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

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))), ….

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 $$ be the two-sided infinite direct product measure space of (Ω, ℱ, μ). Then, for any f ∈ L_p(m)  (p ≥ 1), there exists a $$-null set N^* such that for any $$, there exists a function $$ such that

where ℵ_n(ω^*) denotes the nth coordinate of ω^*, and this holds also in the norm of L_p(m).

Remark 3. In Kakutani’s random ergodic theorem (as well as in Pitt-Ulam-von Neumann’ theorem), the sequence $$ 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

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]).

Theorem 4. Let r₀ be any fixed integer with 1 ≤ r₀ ≤ r. Let $$ be a strongly $$-measurable family of positive linear contractions on L₁(m). Suppose that there exists a strictly positive L₁(m)-function invariant under $$. Then, for every f ∈ L₁(m), there exists a μ^*-null set N^* ∈ ℱ^* such that for any ω^* ∈ Ω^* − N^*, there exists a function $$ such that

Proof. We need the following lemmas.

Lemma 5 (see [10, 14]). The strong $$-measurability of the operator family $$ guarantees that for any g ∈ L₁(m ⊗ μ^*), there exists a uniquely determined β ⊗ ℱ^*-measurable version $$ of $$ such that excepting a μ^*-null set,

Using the measurable version appearing in Lemma 5, we define

From the norm conditions of $$ in Theorem 4, it turns out that T is a linear operator on L₁(m ⊗ μ^*) with $$. 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 $$ such that

Moreover, as is easily checked, we find that

One can easily verify that excepting a suitable μ^*-null set $$,

for all k = 1,2, …. Next we wish to show that $$ does depend essentially only on $$. To do this, we define sub-σ-fields ℘_n, n = 1,2, …, of β ⊗ ℱ^* by

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

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 $$ only for the case when $$, is of the form (15).

Lemma 6. It holds that $$.

Proof. It follows that for a sufficiently large ℓ and n < ℓ, E(h∣℘_n) is such that

Thus, excepting a μ^*-null set $$, we get

On the other hand, since excepting a μ^*-null set we have that if n ≥ r − r₀, then Consequently, and by approximation, and so by iteration, In particular, Since T is the adjoint operator of T^*, we have thus

We return to the proof of the theorem. By Lemma 6 and the martingale convergence theorem (cf. [16, 17]), we have

as m → ∞, and thus which implies that $$ depends essentially only on $$. Hence, the theorem follows from (11), (26), and Fubini’s theorem. The proof of Theorem 4 has hereby been completed.

Theorem 7. Let r₀ be any fixed integer with 1 ≤ r₀ ≤ r. Let $$ be a strongly $$-measurable family of linear contractions on L₁(m) with $$ for all ω^* ∈ Ω^*. Then, for every f ∈ L_p(m ⊗ μ^*) with 1 ≤ p < ∞, there exists a μ^*-null set N^* ∈ ℱ^* such that for any ω^* ∈ Ω^* − N^*, there exists a function $$ such that

and that if 1 < p < ∞, then and that if m(S) < ∞ and f ∈ L(S × Ω^*)log ⁺L(S × Ω^*), then

Proof. As before, we define $$ for g ∈ L₁(m ⊗ μ^*). From the norm conditions of $$ in Theorem 7, it turns out that T is a linear operator on L₁(m ⊗ μ^*) with $$ and $$. Then, It follows from the Riesz convexity theorem that $$ for all p with 1 < p < ∞. Thus, for any f ∈ L_p(m ⊗ μ^*)  (1 ≤ p < ∞), we can apply Dunford and Schwartz’s ergodic theorem [18] to ensure the existence of a function $$ such that

and that if 1 < p < ∞, then and that if m(S) < ∞ and f ∈ L(S × Ω^*)log ⁺L(S × Ω^*), then

Now, adapting the (almost) same argument as used in the proof of Theorem 4, we can find that

Note here that if f ∈ L_p(m ⊗ μ^*), 1 < p < ∞, (or f ∈ L(S × Ω^*)log ⁺L(S × Ω^*)), then 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.

Theorem 8. Let r₁, …, r_ν be ν positive integers. Let $$, i = 1,2, …, ν, be strongly $$-measurable families of linear contractions on L₁(m) with $$. For each i = 1,2, …, ν, we set

Then, for every f ∈ L₁(m), there exists a μ^*-null set N^* such that for every ω^* ∈ Ω^* − N^*, there exists a function $$ with q = max (r₁, …, r_ν) such that the multiple averages converge to $$. as n₁ → ∞, …, n_ν → ∞ independently. Here, $$ 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^*)

Each T_i turns out to be a linear contraction on L₁(m ⊗ μ^*) with $$. So, it follows from the Riesz convexity theorem that $$ for 1 < p < ∞. Hence, from Dunford-Schwartz’s ergodic theorem [18], we have that for every $$ the multiple averages converge to $$ 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 Note here that Thus, by Lemma 6 and the (L_p) martingale convergence theorem, we find that excepting a μ^*-null set Finally, observe that excepting a μ^*-null set, we get In fact, we have for two operators T₁ and T₂ 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 $$, the theorem follows from the above arguments.

