On the Dependence of the Limit Functions on the Random Parameters in Random Ergodic Theorems

Abstract

We study the structure of the ergodic limit functions determined in random ergodic theorems. When the r random parameters are shifted by the 𝑟₀-shift transformation with 𝑟₀∈{1,2,…,𝑟}, the major finding is that the (random) ergodic limit functions determined in random ergodic theorems depend essentially only on the 𝑟---𝑟₀ 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 𝐿₁ 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 [1] and Ulam-von Neumann’s random ergodic theorem [2] concerning a finite number of measure-preserving transformations and Kakutani’s random ergodic theorem [3] 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 [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 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 [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 𝐿₁ with an additional condition. Yoshimoto [7] extended both Gladysz’s theorem and Cairoli’s theorem to the case of positive linear contractions on 𝐿₁ 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 𝜎-finite measure space (𝑆,𝛽,𝑚) and a probability space (Ω,ℱ,𝜇). Let 𝐿_𝑝(𝑚)=𝐿_𝑝(𝑆,𝛽,𝑚), 1≤𝑝≤∞ be the usual Banach spaces of eqivalence classes of β-measurable functions defined on 𝑆. From now on, we shall write 𝑓_𝜔(𝑠) for 𝑓(𝑠,𝜔) if we wish to regard 𝑓(𝑠,𝜔) as a function of 𝑠 defined on 𝑆 for 𝜔 arbitrarily fixed in Ω.

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 𝑈(𝑠), 𝑉(𝑠) be two given measure-preserving transformations of 𝑆 into itself, which generate all the combinations of the transformations: 𝑈, 𝑉, 𝑈(𝑈), 𝑈(𝑉), 𝑉(𝑈), 𝑉(𝑉), 𝑈(𝑈(𝑉)), 𝑈(𝑉(𝑈)),…. The ergodic limit exists then for almost every point 𝑠 of 𝑆 and almost every choice of the infinite sequence obtained by applying 𝑈 and 𝑉 in turn at random, for example, 𝑈(𝑠), 𝑉(𝑈(𝑠)), 𝑉(𝑉(𝑈(𝑠))),….

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 𝐿_𝑝(𝑚)(𝑝≥1, 𝑚(𝑆)<∞) 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 [3] to the case of an infinite number of measure-preserving transformations as follows.

Theorem 2 (Kakutani (1948–1950) [3]). Let Φ={𝜑_𝜔∶𝜔∈Ω} be a 𝛽⊗ℱ-measurable family of measure-preserving transformations 𝜑_𝜔 defined on 𝑆, where 𝑚(𝑆)=𝜇(𝑆)=1. Let (Ω^∗_±,ℱ^∗_±,𝜇^∗_±) be the two-sided infinite direct product measure space of (Ω,ℱ,𝜇). Then, for any 𝑓∈𝐿_𝑝(𝑚)(𝑝≥1), there exists a (ℱ^∗_±,𝜇^∗_±)-null set 𝑁^∗ such that for any 𝜔^∗∈Ω^∗_±−𝑁^∗, there exists a function 𝑓^∼_𝜔∗(𝑠)∈𝐿_𝑝(𝑚) such that lim_𝑛→∞1𝑛^𝑛−1_𝑘=0𝑓𝜑_{ℵ𝑘(𝜔∗)}⋯𝜑_{ℵ0(𝜔∗)}𝑠=𝑓^∼_𝜔∗(𝑠)𝑚-𝑎.𝑒., (1) where ℵ_𝑛(𝜔^∗) denotes the 𝑛th coordinate of 𝜔^∗, and this holds also in the norm of 𝐿_𝑝(𝑚).

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 (𝜑_{ℵ𝑛(𝜔∗)})_𝑛≥0 of measure-preserving transformations on 𝑆 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 (𝜑_𝑛)_𝑛≥1 at random, not necessarily with the same distribution but independently from a given set Φ of measure-preserving transformations on 𝑆, under what condition does the limit lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑓𝜑_𝑘⋯𝜑₁𝑠 (2) exist 𝑚-a.e. or in 𝐿₂(𝑚)-mean with probability 1? 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 [10].

Jacobs’ General Random Ergodic Theorem [10]. Let 𝜎 be an endomorphism of (Ω,ℱ,𝜇) and let {𝑇_𝜔∶𝜔∈Ω} be a strongly ℱ-measurable family of random linear contractions on 𝐿₁(𝑚). Let {𝑝_𝑛}_𝑛≥0 be a sequence of 𝛽⊗ℱ-measurable functions defined on 𝑆×Ω which is admissible for {𝑇_𝜔∶𝜔∈Ω}. This means that if ℎ∈𝐿₁(𝑚⊗𝜇) and |ℎ(𝑠,𝜔)|≤𝑝_𝑛(𝑠,𝜔)𝑚⊗𝜇-a.e., then |𝑇_𝜔ℎ_𝜎𝜔(𝑠)|≤𝑝_𝑛+1(𝑠,𝜔)𝑚⊗𝜇-a.e. Then, for any function 𝑓∈𝐿₁(𝑚), there exists a 𝜇-null set 𝑁∈ℱ such that for any 𝜔∈Ω−𝑁, lim_𝑛→∞∑^𝑛_𝑘=0𝑇_𝜔𝑇_𝜎𝜔⋯𝑇_𝜎𝑘𝜔𝑓(𝑠)∑^𝑛_𝑘=0𝑝_𝑘()=𝑠,𝜔^∼𝑓(𝑠,𝜔) (3) exists and is finite 𝑚-a.e. on the set {𝑠∶∑^∞_𝑘=0𝑝_𝑘(𝑠,𝜔)>0} (cf. [11] which includes a further weighted generalization of Jacobs’theorem).

If all 𝑇_𝜔 are positive, then Jacobs’ theorem yields the following Chacon-Ornstein’s type random ergodic theorem (cf. [10, 12]); for 𝑓∈𝐿₁(𝑚) and 𝑔∈𝐿⁺₁(𝑚), there exists a 𝜇-null set 𝑁 such that for any 𝜔∈Ω−𝑁, lim_𝑛→∞∑^𝑛_𝑘=0𝑇_𝜔𝑇_𝜎𝜔⋯𝑇_𝜎𝑘𝜔𝑓(𝑠)∑^𝑛_𝑘=0𝑇_𝜔𝑇_𝜎𝜔⋯𝑇_𝜎𝑘𝜔𝑔(𝑠)=𝑓^∼_𝑔(𝑠,𝜔) (4) exists and is finite 𝑚-a.e. on the set {𝑠∶∑^∞_𝑘=0𝑇_𝜔𝑇_𝜎𝜔⋯𝑇_𝜎𝑘𝜔𝑔(𝑠)>0}. Moreover, if the family {𝑇_𝜔,𝜔∈Ω} has a strictly positive invariant function 𝑔∈𝐿⁺₁(𝑚), then for every 𝑓∈𝐿₁(𝑚), there exists a 𝜇-null set 𝑁 such that for any 𝜔∈Ω−𝑁lim_𝑛→∞1𝑛+1^𝑛_𝑘=0𝑇_𝜔𝑇_𝜎𝜔⋯𝑇_𝜎𝑘𝜔𝑓(𝑠)=𝑓^∗(𝑠,𝜔) (5) exists and is finite 𝑚-a.e. Unfortunately, this Cesàro-type result does not hold in general without assuming the existence of a strictly positive invariant function. However, if the family {𝑇_𝜔∶𝜔∈Ω} satisfies the norm conditions ‖𝑇_𝜔‖_𝐿1(𝑚)≤1 and ‖𝑇_𝜔‖_{𝐿∞(𝑚)}≤1 for all 𝜔∈Ω, then the above Cesàro-type random ergodic theorem holds even without assuming the existence of a strictly positive invariant function in 𝐿₁(𝑚). The above limit functions 𝑓^∼(𝑠,𝜔)=[𝑓^∼]_𝜔(𝑠), 𝑓^∼_𝑔(𝑠,𝜔)=[𝑓^∼_𝑔]_𝜔(𝑠), and 𝑓^∗(𝑠,𝜔)=[𝑓^∗]_𝜔(𝑠) depend generally on the random parameter 𝜔. Our particular interest is in the relationship between the limit functions and the random parameters in the case that 𝜎 is the shift transformation of the random parameter space (Ω^∗,ℱ^∗,𝜇^∗) being the one-sided infinite product of the same probability space (Ω,ℱ,𝜇).

2. Random Ergodic Theorems

Throughout all that follows, let (Ω^∗,ℱ^∗,𝜇^∗) be the one-sided infinite product measure space of (Ω,ℱ,𝜇): Ω^∗,ℱ^∗,𝜇^∗=^∞_𝑛=1Ω_𝑛,^∞_𝑛=1ℱ_𝑛,^∞_𝑛=1𝜇_𝑛,Ω^∗_𝑟,ℱ^∗_𝑟,𝜇^∗_𝑟=^𝑟_𝑛=1Ω_𝑛,^𝑟_𝑛=1ℱ_𝑛,^𝑟_𝑛=1𝜇_𝑛,,Ω𝑟:apositiveinteger_𝑛,ℱ_𝑛,𝜇_𝑛=Ω,ℱ,𝜇,𝑛=1,2,…. (6)

Let 𝜑 be the (one-sided) shift transformation defined on Ω^∗ which means that using the coordinate functions ℵ_𝑛(⋅), ℵ_𝑛𝜑𝜔^∗=ℵ_𝑛+1𝜔^∗,𝑛=1,2,…. (7)

Then, 𝜑 is clearly a ℱ^∗-measurable and 𝜇^∗-measure-preserving transformation defined on Ω^∗. Let 𝑟 be a fixed positive integer. For simplicity, we let [𝜔^∗]_𝑟=(𝜔₁,…,𝜔_𝑟) for any 𝜔^∗=(𝜔₁,…,𝜔_𝑟,…)∈Ω^∗. Fix an 𝑟₀∈{1,2,…,𝑟}. Suppose that to each [𝜔^∗]_𝑟∈Ω^∗_𝑟, there corresponds a linear contraction operator 𝑇_[𝜔∗] on 𝐿₁(𝑚). The family {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} is said to be strongly ℱ^∗_𝑟-measurable if for any ℎ∈𝐿₁(𝑚) the function 𝑇_{[𝜔∗]𝑟}ℎ is strongly ℱ^∗_𝑟-measurable as an 𝐿₁(𝑚)-valued function defined on Ω^∗_𝑟, namely, for the mapping Ψ_ℎ∶[𝜔^∗]_𝑟→𝑇_{[𝜔∗]𝑟}ℎ of ℱ^∗_𝑟 into 𝐿₁(𝑚), 𝜇^∗_𝑟∘Ψ_ℎ⁻¹ has a separable support (cf. [13]).

