ 1 (π). Now, let {(π[π∗ ]π , π[π∗ ]π , ππ∗ , ππ0 ) : π∗ ∈ Ω∗ } be a random affine system on πΏ 1 (π) such that {π[π∗ ]π : π∗ ∈ Ω∗ } is a strongly measurable family of linear contractions on πΏ 1 (π) as well as on πΏ ∞ (π) and such that for some π ∈ πΏ 1 (π ⊗ π∗ ), ππ (π, π∗ ) (π ) π+1 Μ ∗ =π [π ]π−π (π ) 0 π-π.π. (69) Proof. It follows that there exists a π∗ -null set π1∗ such that for any π∗ ∈ Ω∗ − π1∗ , ππ (π, π∗ ) (π ) = ππ∗ (π ) π + ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) π=1 (65) converge if and only if β ∈ (πΌ − π)πΏ 1 (π). If β ∈ (πΌ − π)πΏ 1 (π), then there exists a unique π ∈ (πΌ − π)πΏ 1 (π) such that (πΌ − π)π = β, and the limit of (1/π) ∑ππ=1 ππ π is πΈπ+π, where πΈπ is the limit of (1/π) ∑ππ=1 ππ π. Therefore, iterating ππ = ππ+(πΌ− − π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) π−1 + ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) π=1 + ππ∗ (π ) . (70) 10 Abstract and Applied Analysis 5. Examples Moreover, we have ∗ lim π→∞ ∗ π (π , π ) π (π , π ) = lim = 0 π-a.e., π→∞ π + 1 π+1 lim π→∞π 1 π ∗ π π0 ∗ ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) + 1 [π ]π [π π ]π (71) = 0 π-a.e. Therefore, by Theorem 7, there exists a π∗ -null set π2∗ such ∗ , that for every π∗ ∈ Ω∗ − π2∗ , there exist functions π[π ∗] π−π0 ∗ π[π∗ ]π−π ∈ πΏ 1 (π) such that 0 lim ππ (π, π∗ ) (π ) π→∞ π+1 ∗ = π[π (π ) ∗] π−π 0 ∗ + π[π (π ) ∗] π−π 0 (72) π-π.π. Hence, the theorem follows by putting π∗ = π1∗ ∪ π2∗ and Μ (π , [π∗ ] ) = π∗ (π , [π∗ ] ) + π∗ (π , [π∗ ] ) . (73) π π−π0 π−π0 π−π0 As far as we are concerned with the ergodic behaviors of CesaΜro-type processes for nonexpansive operators on πΏ π , one can only expect weak convergence in general. In fact, the pointwise convergence of the (πΆ, 1) averages of nonlinear and nonexpansive operators on πΏ π may fail to hold. In addition, these (πΆ, 1) averages do not need converge in the strong operator topology of πΏ π (see [27, 28]). The socalled nonlinear sums introduced by Wittmann [29] 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. [27]). Under the above setting, observe that Example 1. Let {ππ : π ∈ Ω} be a π½ ⊗ F-measurable family of π-measure-preserving transformations on π, and let π be πmeasure-preserving transformation on Ω. Then, for any π ∈ πΏ 1 (π), there exists a π-null set π∗ ∈ π½ such that for each π ∈ Ω − π∗ , there exists a function ππ∗ ∈ πΏ 1 (π) such that 1 π ∑ π (πππ−1 π ⋅ ⋅ ⋅ ππ π ) = ππ∗ (π ) π→∞π π=1 lim π Φ (π , π) = (ππ π , ππ) , + ∑ [π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ ] (π ) + π[π∗ ]π (π ) , π {π 1 } = ⋅ ⋅ ⋅ = π {π π } = π {π1 } = ⋅ ⋅ ⋅ = π {π2π } = ππ2π−1 π π = { 1 π ∑ π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[π(π−1)π0 π∗ ]π ππππ0 π∗ (π ) π→∞π π=1 lim = ∗ π[π ∗] π−π0 (π ) π-π.