Theorem 9. Let r₁, …, r_ν be ν positive integers. Let $$, i = 1,2, …, ν, be a strongly $$-measurable commuting family of linear contractions on L₁(m) with $$. Then, for every f ∈ L₁(m), there exists a μ^*-null set N^* such that for every ω^* ∈ Ω^* − N^*, there exists a function $$ with q = max (r₁, …, r_ν) such that

Theorem 10. Let r₁, …, r_ν be ν positive integers. Let $$, i = 1,2, …, ν, be strongly $$-measurable commuting family of linear contractions on L₁(m) with $$. Then, for every f ∈ L₁(m), there exists a μ^*-null set N^* such that for every ω^* ∈ Ω^* − N^*, there exists a function $$ with q = max (r₁, …, r_ν) such that the multiple averages

converge to $$. as u(n)  ( = (n₁, …, n_ν)≥(1, …, 1))→∞ in a sector of $$.

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

so that 1 is also U-subinvariant. Thus, for f ∈ L₁(m ⊗ μ^*)∩L_∞(m ⊗ μ^*), Since U is a positive contraction on L₁(m ⊗ μ^*), and U is a positive Dunford-Schwartz operator on L₁(m ⊗ μ^*).

Theorem 12. Let $$ be a strongly $$-measurable family of positive linear contractions on L₁(m), where each $$ is the adjoint operator of the corresponding sub-Markovian operator $$. Assume either of the following conditions:

• (i)

  m  is finite and E(U · ∣℘₀)-subinvariant,

• (ii)

  m is σ-finite and $$ for all ω^* ∈ Ω^*.

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.

Theorem 13. Let r₀ be any fixed integer with 1 ≤ r₀ ≤ r. Let $$ be a strongly $$-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 $$ such that

• (i)

  $$⋯$$   for m-almost all s ∈ S,

• (ii)

  $$⋯$$,

• (iii)

  $$⋯$$(s) = 0,

Theorem 14. Let r₀ be any fixed integer with 1 ≤ r₀ ≤ r and $$ be a $$-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 $$ such that

Proof. Define the skew product Φ^* of ϕ (the one-sided shift transformation of Ω^*) and $$ as follows:

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

Furthermore, applying Lemma 6 to the operator T^* induced by Φ^*, it follows that so that to complete the proof of the theorem, we may take

Remark 15. For example, if r₀ = r = 1 in Theorem 14, then $$. 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].

Theorem 16. Let $$ be a strongly $$-measurable family of positive linear contractions on L₁(m), where each $$ is the adjoint operator of the corresponding sub-Markovian operator $$. 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 $$ such that

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 $$ 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 $$ 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

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 $$ given in L₁(m) with $$. Then, there exists a μ^*-null set N^* such that for any ω^* ∈ Ω^* − N^*, there exists a function $$ such that

Proof. It follows that there exists a μ^*-null set $$ such that for any $$,

Moreover, we have Therefore, by Theorem 7, there exists a μ^*-null set $$ such that for every $$, there exist functions $$, $$ such that Hence, the theorem follows by putting $$ and

Theorem 19. Let $$ be a random affine system given in L₁(m) with $$. If f ∈ L₁(m  ⊗  μ^*), then there exists a μ^*-null set N^* such that for any ω^* ∈ Ω^* − N^*, there exists a function $$ such that

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 $$ such that

which follows from Birkhoff’s ergodic theorem applied to the so-called skew product Φ of ϕ and {τ_ω} defined by If the skew product transformation Φ is ergodic, then the function $$ is constant almost everywhere on S × Ω. It is worthwhile to notice that, in general, the limit function $$ 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 Thus, for instance, if we consider a function f(s, ω) defined by where δ_ji denotes the Kronecker delta, then the limit function $$ determined by (76) depends essentially on the two variables s and ω (therefore, the limit function $$ is not necessarily independent of the random variable ω.) In fact, one can easily find that under the above setting See also Example 3 below.

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

Define a $$-measurable family $$ of measure-preserving transformations on S by 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 $$ such that where

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 $$ such that

for m-almost all s ∈ S. In fact, letting Q(s, ω^*) = q(s) · e(ω^*) with e = 1 ∈ L_p(μ^*) and we have by Gladysz’s theorem ([5], Satz 5) that there exists a function F ∈ L_p(m ⊗ μ^*) such that almost everywhere on S × Ω^*. Hence, we may take and then Moreover, since q is also $$-measurable and H ∈ L₁(m ⊗ μ^*) if we consider the function Then, by Gladysz’s lemma ([5], Hilfssatz 3) there exists a $$-measurable function g = g(s, ω₁, …, ω_r−1) such that Consequently, we have for any ω^* ∈ Ω^* − N^*,

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 $$ Unfortunately, we cannot expect any convergence for every random sequence chosen from the stock. Therefore, it is desirable to consider how to choose almost every (not every) random sequence from the stock (cf. Revesz [30] and Yoshimoto [31]). In Pitt [1] and Ulam and von Neumann [2], the random ergodic theorem for two-measure-preserving transformations U, V (cited in Section 1) means the existence of the a.e. limit of the form $$ 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 $$ of linear contractions on L_p(m) as given in Theorem 4 plays a role of such an advance stock of linear contractions on L₁(m). To illustrate this, we let S = X = [0,1) and consider the $$-measurable, m-measure-preserving transformations $$, ω^* ∈ Ω^*, defined by

In this case, the random system $$ is taken as a stock of measure-preserving transformations on S. If ω₁, …, ω_r are linearly independent irrational numbers, then $$ is ergodic. Thus, for any f ∈ L₁(m) with period 1,