Our main result is stated as follows.

Theorem 4. Let 𝑟₀ be any fixed integer with 1≤𝑟₀≤𝑟. Let {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} be a strongly ℱ^∗_𝑟-measurable family of positive linear contractions on 𝐿₁(𝑚). Suppose that there exists a strictly positive 𝐿₁(𝑚)-function invariant under {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗}. Then, for every 𝑓∈𝐿₁(𝑚), there exists a 𝜇^∗-null set 𝑁^∗∈ℱ^∗ such that for any 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∼_{[𝜔∗]0𝑟−𝑟}∈𝐿₁(𝑚) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(𝑠)=𝑓^∼_{[𝜔∗]0𝑟−𝑟}(𝑠)𝑚-𝑎.𝑒. (8)

Proof. We need the following lemmas.
Lemma 5 (see [10, 14]). The strong ℱ^∗_𝑟-measurability of the operator family {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} guarantees that for any 𝑔∈𝐿₁(𝑚⊗𝜇^∗), there exists a uniquely determined 𝛽⊗ℱ^∗-measurable version [𝑇_{[𝜔∗]𝑟}𝑔_{𝜑𝑟0𝜔∗}](𝑠) of 𝑇_{[𝜔∗]𝑟}𝑔_{𝜑𝑟0𝜔∗}(𝑠) such that excepting a 𝜇^∗-null set, 𝑇_{[𝜔∗]𝑟}𝑔_{𝜑𝑟0𝜔∗}(𝑠)=𝑇_{[𝜔∗]𝑟}𝑔_{𝜑𝑟0𝜔∗}(𝑠)𝑚-a.e. (9)
Using the measurable version appearing in Lemma 5, we define 𝑇𝑔𝑠,𝜔^∗=𝑇_{[𝜔∗]𝑟}𝑔_{𝜑𝑟0𝜔∗}(𝑠). (10)
From the norm conditions of {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} in Theorem 4, it turns out that 𝑇 is a linear operator on 𝐿₁(𝑚⊗𝜇^∗) with ‖𝑇‖_{𝐿1(𝑚⊗𝜇∗)}≤1. Moreover, it is easy to check that there exists a strictly positive 𝐿₁(𝑚⊗𝜇^∗)-function invariant under 𝑇. Thus, for any 𝑓∈𝐿₁(𝑚⊗𝜇^∗), we can apply Chacon-Ornstein’s ergodic theorem [12] (cf. Hopf’s ergodic theorem [15]) to ensure the existence of a function 𝑓^∼∈𝐿₁(𝑚⊗𝜇^∗) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑇^𝑘𝑓𝑠,𝜔^∗=𝑓^∼𝑠,𝜔^∗𝑚⊗𝜇-a.e. (11)
Moreover, as is easily checked, we find that 𝑇𝑓^∼=𝑓^∼. (12)
One can easily verify that excepting a suitable 𝜇^∗-null set 𝑁^∗₁∈ℱ^∗, 𝑇^𝑘𝑓𝑠,𝜔^∗=𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(𝑠)𝑚-a.e., (13) for all 𝑘=1,2,…. Next we wish to show that 𝑓^∼(𝑠,𝜔^∗) does depend essentially only on (𝑠,[𝜔^∗]_𝑟−𝑟0). To do this, we define sub-𝜎-fields ℘_𝑛, 𝑛=1,2,…, of 𝛽⊗ℱ^∗ by ℘_𝑛=𝛽⊗ℱ₁⊗⋯⊗𝛽_𝑛⊗ℱ_𝑛⊗∅,Ω_𝑛+1℘⊗⋯,𝑛=1,2,…,₀=𝛽⊗∅,Ω₁⊗∅,Ω₂⊗⋯. (14)
It is clear that if 𝑚<𝑛, then ℘_𝑚⊂℘_𝑛 and that if 𝑓∈𝐿₁(𝑚⊗𝜇^∗), 𝑓_𝑛=℘_𝑛𝑓, then ℘_𝑚𝑓_𝑛=𝑓_𝑚 whenever 𝑚≤𝑛. Therefore, the system {𝑓_𝑛,℘_𝑛∶𝑛=1,2,…} forms a martingale. For each 𝑛, let 𝐸(⋅∣℘_𝑛) denote the conditional expectation operator with respect to the sub-𝜎-field ℘_𝑛. Let ℎ(𝑠,𝜔^∗) be of the form ℎ𝑠,𝜔^∗=𝜉(𝑠)⋅^ℓ_𝑖=1𝜂_𝑖𝜔_𝑖, (15) where 𝜉∈𝐿₁(𝑚)∩𝐿_∞(𝑚), 𝜂_𝑖∈𝐿_∞(Ω_𝑖), 𝑖=1,2,…,ℓ. Then, the linear combinations of functions of the form (15) are everywhere dense in 𝐿₁(𝑚⊗𝜇^∗). Thus, for the question confronting us, it suffices to prove the relation 𝐸(𝑓^∼∣℘_𝑟−𝑟0)=𝑓^∼ only for the case when 𝑓^∼, is of the form (15).
Lemma 6. It holds that 𝐸(𝑇^𝑚⋅∣℘_𝑟−𝑟0)=𝑇^𝑚𝐸(⋅∣℘_{(𝑚−1)𝑟0+𝑟}),𝑚=1,2,….
Proof. It follows that for a sufficiently large ℓ and 𝑛<ℓ, 𝐸(ℎ∣℘_𝑛) is such that 𝐸ℎ∣℘_𝑛𝑠,𝜔^∗=𝜉(𝑠)⋅^𝑛_𝑖=1𝜂_𝑖𝜔_𝑖⋅^ℓ_{𝑗=𝑛+1Ω}𝜂_𝑗(𝜔)𝑑𝜇(𝜔)𝑚⊗𝜇^∗-a.e. (16)
Thus, excepting a 𝜇^∗-null set 𝑁^∗₂, we get 𝑇^∗𝐸ℎ∣℘_𝑛𝜔∗(𝑠)=𝑇^∗_{[𝜔∗]𝑟}𝜉(𝑠)^𝑛_𝑖=1𝜂_𝑖𝜔_𝑖+𝑟0×^ℓ_{𝑗=𝑛+1Ω}𝜂_𝑗(𝜔)𝑑𝜇(𝜔)𝑚-a.e. (17) On the other hand, since excepting a 𝜇^∗-null set 𝑇^∗ℎ_𝜔∗(𝑠)=𝑇^∗_{[𝜔∗]𝑟}𝜉(𝑠)^ℓ_𝑖=1𝜂_𝑖𝜔_𝑖+𝑟0𝑚-a.e., (18) we have that if 𝑛≥𝑟−𝑟₀, then 𝐸𝑇^∗ℎ∣℘_{𝑛+𝑟0𝜔∗}(𝑠)=𝑇_{[𝜔∗]𝑟}𝜉(𝑠)^𝑛_𝑖=1𝜂_𝑖𝜔_𝑖+𝑟0×^ℓ_𝑗=𝑛+1𝜂_𝑗(𝜔)𝑑𝜇(𝜔)𝑚-a.e. (19) Consequently, 𝑇^∗𝐸ℎ∣℘_𝑛=𝐸𝑇^∗ℎ∣℘_𝑛+𝑟0𝑚⊗𝜇^∗-a.e., (20) and by approximation, 𝑇^∗𝐸⋅∣℘_𝑛=𝐸𝑇^∗⋅∣℘_𝑛+𝑟0, (21) and so by iteration, 𝑇^∗𝑚𝐸⋅∣℘_𝑛=𝐸𝑇^∗𝑚⋅∣℘_{𝑛+𝑚𝑟0},𝑚=0,1,2,…,𝑛≥𝑟−𝑟₀. (22) In particular, 𝑇^∗𝑚𝐸⋅∣℘_𝑟−𝑟0=𝐸𝑇^∗𝑚⋅∣℘_{(𝑚−1)𝑟0+𝑟},𝑚=1,2,…. (23) Since 𝑇 is the adjoint operator of 𝑇^∗, we have thus 𝐸𝑇^𝑚⋅∣℘_𝑟−𝑟0=𝑇^𝑚𝐸⋅∣℘_{(𝑚−1)𝑟0+𝑟},𝑚=1,2,…. (24)
We return to the proof of the theorem. By Lemma 6 and the martingale convergence theorem (cf. [16, 17]), we have 𝐸𝑓^∼∣℘_𝑟−𝑟0−𝑓^∼_{𝐿1(𝑚⊗𝜇∗)}=𝐸𝑇𝑓^𝑚∼∣℘_𝑟−𝑟0−𝑇𝑓^𝑚∼_{𝐿1(𝑚⊗𝜇∗)}≤𝑇^𝑚𝐸𝑓^∼∣℘_{(𝑚−1)𝑟0+𝑟}−𝑇𝑓^𝑚∼_{𝐿1(𝑚⊗𝜇∗)}≤𝐸𝑓^∼∣℘_{(𝑚−1)𝑟0+𝑟}−𝑓^∼_{𝐿1(𝑚⊗𝜇∗)}→0 (25) as 𝑚→∞, and thus 𝐸𝑓^∼∣℘_𝑟−𝑟0=𝑓^∼, (26) which implies that 𝑓^∼(𝑠,𝜔^∗) depends essentially only on (𝑠,[𝜔^∗]_𝑟−𝑟0). Hence, the theorem follows from (11), (26), and Fubini’s theorem. The proof of Theorem 4 has hereby been completed.