π. π π+1 , π 1 , (75) (π: integer, π ≥ 2) , 1 2π (π: integer, 2π ≥ π) , if 1 ≤ π ≤ π − 1, , if π = π, ππ2π = ππ−12π−1 , (78) π = 1, 2, . . . , π, π = 1, 2, . . . , π, if 1 ≤ π ≤ 2π − 1, if π = 2π. Thus, for instance, if we consider a function π(π , π) defined by = 0, Theorem 19. Let {(π[π∗ ]π , π[π∗ ]π , [(πΌ − π)π]π∗ , ππ0 ) : π∗ ∈ Ω∗ } be a random affine system given in πΏ 1 (π) with βπ[π∗ ]π βπΏ (π) ≤ ∞ 1. If π ∈ πΏ 1 (π ⊗ π∗ ), then there exists a π∗ -null set π∗ such ∗ ∈ that for any π∗ ∈ Ω∗ − π∗ , there exists a function π[π ∗] π−π0 πΏ 1 (π) such that 1 π Ω = {π1 , . . . , π2π } , π = 1, 2, . . . . Then, Theorem 7, together with Theorems 2.1 and 2.2 of [27], yields the following theorem. (77) π = {π 1 , . . . , π π } , πππ (π π ) = π ⋅ πΏππ , π=1 (π , π) ∈ π × Ω. If the skew product transformation Φ is ergodic, then the function ππ∗ (π ) is constant almost everywhere on π × Ω. It is worthwhile to notice that, in general, the limit function ππ∗ (π ) depends on the two variables π and π. To illustrate this, we consider the measure spaces (π, π½, π) and (Ω, F, π) and transformations given by π , πππ = { π+1 π1 , (74) (76) which follows from Birkhoff ’s ergodic theorem applied to the so-called skew product Φ of π and {ππ } defined by [π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[πππ0 π∗ ]π ππ(π+1)π0 π∗ ] (π ) = [π[π∗ ]π π[ππ0 π∗ ]π ⋅ ⋅ ⋅ π[πππ0 π∗ ]π ππ(π+1)π0 π∗ ] (π ) π-a.e., if π = π (π, π = 1, 2, . . . , π) , if π = 1, 2, . . . , π; and π = π + 1, . . . , 2π (79) where πΏππ denotes the Kronecker delta, then the limit function ππ∗ (π ) determined by (76) depends essentially on the two variables π and π (therefore, the limit function ππ∗ (π ) is not necessarily independent of the random variable π.) In fact, one can easily find that under the above setting 1 2π 1 ∑ ππ1 (ππ2π−1 π1 ⋅ ⋅ ⋅ ππ1 π π ) = ππ∗1 (π π ) = , π → ∞ 2π 2 π=1 lim π 1 2π lim ∑ πππ (ππ2π−1 π1 ⋅ ⋅ ⋅ ππ1 π π ) = ππ∗π (π π ) = . π → ∞ 2π 2 π=1 See also Example 3 below. (80) Abstract and Applied Analysis 11 The following example given by GΕadysz [5] will be a great help to understand the subject of this paper. Example 2 (see Gladysz [5]). In this example, we consider the measure spaces (π, π½, π) and (Ω, F, π) taken to be π = Ω = [0, 1), π½ = F = the π-field of Borel sets, and π = π = the Lebesgue measure. Let π be a fixed integer with π ≥ 2, and let π½π , π = 1, 2, . . . , π, be real constants such that 1 π½1 = ⋅ ⋅ ⋅ = π½π−1 = , π 1 π½π = −1 + . π π[π∗ ]π π = π + ∑π½π ππ (82) Take π0 = 1. Then, for a function π ∈ πΏ 1 (π) given by π(π ) = π2πππ = exp(π ), there exists a π∗ -null set π∗ such that for each ∗ (⋅) ∈ πΏ 1 (π) such π∗ ∈ Ω∗ − π∗ , there exists a function π[π ∗] π−1 that π-a.e., (83) where π-a.e. (85) Next, if (for example) π = 5 and π0 = 3, we let π½1 , . . . , π½5 be real constants such that π½1 =ΜΈ 0, π½2 =ΜΈ 0, π½3 = 0, π½4 = −π½1 , π½5 = −π½2 . (86) Using the skew product transformation Θ defined by (87) we have for the function π(π ) = exp(π ), π (Θπ (π , π∗ )) 5 5 5 = exp [π + ∑ π½π ππ + ∑π½π π3+π + ⋅ ⋅ ⋅ + ∑ π½π π3π+π ] (88) π=1 π=1 [ π=1 ] = exp [π + π½1 π1 + π½2 π2 ] , π = 1, 2, . . . . Hence it follows at once that for almost all π∗ ∈ Ω∗ , there exists a function π∗ ∈ πΏ 1 (π ⊗ π∗ ) such that 1 π πΌ−1 ∑ π΄ π−π π (π[π3π π∗ ]5 ⋅ ⋅ ⋅ π[π3 π∗ ]5 π[π∗ ]π π ) π → ∞ π΄πΌ π π=0 lim π∗ (π , [π∗ ]π−1 ) = πΎ ⋅ exp (π ) ⋅ exp [π½1 π1 + ⋅ ⋅ ⋅ + (π½1 + ⋅ ⋅ ⋅ + π½π−1 ) ππ−1 ] , π» (π∗ ) = exp [(π½2 + ⋅ ⋅ ⋅ + π½π ) π1 + ⋅ ⋅ ⋅ + π½π ππ−1 ] , 1 π ∑ π» (ππ π∗ ) π→∞π π=1 ∗ ∗ ∗ = ∫ π» (π ) ππ (π ) 1 π ∑ π (π[π3π π∗ ]5 ⋅ ⋅ ⋅ π[π3 π∗ ]5 π[π∗ ]π π ) π→∞π + 1 π=0 = lim = πΎ = lim Ω∗ (π ) Θ (π , π∗ ) = (π[π∗ ]5 π , π3 π∗ ) , (mod1) , π ∈ π. 1 π ∗ ∑ π (π[ππ−1 π∗ ]π ⋅ ⋅ ⋅ π[π∗ ]π π ) = π[π (π ) ∗] π−1 π→∞π π=1 = π=1 ∗ π[π ∗] π−1 π½1 + π½2 =ΜΈ 0, π=1 lim π → 1+0 (81) Define a π½ ⊗ F∗π -measurable family {π[π∗ ]π : π∗ ∈ Ω∗ } of measure-preserving transformations on π by π ∞ = lim (π − 1) ∑ π−(π+1) π (π[ππ−1 π∗ ]π ⋅ ⋅ ⋅ π[ππ∗ ]π π[π∗ ]π π ) π → 1+0 π=0 ∗ = exp [π + π½1 π1 + π½2 π2 ] = π[π (π ) ∗] 5−3 (π is ergodic) 1 = exp [ (π1 + 2π2 + ⋅ ⋅ ⋅ + (π − 1) ππ−1 )] π ∞ lim (π − 1) ∑ π−(π+1) π (π[π3π π∗ ]5 ⋅ ⋅ ⋅ π[π3 π∗ ]5 π[π∗ ]π π ) ∗ = π[π ∗ ] (π ) 2 π-a.e. ( =ΜΈ 0) . (84) Supplement. Let 0 < πΌ < 1, πΌπ > 1, and π0 = 1. Then, for the function π(π ) = exp(π ) (∈ πΏ π (π)), there exists a π∗ -null set π∗ such that for any π∗ ∈ Ω∗ − π∗ , there exists a function π∗ ∈ πΏ π (π⊗π∗ ) such that (if necessary, apply Hille’s theorem [22]) 1 π πΌ−1 ∑ π΄ π−π π (π[ππ π∗ ]π ⋅ ⋅ ⋅ π[ππ∗ ]π π[π∗ ]π π ) π → ∞ π΄πΌ π π=0 lim 1 π πΌ−1 = lim ∑ π΄ π−π π (π[ππ π∗ ]π ⋅ ⋅ ⋅ π[ππ∗ ]π π[π∗ ]π π ) π→∞π + 1 π=0 (89) Example 3. In the setting of Example 2, let π be an π½measurable function with |π(π )| = 1. Then, for π ∈ πΏ π (π ⊗ π∗ ), 1 ≤ π < ∞, there exists a π∗ -null set π∗ such that for any π∗ ∈ Ω∗ − π∗ , there exists a function π»π∗ ∈ πΏ π (π) such that 1 π ∑ π (π[π∗ ]π π ) π (π[ππ∗ ]π π[π∗ ] π ) ⋅ ⋅ ⋅ π→∞π π=1 lim π (π[ππ−1 π∗ ]π ⋅ ⋅ ⋅ π[π∗ ]π π ) × π (π[ππ π∗ ]π ⋅ ⋅ ⋅ π[π∗ ]π π ) = π»π∗ (π ) , π»π∗ (π ) = π (π[π∗ ]π π ) π»ππ∗ (π[π∗ ]π π ) , (90) 12 Abstract and Applied Analysis for π-almost all π ∈ π. In fact, letting π(π , π∗ ) = π(π ) ⋅ π(π∗ ) with π = 1 ∈ πΏ π (π∗ ) and Φ∗ (π , π∗ ) = (π[π∗ ]π π , ππ∗ ) , (91) we have by Gladysz’s theorem ([5], Satz 5) that there exists a function πΉ ∈ πΏ π (π ⊗ π∗ ) such that π 1 π ∑ π (Φ∗ (π , π∗ )) π (Φ∗2 (π , π∗ )) ⋅ ⋅ ⋅ π→∞π π=1 π[π∗ ]π π = π + ∑ πΏπ ππ lim π=1 π (Φ∗π (π , π∗ )) π (Φ∗π+1 (π , π∗ )) = [π (π , π∗ )] −1 ⋅ πΉ (π , π∗ ) (= π(π )−1 πΉ (π , π∗ )) , are chosen at random, not necessarily with the same distribution but independently). In general the random system {(π[π∗ ]π , π) : π∗ ∈ Ω∗ } of linear contractions on πΏ π (π) as given in Theorem 4 plays a role of such an advance stock of linear contractions on πΏ 1 (π). To illustrate this, we let π = π = [0, 1) and consider the π½ ⊗ F∗π -measurable, π-measurepreserving transformations π[π∗ ]π , π∗ ∈ Ω∗ , defined by (92) πΉ (π , π∗ ) = π (π , π∗ ) πΉ (Φ∗ (π , π∗ )) In this case, the random system {(π[π∗ ]π , π) : π∗ ∈ Ω∗ } is taken as a stock of measure-preserving transformations on π. If π1 , . . . , ππ are linearly independent irrational numbers, then {π[π∗ ]π } is ergodic. Thus, for any π ∈ πΏ 1 (π) with period 1, lim almost everywhere on π × Ω∗ . Hence, we may take π» (π , π∗ ) = π(π )−1 πΉ (π , π∗ ) , (93) π» (π , π∗ ) = π (π[π∗ ]π π ) π» (Φ∗ (π , π∗ )) π → 1+0 π ⊗ π∗ -a.e. (94) Moreover, since π is also π½ ⊗ F∗π -measurable and π» ∈ πΏ 1 (π ⊗ π∗ ) if we consider the function π π (π , π1 , . . . , ππ ) = π (π + ∑ π½π ππ ) (95) π=1 Then, by Gladysz’s lemma ([5], Hilfssatz 3) there exists a π½ ⊗ F∗π−1 -measurable function π = π(π , π1 , . . . , ππ−1 ) such that π ⊗ π∗ -a.e. ∗ (96) ∗ Consequently, we have for any π ∈ Ω − π , π»π∗ (π ) = π[π∗ ]π−1 (π ) π-a.e. (97) Remark 20. It is an interesting problem to ask what happens if we transform a function π ∈ πΏ π with a random sequence π1 , π2 , . . . , ππ , . . ., of operators chosen at random from some stock of linear operators on πΏ π given in advance. What can we say about the limit limπ → ∞ (1/π) ∑ππ=1 π1 π2 ⋅ ⋅ ⋅ ππ π? Unfortunately, we cannot expect any convergence for every random sequence chosen from the stock. Therefore, it is desirable to consider how to choose almost every (not every) random sequence from the stock (cf. Revesz [30] and Yoshimoto [31]). In Pitt [1] and Ulam and von Neumann [2], the random ergodic theorem for two-measure-preserving transformations π, π (cited in Section 1) means the existence of the a.e. limit of the form limπ → ∞ (1/π) ∑ππ=1 π(π1 π2 ⋅ ⋅ ⋅ ππ π )) for almost every sequence (ππ )π≥1 of the infinite sequences obtained by applying π and π in turn at random. This is just the case that the transformations ππ , π ≥ 1, are chosen at random with the same distribution and independently. See also Remark 3 (the case that the transformations ππ , π ≥ 1, ∞ π π=1 π=1 = lim (π − 1) ∑ π−(π+1) π (π + π ∑ πΏπ ππ ) and then ∗ (98) π 1 π ∑ π (π + π ∑ πΏπ ππ ) π→∞π π=1 π=1 = π (π ) πΉ (Φ∗ (π , π∗ )) , π» (π , π∗ ) = π (π , π1 , . . . , ππ−1 ) (mod1) , (π ∈ π, πΏπ =ΜΈ 0) . 