If we take 𝑟₀=𝑟 in Theorem 4, we have Cairoli’s theorem in which 𝑓^∼(𝑠,𝜔^∗) does not depend essentially on 𝜔^∗. The 𝑟 random parameter generalization (see [7]) of Cairoli’s theorem is obtained by taking 𝑟₀=1 in Theorem 4. Adapting the (almost) same argument as used in the proof of Theorem 4, we have the following.

Theorem 7. Let 𝑟₀ be any fixed integer with 1≤𝑟₀≤𝑟. Let {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} be a strongly ℱ^∗_𝑟-measurable family of linear contractions on 𝐿₁(𝑚) with ‖𝑇_{[𝜔∗]𝑟}‖_{𝐿∞(𝑚)}≤1 for all 𝜔^∗∈Ω^∗. Then, for every 𝑓∈𝐿_𝑝(𝑚⊗𝜇^∗) with 1≤𝑝<∞, there exists a 𝜇^∗-null set 𝑁^∗∈ℱ^∗ such that for any 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∼_{[𝜔∗]0𝑟−𝑟}∈𝐿_𝑝(𝑚) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(𝑠)=𝑓^∼_{[𝜔∗]0𝑟−𝑟}(𝑠)𝑚-𝑎.𝑒., (27) and that if 1<𝑝<∞, then lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(⋅)−𝑓^∼_{[𝜔∗]0𝑟−𝑟}(⋅)_{𝐿𝑝(𝑚)}=0, (28) and that if 𝑚(𝑆)<∞ and 𝑓∈𝐿(𝑆×Ω^∗)log⁺𝐿(𝑆×Ω^∗), then lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(⋅)−𝑓^∼_{[𝜔∗]0𝑟−𝑟}(⋅)_𝐿1(𝑚)=0. (29)

Proof. As before, we define 𝑇𝑔(𝑠,𝜔^∗)=[𝑇_{[𝜔∗]𝑟}𝑔_{𝜑𝑟0𝜔∗}](𝑠) for 𝑔∈𝐿₁(𝑚⊗𝜇^∗). From the norm conditions of {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} in Theorem 7, it turns out that 𝑇 is a linear operator on 𝐿₁(𝑚⊗𝜇^∗) with ‖𝑇‖_{𝐿1(𝑚⊗𝜇∗)}≤1 and ‖𝑇‖_{𝐿∞(𝑚⊗𝜇∗)}≤1. Then, It follows from the Riesz convexity theorem that ‖𝑇‖_{𝐿𝑝(𝑚⊗𝜇∗)}≤1 for all 𝑝 with 1<𝑝<∞. Thus, for any 𝑓∈𝐿_𝑝(𝑚⊗𝜇^∗)(1≤𝑝<∞), we can apply Dunford and Schwartz’s ergodic theorem [18] to ensure the existence of a function 𝑓^∼∈𝐿_𝑝(𝑚⊗𝜇^∗) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑇^𝑘𝑓𝑠,𝜔^∗=𝑓^∼𝑠,𝜔^∗𝑇𝑚⊗𝜇-a.e.,_∼𝑓=𝑓^∼, (30) and that if 1<𝑝<∞, then lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑇^𝑘𝑓−𝑓^∼_{𝐿𝑝(𝑚⊗𝜇∗)}=0, (31) and that if 𝑚(𝑆)<∞ and 𝑓∈𝐿(𝑆×Ω^∗)log⁺𝐿(𝑆×Ω^∗), then lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑇^𝑘𝑓−𝑓^∼_{𝐿1(𝑚⊗𝜇∗)}=0. (32)
Now, adapting the (almost) same argument as used in the proof of Theorem 4, we can find that 𝑓^∼𝑠,𝜔^∗=𝑓^∼𝑠,𝜔^∗_𝑟−𝑟0𝑚⊗𝜇^∗-a.e. (33) Note here that if 𝑓∈𝐿_𝑝(𝑚⊗𝜇^∗), 1<𝑝<∞, (or 𝑓∈𝐿(𝑆×Ω^∗)log⁺𝐿(𝑆×Ω^∗)), then sup_𝑛≥11𝑛^𝑛_𝑘=1|𝑇|^𝑘𝑓∈𝐿_𝑝𝑚⊗𝜇^∗or,∈𝐿₁𝑚⊗𝜇^∗, (34) where |𝑇| denotes the linear modulus of 𝑇 (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 𝑟₀=𝑟=1 and to the 𝑟 random parameter result of Gładysz [5] by taking 𝑟₀=1.

Theorem 8. Let 𝑟₁,…,𝑟_𝜈 be 𝜈 positive integers. Let {𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}∶𝜔^∗∈Ω^∗}, 𝑖=1,2,…,𝜈, be strongly ℱ_𝑟^(𝑖)_𝑖(𝑖=1,2,…,𝜈)-measurable families of linear contractions on 𝐿₁(𝑚) with ‖𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}‖_{𝐿∞(𝑚)}≤1. For each 𝑖=1,2,…,𝜈, we set 𝑇_𝑖𝑘,𝜔^∗_𝑟𝑖=𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}𝑇^(𝑖)_{[𝜑𝜔∗]𝑟𝑖}⋯𝑇^(𝑖)_{[𝜑𝑘−1𝜔∗]𝑟𝑖}𝑇,𝑘=1,2,⋯,_𝑖0,𝜔^∗_𝑟𝑖=theidentityoperator. (35) Then, for every 𝑓∈𝐿₁(𝑚), there exists a 𝜇^∗-null set 𝑁^∗ such that for every 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∼_{[𝜔∗]𝑞−1}∈𝐿₁(𝑚) with 𝑞=max(𝑟₁,…,𝑟_𝜈) such that the multiple averages 1𝑛₁⋯𝑛_𝜈^𝑛1_𝑘1=1⋯^𝑛𝜈_𝑘𝜈=1𝑇₁𝑘₁,𝜔^∗_𝑟1⋯𝑇_𝜈×𝑘_𝜈,𝜑^{𝑘1+⋯+𝑘𝜈−1}𝜔^∗_𝑟𝜈𝑓(𝑠) (36) converge to 𝑓^∼_{[𝜔∗]𝑞−1}(𝑠)𝑚-a.e. as 𝑛₁→∞,…,𝑛_𝜈→∞ independently. Here, 𝑓^∼_{[𝜔∗]𝑞−1}(⋅) means that the number of random parameters is at most 𝑞−1.

Proof. As already seen above, we can define the operators 𝑇₁,…,𝑇_𝜈 on 𝐿₁(𝑚⊗𝜇^∗) as follows∶ for ℎ∈𝐿₁(𝑚⊗𝜇^∗)𝑇_𝑖ℎ𝑠,𝜔^∗=𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}ℎ_{𝜑𝑟𝑖𝜔∗}(𝑠),𝑖=1,2,…,𝜈. (37) Each 𝑇_𝑖 turns out to be a linear contraction on 𝐿₁(𝑚⊗𝜇^∗) with ‖𝑇_𝑖‖_{𝐿∞(𝑚⊗𝜇∗)}≤1. So, it follows from the Riesz convexity theorem that ‖𝑇_𝑖‖_{𝐿𝑝(𝑚⊗𝜇∗)}≤1 for 1<𝑝<∞. Hence, from Dunford-Schwartz’s ergodic theorem [18], we have that for every 𝑓∈𝐿_𝑝(𝑚⊗𝜇^∗) the multiple averages 1𝑛₁⋯𝑛_𝜈^𝑛1_𝑘1=1⋯^𝑛𝜈_𝑘𝜈=1𝑇^𝑘1₁⋯𝑇^𝑘𝜈_𝜈𝑓=𝐸₁⋯𝐸_𝜈𝑓𝑚⊗𝜇^∗-a.e. (38) converge to 𝐸₁⋯𝐸_𝜈𝑓 almost everywhere on 𝑆×Ω^∗ as 𝑛₁→∞,…,𝑛_𝜈→∞ independently, where each 𝐸_𝑖 is a projection of 𝐿_𝑝(𝑚⊗𝜇^∗) onto the manifold 𝑁(𝐼−𝑇_𝑖)={𝑔∈𝐿_𝑝(𝑚⊗𝜇^∗)∶𝑇_𝑖𝑔=𝑔} with lim_{𝑛𝑖→∞}1𝑛_𝑖^𝑛𝑖_𝑘𝑖=1𝑇^𝑘𝑖_𝑖𝑓−𝐸_𝑖𝑓_{𝐿𝑝(𝑚⊗𝜇∗)}=0,𝑖=1,2,…,𝜈. (39) Note here that 𝐸_𝜈𝑓∈𝑁𝐼−𝑇_𝜈,𝐸_𝜈−1𝐸_𝜈𝑓∈𝑁𝐼−𝑇_𝜈−1𝐸,…,₁⋯𝐸_𝜈𝑓∈𝑁𝐼−𝑇₁. (40) Thus, by Lemma 6 and the (𝐿_𝑝) martingale convergence theorem, we find that excepting a 𝜇^∗-null set 𝐸_𝜈𝑓𝑠,𝜔^∗=𝐸_𝜈𝑓_{[𝜔∗]𝑟𝜈−1}(𝑠)𝐸𝑚-a.e.,_𝜈−1𝐸_𝜈𝑓𝑠,𝜔^∗=𝐸_𝜈−1𝐸_𝜈𝑓_{[𝜔∗]𝜈max(𝑟𝜈−1,𝑟)−1}(𝑠)𝐸𝑚-a.e.,₁⋯𝐸_𝜈𝑓𝑠,𝜔^∗=𝐸₁⋯𝐸_𝜈𝑓_{[𝜔∗]1𝜈max(𝑟,…,𝑟)−1}(𝑠)𝑚-a.e. (41) Finally, observe that excepting a 𝜇^∗-null set, we get 𝑇^𝑘_𝑖𝑓𝑠,𝜔^∗=𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}𝑇^(𝑖)_{[𝜑𝜔∗]𝑟𝑖}⋯𝑇^(𝑖)_{[𝜑𝑘−1𝜔∗]𝑟𝑖}𝑓_{𝜑𝑘𝜔∗}(𝑠)=𝑇_𝑖𝑘,𝜔^∗_𝑟𝑖𝑓_𝜔∗(𝑠)𝑇𝑚-a.e.,𝑘=1,2,…,𝑖=1,2,…,𝜈,_𝑘11⋯𝑇^𝑘𝜈_𝜈𝑓𝑠,𝜔^∗=𝑇₁𝑘₁,𝜔^∗_𝑟1⋯𝑇_𝜈𝑘_𝜈,𝜑^{𝑘1+⋯+𝑘𝜈−1}𝜔^∗_𝑟𝜈×𝑓_𝜔∗(𝑠)𝑚-a.e. (42)(43) In fact, we have for two operators 𝑇₁ and 𝑇₂𝑇^𝑘1₁𝑇^𝑘2₂𝑓𝑠,𝜔^∗=𝑇⁽¹⁾_{[𝜔∗]𝑟1}⋯𝑇⁽¹⁾_{[𝜑𝑘1−1𝜔∗]𝑟1}𝑇^𝑘2₂𝑓_{𝜑𝑘1𝜔∗}(𝑠)=𝑇⁽¹⁾_{[𝜔∗]𝑟1}⋯𝑇⁽¹⁾_{[𝜑𝑘1−1𝜔∗]𝑟1}𝑇⁽²⁾_{[𝜑𝑘1𝜔∗]𝑟2}⋯𝑇⁽²⁾_{[𝜑𝑘12+𝑘−1𝜔∗]𝑟2}𝑓_{𝜑𝑘12+𝑘𝜔∗}(𝑠)=𝑇₁𝑘₁,𝜔^∗_𝑟1𝑇₂𝑘₂,𝜑^𝑘1𝜔^∗_𝑟2×𝑓𝑠,𝜔^∗𝑚-a.e. (44) To complete the proof of the above equality, assume that (43) has already been established for the 𝜈−1 operators 𝑇₂,…,𝑇_𝜈. Then, it is easily verified by the induction hypothesis that (43) holds for the 𝜈 operators 𝑇₁,…,𝑇_𝜈. Hence, taking 𝑓(𝑠,𝜔^∗)=𝑓(𝑠)𝑒(𝜔^∗)(𝑒(⋅)=1), the theorem follows from the above arguments.

In particular, if the operators in question are commutative then we have the following.

Theorem 9. Let 𝑟₁,…,𝑟_𝜈 be 𝜈 positive integers. Let {𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}∶𝜔^∗∈Ω^∗}, 𝑖=1,2,…,𝜈, be a strongly ℱ_𝑟^(𝑖)_𝑖(𝑖=1,2,…,𝜈)-measurable commuting family of linear contractions on 𝐿₁(𝑚) with ‖𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}‖_{𝐿∞(𝑚)}≤1. Then, for every 𝑓∈𝐿₁(𝑚), there exists a 𝜇^∗-null set 𝑁^∗ such that for every 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∼_{[𝜔∗]𝑞−1}∈𝐿₁(𝑚) with 𝑞=max(𝑟₁,…,𝑟_𝜈) such that lim_𝑛→∞1𝑛^𝜈𝑛−1_𝑘1=0⋯^𝑛−1_𝑘𝜈=0𝑇₁𝑘₁,𝜔^∗_𝑟1⋯𝑇_𝜈𝑘_𝜈,𝜑^{𝑘1+⋯+𝑘𝜈−1}𝜔^∗_𝑟𝜈𝑓(𝑠)=𝑓^∼_{[𝜔∗]𝑞−1}(𝑠)𝑚-a.e. (45)

In passing, we make mention of the a.e. convergence for sectorial restricted random averages. We say that a sequence 𝑢(𝑛)⊂𝐙^𝜈₊ remains in a sector of 𝐙^𝜈₊ if there is a constant 𝐶₀>0 such that the ratios 𝑛_𝑖/𝑛_𝑗 are bounded by 𝐶₀ for 1≤𝑖,𝑗≤𝜈 and all 𝑢(𝑛)=(𝑛₁,…,𝑛_𝜈) (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 𝑚-a.e. in a sector 𝐙^𝜈₊. In addition, in this case, Theorem 8 implies that the limit function depends essentially only on the 𝑞−1 random parameters. Hence, we have the following.

Theorem 10. Let 𝑟₁,…,𝑟_𝜈 be 𝜈 positive integers. Let {𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}∶𝜔^∗∈Ω^∗}, 𝑖=1,2,…,𝜈, be strongly ℱ_𝑟^(𝑖)_𝑖(𝑖=1,2,…,𝜈)-measurable commuting family of linear contractions on 𝐿₁(𝑚) with ‖𝑇^(𝑖)_{[𝜔∗]𝑟𝑖}‖_{𝐿∞(𝑚)}≤1. Then, for every 𝑓∈𝐿₁(𝑚), there exists a 𝜇^∗-null set 𝑁^∗ such that for every 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∼_{[𝜔∗]𝑞−1}∈𝐿₁(𝑚) with 𝑞=max(𝑟₁,…,𝑟_𝜈) such that the multiple averages 1𝑛₁⋯𝑛_𝜈^𝑛1_𝑘1=1⋯^𝑛𝜈_𝑘𝜈=1𝑇₁𝑘₁,𝜔^∗_𝑟1⋯𝑇_𝜈𝑘_𝜈,𝜑^{𝑘1+⋯+𝑘𝜈−1}𝜔^∗_𝑟𝜈𝑓(𝑠) (46) converge to 𝑓^∼_{[𝜔∗]𝑞−1}(𝑠)𝑚-a.e. as 𝑢(𝑛)  (=(𝑛₁,…,𝑛_𝜈)≥(1,…,1))→∞ in a sector of 𝐙^𝜈₊.

A sub-Markovian operator 𝑃^∗ on 𝐿_∞(𝑚) means that 𝑃^∗ is a positive linear contraction on 𝐿_∞(𝑚) with 1 subinvariant under 𝑃^∗ (i.e., 𝑃^∗1≤1) such that lim_𝑛→∞𝑃^∗ℎ_𝑛=0𝑚-a.e. for any (decreasing) sequence (ℎ_𝑛)_𝑛≥1⊂𝐿_∞(𝑚) with lim_𝑛→∞ℎ_𝑛=0𝑚-a.e. Let 𝑃 denote the positive linear contraction on 𝐿₁(𝑚) with 𝑃^∗ as the adjoint operator of 𝑃. When 𝜐=𝑝𝜇 with a 𝑃-subinvariant function 𝑝≥0 (𝑝≠0), 𝜐 is called a 𝑃-subinvariant measure. Here, the subinvariant function is not necessarily integrable.

Now, let Π^∗={𝑃^∗_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} be a strongly ℱ^∗_𝑟-measurable family of sub-Markovian operators on 𝐿_∞(𝑚). Let Π={𝑃_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} be a strongly ℱ^∗_𝑟-measurable family of positive linear contractions on 𝐿₁(𝑚), where each 𝑃_{[𝜔∗]𝑟} is the adjoint operator of the corresponding operator 𝑃^∗_{[𝜔∗]𝑟}. Then, there exists a positive linear contraction 𝑈 on 𝐿₁(𝑚⊗𝜇^∗) such that for any 𝑓∈𝐿₁(𝑚⊗𝜇^∗), 𝑈𝑓𝑠,𝜔^∗=𝑃_{[𝜔∗]𝑟}𝑓_{𝜑𝑟0𝜔∗}(𝑠)𝑚×𝜇^∗-a.e (47)

In addition, if ‖𝑃_{[𝜔∗]𝑟}‖_{𝐿∞(𝑚)}≤1 for all 𝜔^∗∈Ω^∗, 𝑈 is also a contraction on 𝐿_∞(𝑚⊗𝜇^∗) (cf. [10, 14]). The operator 𝑈 extends uniquely to a positive linear transformation on the class of nonnegative 𝛽⊗ℱ^∗-measurable functions defined on 𝑆×Ω^∗ (see [20, page 51]).

Lemma 11. Let the measure 𝑚 be finite. Assume that 𝑚 is 𝐸(𝑈⋅∣℘₀)-subinvariant. Then, 𝑈 turns out to be a positive Dunford-Schwartz operator on 𝐿₁(𝑚⊗𝜇^∗).

Proof. Since 𝑚 is assumed to be 𝐸(𝑈⋅∣℘₀)-subinvariant, the function 1(∈𝐿₁(𝑚⊗𝜇^∗)) is 𝐸(𝑈⋅∣℘₀)-subinvariant. According to Lemma 1.5 of Woś [21], we get 𝑈1=𝐸𝑈1∣℘₀, (48) so that 1 is also 𝑈-subinvariant. Thus, for 𝑓∈𝐿₁(𝑚⊗𝜇^∗)∩𝐿_∞(𝑚⊗𝜇^∗), 𝑈𝑓_∞≤𝑈𝑓_∞=𝑓_∞𝑈1≤𝑓_∞. (49) Since 𝑈 is a positive contraction on 𝐿₁(𝑚⊗𝜇^∗), and 𝑈 is a positive Dunford-Schwartz operator on 𝐿₁(𝑚⊗𝜇^∗).

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 [21] (In proving the key Lemma 1.5 of [21], Woś showed that if 𝑓∈𝐿₁(𝑚⊗𝜇^∗) does not depend essentially on parameter 𝜔^∗, then 𝑈𝑓=𝐸(𝑈𝑓∣℘₀). Using this equality, he deduced 𝑈1≤1 from 𝐸(𝑈1∣℘₀)≤1 in order to prove Theorems 2.8 and 3.4 of [21]. But obviously, 1 does not necessarily belong to 𝐿₁(𝑚⊗𝜇^∗) in case where 𝑚 is 𝜎-finite. His arguments are of course correct in the case that 𝑚 is finite (see, e.g., conditions (i) and (ii) in Theorem 12)).