1 = ∫ π (π ) ππ 0 (99) π-a.e. This is an immediate consequence of the ergodicity of the family {π[π∗ ]π }. We can state this fact in terms of stochastic processes. For example, see GΕadysz [5], Satz 3. Acknowledgment The author is certainly indebted and very grateful to Mrs. Caroline Nashat for her kind assistance. References [1] H. R. Pitt, “Some generalizations of the ergodic theorem,” Proceedings of the Cambridge Philosophical Society, vol. 38, pp. 325– 343, 1942. [2] S. M. Ulam and J. von Neumann, “Random ergodic theorem (abstract),” Bulletin of the American Mathematical Society, vol. 51, p. 660, 1945. [3] S. Kakutani, “Random ergodic theorems and Markoff processes with a stable distribution,” in Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, pp. 247– 261, University of California Press, Berkeley, Calif, USA, 1950. [4] C. Ryll-Nardzewski, “On the ergodic theorems. III. The random ergodic theorem,” Studia Mathematica, vol. 14, pp. 298–301, 1954. [5] S. GΕadysz, “UΜber den stochastischen Ergodensatz,” Studia Mathematica, vol. 15, pp. 158–173, 1956. [6] R. Cairoli, “Sur le theΜoreΜme ergodique aleΜatoire,” Bulletin des Sciences MatheΜmatiques, vol. 88, pp. 31–37, 1964. [7] T. Yoshimoto, “On the random ergodic theorem,” Studia Mathematica, vol. 61, no. 3, pp. 231–237, 1977. [8] P. Revesz, “Some remarks on the random ergodic theorem I,” Publications of the Mathematical Institute of the Hungarian Academy of Sciences, vol. 5, pp. 375–381, 1960. Abstract and Applied Analysis [9] P. Revesz, “Some remarks on the random ergodic theorem II,” Publications of the Mathematical Institute of the Hungarian Academy of Sciences, vol. 6, pp. 205–213, 1961. [10] K. Jacobs, Lecture Notes on Ergodic Theory, Aarhus University, Mathematisches Institut, 1962. [11] T. Yoshimoto, “On multiparameter Chacon’s type ergodic theorems,” Acta Scientiarum Mathematicarum, vol. 78, pp. 489–515, 2012. [12] R. V. Chacon and D. S. Ornstein, “A general ergodic theorem,” Illinois Journal of Mathematics, vol. 4, pp. 153–160, 1960. [13] E. Hille and R. S. Phillips, Functional Analysis and SemiGroups, vol. 31, American Mathematical Society Colloquium, Providence, RI, USA, 1957. [14] T. Yoshimoto, Induced contraction semigroups and random ergodic theorems [Dissertationes Mathematicae], 1976. [15] E. Hopf, “The general temporally discrete Markoff process,” vol. 3, pp. 13–45, 1954. [16] S. D. Chatterji, “Martingale convergence and the RadonNikodym theorem in Banach spaces,” Mathematica Scandinavica, vol. 22, pp. 21–41, 1968. [17] S. D. Chatterji, “Les martingales et leurs applications analytiques,” in EΜcole d’EΜteΜ de ProbabiliteΜs: Processus Stochastiques, vol. 307 of Lecture Notes in Mathematics, pp. 27–164, Springer, Berlin, Germany, 1973. [18] N. Dunford and J. T. Schwartz, Linear Operators. I, Interscience, New York, NY, USA, 1957. [19] R. V. Chacon and U. Krengel, “Linear modulus of linear operator,” Proceedings of the American Mathematical Society, vol. 15, pp. 553–559, 1964. [20] U. Krengel, Ergodic Theorems: With a Supplement by Antoine Brunel, vol. 6 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1985. [21] J. WosΜ, “Random ergodic theorems for sub-Markovian operators,” Studia Mathematica, vol. 74, no. 2, pp. 191–212, 1982. [22] E. Hille, “Remarks on ergodic theorems,” Transactions of the American Mathematical Society, vol. 57, pp. 246–269, 1945. [23] K. Yosida and S. Kakutani, “Operator-theoretical treatment of Markoff ’s process and mean ergodic theorem,” Annals of Mathematics, vol. 42, pp. 188–228, 1941. [24] Y. DeΜniel, “On the a.s. CesaΜro-πΌ convergence for stationary or orthogonal random variables,” Journal of Theoretical Probability, vol. 2, no. 4, pp. 475–485, 1989. [25] R. Irmisch, Punktweise ergodensaΜtze fuΜr (C, πΌ)—Verfahren, 0< πΌ <1 [Dissertation, Fachbereich Mathematik], TH Darmstadt, 1980. [26] H. Anzai, “Mixing up property of Brownian motion,” Osaka Mathematical Journal, vol. 2, no. 1, pp. 51–58, 1950. [27] T. Yoshimoto, “Nonlinear random ergodic theorems for affine operators,” Journal of Functional Analysis, vol. 256, no. 6, pp. 1708–1730, 2009. [28] U. Krengel and M. Lin, “Order preserving nonexpansive operators in πΏ 1 ,” Israel Journal of Mathematics, vol. 58, no. 2, pp. 170– 192, 1987. [29] R. Wittmann, “Hopf ’s ergodic theorem for nonlinear operators,” Mathematische Annalen, vol. 289, no. 2, pp. 239–253, 1991. [30] P. Revesz, “A random ergodic theorem and its application to the theory of Markov chains,” in Ergodic Theory, F. B. Wright, Ed., pp. 217–250, Academic Press, 1963. [31] T. Yoshimoto, “Random ergodic theorem with weighted averages,” Zeitschrift fuΜr Wahrscheinlichkeitstheorie und Verwante Gebiete, vol. 30, no. 2, pp. 149–165, 1974. 13 Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Algebra Hindawi Publishing Corporation http://www.hindawi.com Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Journal of Hindawi Publishing Corporation http://www.hindawi.com Stochastic Analysis Abstract and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com International Journal of Mathematics Volume 2014 Volume 2014 Discrete Dynamics in Nature and Society Volume 2014 Volume 2014 Journal of Journal of Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Applied Mathematics Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014