Theorem 12. Let Π={𝑃_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} be a strongly ℱ^∗_𝑟-measurable family of positive linear contractions on 𝐿₁(𝑚), where each 𝑃_{[𝜔∗]𝑟} is the adjoint operator of the corresponding sub-Markovian operator 𝑃^∗_{[𝜔∗]𝑟}. Assume either of the following conditions:(i)𝑚is finite and 𝐸(𝑈⋅∣℘₀)-subinvariant,(ii)𝑚 is 𝜎-finite and ‖𝑃_{[𝜔∗]𝑟}‖_{𝐿∞(𝑚)}≤1 for all 𝜔^∗∈Ω^∗.

Then, for any 𝑓∈𝐿_𝑝(𝑚⊗𝜇^∗), 1≤𝑝<∞, there exists a 𝜇^∗-null set 𝑁^∗ such that for every 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∼_{[𝜔∗]0𝑟−𝑟}∈𝐿_𝑝(𝑚) such thatlim_𝑛→∞1𝑛^𝑛_𝑘=1𝑃_{[𝜔∗]𝑟}𝑃_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑃_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(𝑠)=𝑓^∼_{[𝜔∗]0𝑟−𝑟}(𝑠)𝑚-a.e., (50) and that if 1<𝑝<∞, then lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑃_{[𝜔∗]𝑟}𝑃_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑃_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(⋅)−𝑓^∼_{[𝜔∗]0𝑟−𝑟}(⋅)_{𝐿𝑝(𝑚)}=0 (51) and that if 𝑚 is finite and 𝑓∈𝐿(𝑆×Ω^∗)log⁺𝐿(𝑆×Ω^∗), then lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑃_{[𝜔∗]𝑟}𝑃_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑃_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(⋅)−𝑓^∼_{[𝜔∗]0𝑟−𝑟}(⋅)_𝐿1(𝑚)=0. (52)

Proof of Theorem 12. In view of Lemma 11, either condition (i) or condition (ii) guarantees that 𝑈 is a positive Dunford-Schwartz operator on 𝐿₁(𝑚⊗𝜇^∗). Hence, we can apply Theorem 7 to conclude that Theorem 12 follows.

3. (𝐶,𝛼)-Type Random Ergodic Theorems

In this section, we establish a (𝐶,𝛼)-type random ergodic theorem for measure-preserving transformations on 𝑆. In this section, we assume that (𝑆,𝛽,𝑚) is a probability space. For real 𝛼>−1 and 𝑛(=0,1,2,…), let 𝐴^𝛼_𝑛 be the (𝐶,𝛼) coefficient of order 𝛼, which is defined by the generating function 1(1−𝑡)^𝛼+1=^∞_𝑛=0𝐴^𝛼_𝑛𝑡^𝑛,0<𝑡<1. (53) Then, we can easily check that 𝐴^𝛼_𝑛 is decreasing in 𝑛 for −1<𝛼<0 and increasing in 𝑛 for 𝛼>0. For 𝛼>−1, we have 𝐴^𝛼_𝑛>0,𝐴^𝛼₀=1,𝐴⁰_𝑛𝐴=1,_𝑛𝛼−1=𝐴^𝛼_𝑛−𝐴^𝛼_𝑛−1,𝐴^𝛼_𝑛=^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘=𝑛𝑛+𝛼,𝐴^𝛼_𝑛∼𝑛^𝛼Γ()(),𝛼+1𝑛→∞ (54) and moreover, 𝑛^𝛼Γ(𝛼+1)≤𝐴^𝛼_𝑛≤(𝑛+1)^𝛼Γ()𝐴𝛼+1for0<𝛼≤1,_𝑛𝛼−1≤𝑛^𝛼−1Γ(𝛼)for𝑛>0. (55) In general, using Hille’s theorem (see [22, Theorem 8]), we have the following.

Theorem 13. Let 𝑟₀ be any fixed integer with 1≤𝑟₀≤𝑟. Let {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} be a strongly ℱ^∗_𝑟-measurable family of linear contractions on 𝐿_𝑝(𝑚)(1≤𝑝<∞). Let 𝛼>0 be fixed. If for all 𝑓∈𝐿_𝑝(𝑚⊗𝜇^∗) there exists a 𝜇^∗-null set 𝑁^∗ such that for any 𝜔^∗∈Ω^∗−𝑁^∗ there exists a function 𝑓^∗_{[𝜔∗]0𝑟−𝑟}∈𝐿_𝑝(𝑚) such that(i)lim_𝑛→∞(1/𝐴^𝛼_𝑛)∑^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0𝑘𝑟𝜔∗]𝑟}𝑓_{𝜑0(𝑘+1)𝑟𝜔∗}(𝑠)=𝑓^∼_{[𝜔∗]0𝑟−𝑟}(𝑠)   for 𝑚-almost all 𝑠∈𝑆,(ii)lim_𝜆→1+0(𝜆−1)∑^∞_𝑛=0𝜆^(𝑛+1)𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0𝑘𝑟𝜔∗]𝑟}𝑓_{𝜑0(𝑘+1)𝑟𝜔∗}(𝑠)=𝑓^∼_{[𝜔∗]0𝑟−𝑟}(𝑠),(iii)lim_𝑛→∞(1/𝑛^𝛼)𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0𝑛𝑟𝜔∗]𝑟}𝑓_{𝜑0(𝑛+1)𝑟𝜔∗}(𝑠)=0,

for 𝑚-almost all 𝑠∈𝑆. Conversely, if (ii) holds but (iii) be replaced by the condition that there exists a 𝛽⊗ℱ^∗-measurable function 𝑀(𝑠,𝜔^∗) finite almost everywhere such that excepting a 𝜇^∗-null set1𝐴^𝛽_𝑛^𝑛_𝑛=0𝐴^𝛽−1_𝑛−𝑘𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0𝑘𝑟𝜔∗]𝑟}𝑓_{𝜑0(𝑘+1)𝑟𝜔∗}(𝑠)≤𝑀_𝜔∗(𝑠)𝑚-a.e., (56)for all 𝑛≥0, where 𝛽≥0, then (i) holds for all 𝛼>𝛽.

Note here that in Theorem 13, (ii) and (iii) do not necessarily imply (i) in general.

Theorem 14. Let 𝑟₀ be any fixed integer with 1≤𝑟₀≤𝑟 and {𝜓_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} be a 𝛽⊗ℱ^∗_𝑟-measurable family of measure-preserving transformations on 𝑆. Let 0<𝛼<1, 𝛼𝑝>1, and 𝑓∈𝐿_𝑝(𝑚). Then, there exists a 𝜇^∗-null set 𝑁^∗ such that for every 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∗_{[𝜔∗]0𝑟−𝑟}(⋅)∈𝐿_𝑝(𝑚) such that lim_𝑛→∞1𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑓𝜓_{[𝜑0𝑘𝑟𝜔∗]𝑟}⋯𝜓_{[𝜔∗]𝑟}𝑠=lim_𝜆→1+0(𝜆−1)^∞_𝑛=0𝜆^−(𝑛+1)𝑓𝜓_{[𝜑0𝑘𝑟𝜔∗]𝑟}⋯𝜓_{[𝜔∗]𝑟}𝑠=𝑓^∗_{[𝜔∗]0𝑟−𝑟}(𝑠)𝑚-𝑎.𝑒.,lim_{𝑛→∞𝑆}1𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑓𝜓_{[𝜑0𝑘𝑟𝜔∗]𝑟}⋯𝜓_{[𝜔∗]𝑟}𝑠−𝑓^∗_{[𝜔∗]0𝑟−𝑟}(𝑠)^𝑝𝑑𝑚^1/𝑝=0. (57)

Proof. Define the skew product Φ^∗ of 𝜙 (the one-sided shift transformation of Ω^∗) and {𝜓_{[𝜔∗]𝑟}} as follows: Φ^∗𝑠,𝜔^∗=𝜓_{[𝜔∗]𝑟}𝑠,𝜑^𝑟0𝜔^∗,𝑠,𝜔^∗∈𝑆×Ω^∗. (58)
Let (𝑓⋅𝑒)(𝑠,𝜔^∗)=𝑓(𝑠)⋅𝑒(𝜔^∗), where 𝑒(𝜔^∗)=1 for all 𝜔^∗∈Ω^∗. Then, (𝑓⋅𝑒)∈𝐿_𝑝(𝑚⊗𝜇^∗). Since 𝐿_𝑝(𝑚⊗𝜇^∗) is reflexive, we see from Yosida-Kakutani’s mean ergodic theorem [23] and Déniel’s theorem [24] that there exists a function (𝑓⋅𝑒)^∗∈𝐿_𝑝(𝑚⊗𝜇^∗) such that lim_{𝑛→∞𝑆×Ω∗}1𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑓⋅𝑒Φ^∗𝑘𝑠,𝜔^∗−𝑓⋅𝑒^∗𝑠,𝜔^∗𝑝𝑑𝑚⊗𝜇^∗1/𝑝=0,𝑓⋅𝑒^∗Φ^∗𝑠,𝜔^∗=𝑓⋅𝑒^∗𝑠,𝜔^∗𝑚⊗𝜇^∗-a.e.,sup_𝑛≥01𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑓⋅𝑒𝑠,𝜔^∗∈𝐿_𝑝𝑚⊗𝜇^∗⊂𝐿₁𝑚⊗𝜇^∗,lim_𝑛→∞1𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑓⋅𝑒Φ^∗𝑘𝑠,𝜔^∗=𝑓⋅𝑒^∗𝑠,𝜔^∗𝑚⊗𝜇^∗-a.e. (59) Furthermore, applying Lemma 6 to the operator 𝑇^∗ induced by Φ^∗, it follows that 𝑓⋅𝑒^∗𝑠,𝜔^∗=𝑓⋅𝑒^∗𝑠,𝜔^∗_𝑟−𝑟0, (60) so that to complete the proof of the theorem, we may take 𝑓^∗_{[𝜔∗]0𝑟−𝑟}(𝑠)=𝑓⋅𝑒^∗_{[𝜔∗]0𝑟−𝑟}(𝑠). (61)

Remark 15. For example, if 𝑟₀=𝑟=1 in Theorem 14, then 𝑓^∗_{[𝜔∗]𝑟−𝑟0}(𝑠)=𝑓^∗(𝑠). It is worthwhile to note that if 0<𝛼<1 and 𝛼𝑝=1 (so, 𝑝=𝛼⁻¹>1), then the pointwise (𝐶,𝛼)-convergence for Φ^∗ does not hold in general (see [24]). For the case of a positive linear contraction on 𝐿_1/𝛼(𝑚⊗𝜇^∗), see Irmisch [25].

In particular, applying Irmisch’s theorem to sub-Markovian operators, we have the following.

Theorem 16. Let Π={𝑃_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} be a strongly ℱ^∗_𝑟-measurable family of positive linear contractions on 𝐿₁(𝑚), where each 𝑃_{[𝜔∗]𝑟} is the adjoint operator of the corresponding sub-Markovian operator 𝑃^∗_{[𝜔∗]𝑟}. Assume that 𝑚 is 𝐸(𝑈⋅∣℘₀)-subinvariant. Let 0<𝛼<1, 𝛼𝑝>1 and 𝑓∈𝐿_𝑝(𝑚). Then, there exists a 𝜇^∗-null set 𝑁^∗ such that for every 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∗_{[𝜔∗]0𝑟−𝑟}(⋅)∈𝐿_𝑝(𝑚) such that lim_𝑛→∞1𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑃_{[𝜔∗]𝑟}𝑃_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑃_{[𝜑0𝑘𝑟𝜔∗]𝑟}=lim_𝜆→1+0(𝜆−1)^∞_𝑛=0𝜆^−(𝑛+1)𝑃_{[𝜔∗]𝑟}𝑃_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑃_{[𝜑0𝑘𝑟𝜔∗]𝑟}=𝑓^∗_{[𝜔∗]0𝑟−𝑟}(𝑠)𝑚-𝑎.𝑒. (62)

Proof. In view of Lemma 11, 𝑈 is a positive linear contraction on 𝐿₁(𝑚⊗𝜇^∗) as well as on 𝐿_∞(𝑚⊗𝜇^∗). Thus, it follows from the Riesz convexity theorem that ‖𝑈‖_{𝐿𝑝(𝑚⊗𝜇∗)}≤1 for 1<𝑝<∞. 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 (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 𝜉=(𝜉_𝑡)_𝑡≥0 (𝜉_𝑡=𝜉_𝑡(𝜔),𝜉₀=0) be a Brownian motion (or Wiener process) on a probability space (Ω,ℱ,𝜇). This process has independent increments; that is, for arbitrary 𝑡₁<𝑡₂<⋯<𝑡_𝑛, the random variables 𝜉_𝑡2−𝜉_𝑡1,…,𝜉_𝑡𝑛−𝜉_𝑡𝑛−1 are independent. In fact, since the process is Gaussian with 𝐸(𝜉_𝑡)=0 and 𝐸(𝜉_𝑡𝜉_𝑠)=min(𝑡,𝑠) by definition, it is sufficient to verify only that the increments are uncorrelated. Thus, if 𝑠<𝑡<𝑢<𝑣, then 𝐸𝜉_𝑡−𝜉_𝑠𝜉_𝑣−𝜉_𝑢=12((𝑡−𝑠)+(𝑣−𝑢)−|𝑡−𝑠+𝑣−𝑢|)=0. (63)

Let {𝜓_𝑡∶𝑡≥0} be an arbitrary ergodic measurable semiflow on a finite measure space (𝑆,𝛽,𝑚). Then, Anzai’s result may be stated as follows: for any 𝑓∈𝐿₁(𝑚) and for almost all 𝜔∈Ω, there exists a null set 𝑁=𝑁_𝑓(𝜔)∈ℱ such that lim_𝛼→∞1𝛼^𝛼₀𝑓𝜓_{𝜉𝑡(𝜔)}𝑠𝑑𝑡=_𝑆𝑓(𝑠)𝑑𝑚 (64) holds for any 𝑠∈𝑆−𝑁. This is an immediate consequence of the ergodicity of the measure-preserving skew product semiflow {Ψ_𝑡𝑡≥0} defined by Ψ_𝑡(𝜔,𝑠)=(𝜑_𝑡𝜔,𝜓_{𝜉𝑡(𝜔)}𝑠), where {𝜑_𝑡∶𝑡≥0} is the ergodic semiflow on Ω given by 𝜉_𝑎(𝜑_𝑡𝜔)=𝜉_𝑎+𝑡(𝜔). It is an interesting and important problem to generalize Anzai’s result for Brownian motions to the case of contraction operator quasisemigroups in 𝐿₁(𝑚) associated with {𝜑_𝑡}. 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 [27]). An affine operator 𝑈 on 𝐿₁(𝑚) is an operator of the type 𝑈𝑓=𝑇𝑓+ℎ, where 𝑇 is a linear contraction on 𝐿₁(𝑚), and ℎ is a fixed element of 𝐿₁(𝑚). Then, 𝑈 is nonlinear and nonexpansive. The fixed points of 𝑈 are solutions of Poisson’s equation for 𝑇, which is (𝐼−𝑇)𝑓=ℎ. When we assume 𝑇 to be mean ergodic, the averages 1𝑛^𝑛_𝑘=1𝑈^𝑘1𝑓=𝑛^𝑛_𝑘=1𝑇^𝑘1𝑓+𝑛^𝑛_𝑘=1^𝑘−1_𝑗=0𝑇^𝑗ℎ (65) converge if and only if ℎ∈(𝐼−𝑇)𝐿₁(𝑚). If ℎ∈(𝐼−𝑇)𝐿₁(𝑚), then there exists a unique 𝜉∈(𝐼−𝑇)𝐿₁(𝑚) such that (𝐼−𝑇)𝜉=ℎ, and the limit of (1/𝑛)∑^𝑛_𝑘=1𝑈^𝑘𝑓 is 𝐸𝑓+𝜉, where 𝐸𝑓 is the limit of (1/𝑛)∑^𝑛_𝑘=1𝑇^𝑘𝑓. Therefore, iterating 𝑈𝑓=𝑇𝑓+(𝐼−𝑇)𝜉 will yield almost everywhere convergence of the averages of 𝑈^𝑛𝑓 for any 𝑓∈𝐿₁(𝑚). Now, let {(𝑇_{[𝜔∗]𝑟},𝑈_{[𝜔∗]𝑟},𝜉_𝜔∗,𝜑^𝑟0)∶𝜔^∗∈Ω^∗} be a random affine system on 𝐿₁(𝑚) such that {𝑇_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} is a strongly measurable family of linear contractions on 𝐿₁(𝑚) as well as on 𝐿_∞(𝑚) and such that for some 𝑔∈𝐿₁(𝑚⊗𝜇^∗), 𝑈_{[𝜔∗]𝑟}=𝑇_{[𝜔∗]𝑟}+(𝐼−𝑇)𝑔_𝜔∗,𝜔^∗∈Ω^∗, (66) where 𝑇𝑓𝑠,𝜔^∗=𝑇_{[𝜔∗]𝑟}𝑓_{𝜑𝑟0𝜔∗}(𝑠), (67) for 𝑓∈𝐿₁(𝑚⊗𝜇^∗) (cf. [11, 27]). For each 𝑓∈𝐿₁(𝑚⊗𝜇^∗), we define a sequence of random functions {𝑉_𝑓(𝑛,𝜔^∗)∶𝜔^∗∈Ω^∗}, 𝑛=0,1,…, in 𝐿₁(𝑚) inductively by 𝑉_𝑓0,𝜔^∗=𝑓_𝜔∗,𝑉_𝑓1,𝜔^∗=𝑓_𝜔∗+𝑈_{[𝜔∗]𝑟}𝑉_𝑓0,𝜑^𝑟0𝜔^∗,⋮𝑉_𝑓𝑛+1,𝜔^∗=𝑓_𝜔∗+𝑈_{[𝜔∗]𝑟}𝑉_𝑓𝑛,𝜑^𝑟0𝜔^∗. (68)

Theorem 18. Let {𝑉_𝑓(𝑛,𝜔^∗)∶𝜔^∗∈Ω^∗} be the sequence of random functions associated with 𝑓∈𝐿₁(𝑚⊗𝜇^∗) which is determined by a random affine system {(𝑇_{[𝜔∗]𝑟},𝑈_{[𝜔∗]𝑟},[(𝐼−𝑇)𝑔]_𝜔∗,𝜑^𝑟0)∶𝜔^∗∈Ω^∗} given in 𝐿₁(𝑚) with ‖𝑇_{[𝜔∗]𝑟}‖_{𝐿∞(𝑚)}≤1. Then, there exists a 𝜇^∗-null set 𝑁^∗ such that for any 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∼_{[𝜔∗]0𝑟−𝑟}∈𝐿₁(𝑚) such that lim_𝑛→∞𝑉_𝑓𝑛,𝜔^∗(𝑠)=𝑛+1^∼𝑓_{[𝜔∗]0𝑟−𝑟}(𝑠)𝑚-𝑎.𝑒. (69)

Proof. It follows that there exists a 𝜇^∗-null set 𝑁^∗₁ such that for any 𝜔^∗∈Ω^∗−𝑁^∗₁, 𝑉_𝑓𝑛,𝜔^∗(𝑠)=𝑓_𝜔∗(𝑠)+^𝑛_𝑘=1𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(𝑠)−𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑛−1)𝑟𝜔∗]𝑟}𝑔_{𝜑0𝑛𝑟𝜔∗}(𝑠)+^𝑛−1_𝑘=1𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑔_{𝜑0𝑘𝑟𝜔∗}(𝑠)+𝑔_𝜔∗(𝑠). (70) Moreover, we have lim_𝑛→∞𝑓𝑠,𝜔^∗𝑛+1=lim_𝑛→∞𝑔𝑠,𝜔^∗𝑛+1=0𝑚-a.e.,lim_𝑛→∞1𝑇𝑛+1^{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑛−1)𝑟𝜔∗]𝑟}𝑔_{𝜑0𝑛𝑟𝜔∗}(𝑠)=0𝑚-a.e. (71) Therefore, by Theorem 7, there exists a 𝜇^∗-null set 𝑁^∗₂ such that for every 𝜔^∗∈Ω^∗−𝑁^∗₂, there exist functions 𝑓^∗_{[𝜔∗]0𝑟−𝑟}, 𝑔^∗_{[𝜔∗]0𝑟−𝑟}∈𝐿₁(𝑚) such that lim_𝑛→∞𝑉_𝑓𝑛,𝜔^∗(𝑠)𝑛+1=𝑓^∗_{[𝜔∗]0𝑟−𝑟}(𝑠)+𝑔^∗_{[𝜔∗]0𝑟−𝑟}(𝑠)𝑚-𝑎.𝑒. (72) Hence, the theorem follows by putting 𝑁^∗=𝑁^∗₁∪𝑁^∗₂ and 𝑓^∼𝑠,𝜔^∗_𝑟−𝑟0=𝑓^∗𝑠,𝜔^∗_𝑟−𝑟0+𝑔^∗𝑠,𝜔^∗_𝑟−𝑟0. (73)

As far as we are concerned with the ergodic behaviors of Cesàro-type processes for nonexpansive operators on 𝐿_𝑝, one can only expect weak convergence in general. In fact, the pointwise convergence of the (𝐶,1) averages of nonlinear and nonexpansive operators on 𝐿_𝑝 may fail to hold. In addition, these (𝐶,1) averages do not need converge in the strong operator topology of 𝐿_𝑝 (see [27, 28]). The so-called 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 𝑈_{[𝜔∗]𝑟}𝑈_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑈_{[𝜑0𝑘𝑟𝜔∗]𝑟}𝑓_{𝜑0(𝑘+1)𝑟𝜔∗}𝑠)=𝑇_{[𝜔∗]𝑟}(𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0𝑘𝑟𝜔∗]𝑟}𝑓_{𝜑0(𝑘+1)𝑟𝜔∗}𝑠)+_𝑘𝑗=1(𝑇_{[𝜔∗]𝑟}𝑇_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑇_{[𝜑0(𝑗−1)𝑟𝜔∗]𝑟}𝑔_{𝜑0𝑗𝑟𝜔∗}(𝑠)+𝑔_{[𝜔∗]𝑟}(𝑠),𝑘=1,2,…. (74) Then, Theorem 7, together with Theorems 2.1 and 2.2 of [27], yields the following theorem.

Theorem 19. Let {(𝑇_{[𝜔∗]𝑟},𝑈_{[𝜔∗]𝑟},[(𝐼−𝑇)𝑔]_𝜔∗,𝜑^𝑟0)∶𝜔^∗∈Ω^∗} be a random affine system given in 𝐿₁(𝑚) with ‖𝑇_{[𝜔∗]𝑟}‖_{𝐿∞(𝑚)}≤1. If 𝑓∈𝐿₁(𝑚⊗𝜇^∗), then there exists a 𝜇^∗-null set 𝑁^∗ such that for any 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∗_{[𝜔∗]0𝑟−𝑟}∈𝐿₁(𝑚) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑈_{[𝜔∗]𝑟}𝑈_{[𝜑𝑟0𝜔∗]𝑟}⋯𝑈_{[𝜑0(𝑘−1)𝑟𝜔∗]𝑟}𝑓_{𝜑0𝑘𝑟𝜔∗}(𝑠)=𝑓^∗_{[𝜔∗]0𝑟−𝑟}(𝑠)𝑚-𝑎.𝑒. (75)

5. Examples

Example 1. Let {𝜏_𝜔∶𝜔∈Ω} be a 𝛽⊗ℱ-measurable family of 𝑚-measure-preserving transformations on 𝑆, and let 𝜙 be 𝜇-measure-preserving transformation on Ω. Then, for any 𝑓∈𝐿₁(𝑚), there exists a 𝜇-null set 𝑁^∗∈𝛽 such that for each 𝜔∈Ω−𝑁^∗, there exists a function 𝑓^∗_𝜔∈𝐿₁(𝑚) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑓𝜏_𝜙^𝑘−1_𝜔⋯𝜏_𝜔𝑠=𝑓^∗_𝜔(𝑠)𝑚-a.e., (76) which follows from Birkhoff’s ergodic theorem applied to the so-called skew product Φ of 𝜙 and {𝜏_𝜔} defined by Φ(𝑠,𝜔)=𝜏_𝜔𝑠,𝜙𝜔,(𝑠,𝜔)∈𝑆×Ω. (77) If the skew product transformation Φ is ergodic, then the function 𝑓^∗_𝜔(𝑠) is constant almost everywhere on 𝑆×Ω. It is worthwhile to notice that, in general, the limit function 𝑓^∗_𝜔(𝑠) depends on the two variables 𝑠 and 𝜔. To illustrate this,we consider the measure spaces (𝑆,𝛽,𝑚) and (Ω,ℱ,𝜇) and transformations given by 𝑆=𝑠₁,…,𝑠_𝑝,𝑚𝑠₁=⋯=𝑚𝑠_𝑝=1𝑝,𝜔𝑝:integer,𝑝≥2Ω=₁,…,𝜔_2𝑞,𝜇𝜔₁=⋯=𝜇𝜔_2𝑞=1,𝜏2𝑞𝑞:integer,2𝑞≥𝑝_{𝜔2𝑗−1}𝑠_𝑖=𝑠_𝑖+1𝑠,if1≤𝑖≤𝑝−1,₁𝜏,if𝑖=𝑝,,𝑗=1,2,…,𝑞,_𝜔2𝑗=𝜏_𝜔⁻¹_2𝑗−1,𝑗=1,2,…,𝑞,𝜙𝜔_𝑗=𝜔_𝑗+1𝜔,if1≤𝑗≤2𝑞−1,₁,if𝑗=2𝑞. (78) Thus, for instance, if we consider a function 𝑓(𝑠,𝜔) defined by 𝑓_𝜔𝑗𝑠_𝑖=𝑖⋅𝛿_𝑗𝑖,if𝑗=𝑖𝑖,𝑗=1,2,…,𝑝,=0,if𝑖=1,2,…,𝑝;and𝑗=𝑝+1,…,2𝑞 (79) where 𝛿_𝑗𝑖 denotes the Kronecker delta, then the limit function 𝑓^∗_𝜔(𝑠) determined by (76) depends essentially on the two variables 𝑠 and 𝜔 (therefore, the limit function 𝑓^∗_𝜔(𝑠) is not necessarily independent of the random variable 𝜔.) In fact, one can easily find that under the above setting lim_𝑛→∞12𝑛^2𝑛_𝑘=1𝑓_𝜔1𝜏_𝜙^2𝑘−1_𝜔1⋯𝜏_𝜔1𝑠_𝑝=𝑓^∗_𝜔1𝑠_𝑝=12,lim_𝑛→∞12𝑛^2𝑛_𝑘=1𝑓_𝜔𝑝𝜏_𝜙^2𝑘−1_𝜔1⋯𝜏_𝜔1𝑠_𝑝=𝑓^∗_𝜔𝑝𝑠_𝑝=𝑝2. (80) See also Example 3 below.

The following example given by Gładysz [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 (𝑆,𝛽,𝑚) and (Ω,ℱ,𝜇) taken to be 𝑆=Ω=[0,1),𝛽=ℱ= the 𝜎-field of Borel sets, and 𝑚=𝜇= the Lebesgue measure. Let 𝑟 be a fixed integer with 𝑟≥2, and let 𝛽_𝑗, 𝑗=1,2,…,𝑟, be real constants such that 𝛽₁=⋯=𝛽_𝑟−1=1𝑟,𝛽_𝑟1=−1+𝑟. (81) Define a 𝛽⊗ℱ^∗_𝑟-measurable family {𝜏_{[𝜔∗]𝑟}∶𝜔^∗∈Ω^∗} of measure-preserving transformations on 𝑆 by 𝜏_{[𝜔∗]𝑟}𝑠=𝑠+^𝑟_𝑖=1𝛽_𝑖𝜔_𝑖(mod1),𝑠∈𝑆. (82) Take 𝑟₀=1. Then, for a function 𝑓∈𝐿₁(𝑚) given by 𝑓(𝑠)=𝑒^{2𝜋𝑖𝑠}=exp(𝑠), there exists a 𝜇^∗-null set 𝑁^∗ such that for each 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∗_{[𝜔∗]𝑟−1}(⋅)∈𝐿₁(𝑚) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑓𝜏_{[𝜑𝑘−1𝜔∗]𝑟}⋯𝜏_{[𝜔∗]𝑟}𝑠=𝑓^∗_{[𝜔∗]𝑟−1}(𝑠)𝑚-a.e., (83) where 𝑓^∗𝑠,𝜔^∗_𝑟−1=𝐾⋅exp(𝑠)⋅exp𝛽₁𝜔₁+⋯+𝛽₁+⋯+𝛽_𝑟−1𝜔_𝑟−1,𝐻𝜔^∗=exp𝛽₂+⋯+𝛽_𝑟𝜔₁+⋯+𝛽_𝑟𝜔_𝑟−1,𝐾=lim_𝑛→∞1𝑛^𝑛_𝑘=1𝐻𝜑^𝑘𝜔^∗=_Ω∗𝐻𝜔^∗𝑑𝜇^∗𝜔^∗1𝜑isergodic=exp𝑟𝜔₁+2𝜔₂+⋯+(𝑟−1)𝜔_𝑟−1(≠0). (84)

Supplement. Let 0<𝛼<1, 𝛼𝑝>1, and 𝑟₀=1. Then, for the function 𝑓(𝑠)=exp(𝑠)(∈𝐿_𝑝(𝑚)), there exists a 𝜇^∗-null set 𝑁^∗ such that for any 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝑓^∗∈𝐿_𝑝(𝑚⊗𝜇^∗) such that (if necessary, apply Hille’s theorem [22]) lim_𝑛→∞1𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑓𝜏_{[𝜑𝑘𝜔∗]𝑟}⋯𝜏_{[𝜑𝜔∗]𝑟}𝜏_{[𝜔∗]𝑟}𝑠=lim_𝑛→∞1𝑛+1^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑓𝜏_{[𝜑𝑘𝜔∗]𝑟}⋯𝜏_{[𝜑𝜔∗]𝑟}𝜏_{[𝜔∗]𝑟}𝑠=lim_𝜆→1+0(𝜆−1)^∞_𝑛=1𝜆^−(𝑛+1)𝑓𝜏_{[𝜑𝑛−1𝜔∗]𝑟}⋯𝜏_{[𝜑𝜔∗]𝑟}𝜏_{[𝜔∗]𝑟}𝑠=𝑓^∗_{[𝜔∗]𝑟−1}(𝑠)𝑚-a.e. (85) Next, if (for example) 𝑟=5 and 𝑟₀=3, we let 𝛽₁,…,𝛽₅ be real constants such that 𝛽₁≠0,𝛽₂≠0,𝛽₃𝛽=0,₁+𝛽₂≠0,𝛽₄=−𝛽₁,𝛽₅=−𝛽₂. (86) Using the skew product transformation Θ defined by Θ𝑠,𝜔^∗=𝜏_[𝜔∗]5𝑠,𝜑³𝜔^∗, (87) we have for the function 𝑓(𝑠)=exp(𝑠), 𝑓Θ^𝑘𝑠,𝜔^∗=exp𝑠+⁵_𝑗=1𝛽_𝑗𝜔_𝑗+⁵_𝑗=1𝛽_𝑗𝜔_3+𝑗+⋯+⁵_𝑗=1𝛽_𝑗𝜔_3𝑘+𝑗=exp𝑠+𝛽₁𝜔₁+𝛽₂𝜔₂,𝑘=1,2,…. (88) Hence it follows at once that for almost all 𝜔^∗∈Ω^∗, there exists a function 𝑓^∗∈𝐿₁(𝑚⊗𝜇^∗) such that lim_𝑛→∞1𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑓𝜏_{[𝜑3𝑘𝜔∗]5}⋯𝜏_{[𝜑3𝜔∗]5}𝜏_{[𝜔∗]𝑟}𝑠=lim_𝑛→∞1𝑛+1^𝑛_𝑘=0𝑓𝜏_{[𝜑3𝑘𝜔∗]5}⋯𝜏_{[𝜑3𝜔∗]5}𝜏_{[𝜔∗]𝑟}𝑠=lim_𝜆→1+0(𝜆−1)^∞_𝑛=0𝜆^−(𝑛+1)𝑓𝜏_{[𝜑3𝑘𝜔∗]5}⋯𝜏_{[𝜑3𝜔∗]5}𝜏_{[𝜔∗]𝑟}𝑠=exp𝑠+𝛽₁𝜔₁+𝛽₂𝜔₂=𝑓^∗_{[𝜔∗]5−3}(𝑠)=𝑓^∗_[𝜔∗]2(𝑠)𝑚-a.e. (89)

Example 3. In the setting of Example 2, let 𝑞 be an 𝛽-measurable function with |𝑞(𝑠)|=1. Then, for 𝑓∈𝐿_𝑝(𝑚⊗𝜇^∗),1≤𝑝<∞, there exists a 𝜇^∗-null set 𝑁^∗ such that for any 𝜔^∗∈Ω^∗−𝑁^∗, there exists a function 𝐻_𝜔∗∈𝐿_𝑝(𝑚) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑞𝜏_{[𝜔∗]𝑟}𝑠𝑞𝜏_{[𝜑𝜔∗]𝑟}𝜏_[𝜔∗]𝑠⋯𝑞𝜏_{[𝜑𝑘−1𝜔∗]𝑟}⋯𝜏_{[𝜔∗]𝑟}𝑠×𝑓𝜏_{[𝜑𝑘𝜔∗]𝑟}⋯𝜏_{[𝜔∗]𝑟}𝑠=𝐻_𝜔∗(𝑠),𝐻_𝜔∗(𝑠)=𝑞𝜏_{[𝜔∗]𝑟}𝑠𝐻_𝜑𝜔∗𝜏_{[𝜔∗]𝑟}𝑠, (90) for 𝑚-almost all 𝑠∈𝑆. In fact, letting 𝑄(𝑠,𝜔^∗)=𝑞(𝑠)⋅𝑒(𝜔^∗) with 𝑒=1∈𝐿_𝑝(𝜇^∗) and Φ^∗𝑠,𝜔^∗=𝜏_{[𝜔∗]𝑟}𝑠,𝜑𝜔^∗, (91) we have by Gladysz’s theorem ([5], Satz 5) that there exists a function 𝐹∈𝐿_𝑝(𝑚⊗𝜇^∗) such that lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑄Φ^∗𝑠,𝜔^∗𝑄Φ^∗2𝑠,𝜔^∗⋯𝑄Φ^∗𝑘𝑠,𝜔^∗𝑓Φ^∗𝑘+1𝑠,𝜔^∗=𝑄𝑠,𝜔^∗−1⋅𝐹𝑠,𝜔^∗=𝑞(𝑠)⁻¹𝐹𝑠,𝜔^∗,𝐹𝑠,𝜔^∗=𝑄𝑠,𝜔^∗𝐹Φ^∗𝑠,𝜔^∗=𝑞(𝑠)𝐹Φ^∗𝑠,𝜔^∗, (92) almost everywhere on 𝑆×Ω^∗. Hence, we may take 𝐻𝑠,𝜔^∗=𝑞(𝑠)⁻¹𝐹𝑠,𝜔^∗, (93) and then 𝐻𝑠,𝜔^∗=𝑞𝜏_{[𝜔∗]𝑟}𝑠𝐻Φ^∗𝑠,𝜔^∗𝑚⊗𝜇^∗-a.e. (94) Moreover, since 𝑞 is also 𝛽⊗ℱ^∗_𝑟-measurable and 𝐻∈𝐿₁(𝑚⊗𝜇^∗) if we consider the function 𝑞𝑠,𝜔₁,…,𝜔_𝑟=𝑞𝑠+^𝑟_𝑖=1𝛽_𝑖𝜔_𝑖 (95) Then, by Gladysz’s lemma ([5], Hilfssatz 3) there exists a 𝛽⊗ℱ^∗_𝑟−1-measurable function 𝑔=𝑔(𝑠,𝜔₁,…,𝜔_𝑟−1) such that 𝐻𝑠,𝜔^∗=𝑔𝑠,𝜔₁,…,𝜔_𝑟−1𝑚⊗𝜇^∗-a.e. (96) Consequently, we have for any 𝜔^∗∈Ω^∗−𝑁^∗, 𝐻_𝜔∗(𝑠)=𝑔_{[𝜔∗]𝑟−1}(𝑠)𝑚-a.e. (97)

Remark 20. It is an interesting problem to ask what happens if we transform a function 𝑓∈𝐿_𝑝 with a random sequence 𝑇₁,𝑇₂,…,𝑇_𝑛,…, of operators chosen at random from some stock of linear operators on 𝐿_𝑝 given in advance. What can we say about the limit lim_𝑛→∞(1/𝑛)∑^𝑛_𝑘=1𝑇₁𝑇₂⋯𝑇_𝑘𝑓? 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 𝑈,𝑉 (cited in Section 1) means the existence of the a.e. limit of the form lim_𝑛→∞(1/𝑛)∑^𝑛_𝑘=1𝑓(𝑇₁𝑇₂⋯𝑇_𝑘𝑠)) for almost every sequence (𝑇_𝑛)_𝑛≥1 of the infinite sequences obtained by applying 𝑈 and 𝑉 in turn at random. This is just the case that the transformations 𝑇_𝑛,𝑛≥1, are chosen at random with the same distribution and independently. See also Remark 3 (the case that the transformations 𝑇_𝑛,𝑛≥1, are chosen at random, not necessarily with the same distribution but independently). In general the random system {(𝑇_{[𝜔∗]𝑟},𝜑)∶𝜔^∗∈Ω^∗} of linear contractions on 𝐿_𝑝(𝑚) as given in Theorem 4 plays a role of such an advance stock of linear contractions on 𝐿₁(𝑚). To illustrate this, we let 𝑆=𝑋=[0,1) and consider the 𝛽⊗ℱ^∗_𝑟-measurable, 𝑚-measure-preserving transformations 𝜋_{[𝜔∗]𝑟}, 𝜔^∗∈Ω^∗, defined by 𝜋_{[𝜔∗]𝑟}𝑠=𝑠+^𝑟_𝑗=1𝛿_𝑗𝜔_𝑗(mod1),𝑠∈𝑆,𝛿_𝑗≠0. (98) In this case, the random system {(𝜋_{[𝜔∗]𝑟},𝜑)∶𝜔^∗∈Ω^∗} is taken as a stock of measure-preserving transformations on 𝑆. If 𝜔₁,…,𝜔_𝑟 are linearly independent irrational numbers, then {𝜋_{[𝜔∗]𝑟}} is ergodic. Thus, for any 𝑓∈𝐿₁(𝑚) with period 1, lim_𝑛→∞1𝑛^𝑛_𝑘=1𝑓𝑠+𝑘^𝑟_𝑗=1𝛿_𝑗𝜔_𝑗=lim_𝜆→1+0(𝜆−1)^∞_𝑛=1𝜆^−(𝑛+1)𝑓𝑠+𝑛^𝑟_𝑗=1𝛿_𝑗𝜔_𝑗=¹₀𝑓(𝑠)𝑑𝑚𝑚-a.e. (99)

This is an immediate consequence of the ergodicity of the family {𝜋_{[𝜔∗]𝑟}}. We can state this fact in terms of stochastic processes. For example, see Gładysz [5], Satz 3.

Acknowledgment

The author is certainly indebted and very grateful to Mrs. Caroline Nashat for her kind assistance.