New York Journal of Mathematics

New York J. Math. 3A (1997) 15–30.

Convergence of the p-Series for Stationary Sequences

I. Assani

Abstract. Let (X_{n}) be a stationary sequence. We prove the following

(i) If the variables (X_{n}) are iid and 𝔼(|X_{1}|) < ∞ then

lim_{p→1^{+}}((p − 1)(∑_{n=1}^{∞} \frac{|X_{n}(x)|^{p}}{n^{p}} ))^{1/p} = 𝔼(|X_{1}|), a.e.

(ii) If X_{n}(x) = f(T^{n}x) where (X, F, μ, T) is an ergodic dynamical system, then

lim_{p→1^{+}}((p − 1)(∑_{n=1}^{∞}( \frac{f(T^{n}x)}{n} )^{p}))^{1/p} = ∫ fdμ a.e. for f ≥ 0, f ∈ L log L.

Furthermore the maximal function,

sup_{1<p<∞}(p−1)^{1/p}(∑_{n=1}^{∞}( \frac{f(T^{n}x)}{n} )^{p})^{1/p} is integrable for functions,f ≥ 0, f ∈ L log L.

These limits are linked to the maximal function N^{∗}(x) = ∥( \frac{X_{n}(x)}{n})∥_{1,∞}.

Contents

1. Introduction  15
2. Convergence of the p-series for iid sequences  18
3. Convergence of the p-series for ergodic stationary sequences  23
References  30

1. Introduction

Let Z_{n} be a sequence of independent, identically distributed random variables and (a_{n}) a sequence of positive real numbers. The a.e. convergence of the weighted averages

(∗)  \frac{∑_{n=1}^{N} a_{n}Z_{n}}{A_{n}} ,

Received June 9, 1997. Mathematics Subject Classification. 28D05, 60F15, 60G50. Key words and phrases. pSeries, maximal function, iid random variables and stationary sequences.

\begin{matrix}c \\ ◯ \end{matrix}1997 State University of New York ISSN 1076-9803/97

where A_{n} = ∑_{n=1}^{N} a_{n}, has been characterized by B. Jamison, S. Orey and W. Pruitt ([JOP]). They proved that the condition

(0)  sup_{n} \frac{Ñ_{n}}{n} < ∞

where Ñ_{n} = #\{k : \frac{a_{k}}{A_{k}} ≥ \frac{1}{n}\} is necessary and sufficient for the a.e. convergence of the weighted averages (∗) to 𝔼(Z_{1}). In [A1], interested by the a.e. convergence (y) of averages of the form

\frac{∑_{n=1}^{N} X_{n}(x)g(S^{n}y)}{N},

we considered the maximal function N^{∗}(x) = sup_{n} \frac{N_{n}(x)}{n} where N_{n}(x) = #\{k : \frac{X_{k}(x)}{k} ≥ \frac{1}{n}\}, (X_{k} ≥ 0). We proved the following:

(1) If X_{n} are iid random variables and 𝔼(|X_{1}|) < ∞ then N^{∗}(x) is finite a.e.

(2) If the X_{n} are given by an ergodic dynamical system (i.e., X_{n}(x) = f(T^{n}x) where (X, F, μ, T) is an ergodic dynamical system and f a measurable nonnegative function) then for all p, 1 < p < ∞ there exists a finite constant C_{p} such that

(∗∗)  μ\{x : N^{∗}(x) > λ\} ≤ \frac{C_{p}}{λ^{p}} ∫ |f|^{p}dμ  for all λ > 0.

Furthermore for all p, 1 < p < ∞, for all f ∈ L_{+}^{p} we have lim_{n→∞} \frac{N_{n}(x)}{n} =

∫ fdμ a.e. (A closer inspection of the proof of (∗∗) shows that the constant C_{p} is of the

form \frac{C}{p−1} where C is an absolute constant independent of p.)

If 0 < p < ∞, and (x_{i})_{i≥1} is a sequence of nonnegative real numbers, set

∥(x_{i})∥_{p,∞} = (sup_{λ>0} λ^{p}#{i ≥ 1; |x_{i}| > λ})^{1/p}.

It is easily seen that for r < p

∥(x_{i})∥_{p,∞} ≤ (∑_{i} |x_{i}|^{p})^{1/p} ≤ (\frac{p}{p − r} )^{1/p}∥(x_{i})∥_{r,∞}

(cf. [SW]). In particular, for all p, 1 < p ≤ 2 we have

(3)  (p − 1)^{1/p}(∑_{i} |x_{i}|^{p})^{1/p} ≤ p^{1/p}∥(x_{i})∥_{1,∞}.

As ∥(x_{i})∥_{1,∞} ∼ sup_{n} \frac{#\{k:x_{k}≥1/n\}}{n}, for bounded sequences the previous inequality applied pointwise to a stationary sequence (X_{n}) of integrable functions gives us not only the existence of the p-series

(p − 1)^{1/p}(∑_{i} |\frac{X_{i}(x)}{i}|^{p})^{1/p} for all p, 1 < p ≤ ∞,

Convergence of the p-Series for Stationary Sequences 17

but also the inequality

(4)  sup_{1<p<∞}(p − 1)^{1/p}(∑_{i} |\frac{X_{i}(x)}{i}|^{p})^{1/p} ≤ 2∥(\frac{X_{i}(x)}{i})∥_{1,∞}

if ∥(\frac{X_{i}(x)}{i})∥_{1,∞} < ∞. The inequality (4) and some of our previous results suggest the study of the limit when p tends to 1^{+} of the series

(p − 1)^{1/p}(∑_{i=1}^{∞} |\frac{X_{i}(x)}{i}|^{p})^{1/p}.

Definition. Let (X_{n}) be a stationary sequence of integrable functions. The p series associated to this sequence is the a.e. series (when it exists):

(p − 1)^{1/p}(∑_{i=1}^{∞} |\frac{X_{i}(x)}{i}|^{p})^{1/p}.

In this note, using an elementary lemma on sequence of real numbers, we will show that for (X_{n}) iid with 𝔼(|X_{1}|) < ∞ the p-series

(p − 1)^{1/p}(∑_{i=1}^{∞} |\frac{X_{i}(x)}{i}|^{p})^{1/p} converges a.e. to 𝔼(|X_{1}|)

when p tends to 1^{+}.

The same argument shows that the p series

(p − 1)^{1/p}(∑_{i=1}^{∞} |\frac{∏_{j=1}^{H} X_{j,i}(x_{j})}{i}|^{p})^{1/p} converges a.e. to ∏_{j=1}^{H} 𝔼(|X_{j,1}|)

where (X_{j,n})_{n} are iid random variables satisfying the condition 𝔼(|X_{j,1}|) < ∞, and the variables x_{j} are selected in a universal way specified in [A1].

We can remark that for each p the function G_{p}(x) = (p−1)^{1/p} (∑_{i=1}^{∞} |\frac{X_{i}(x)}{i}|^{p})^{1/p}

is not integrable, as G_{p}(x) ≥ (p − 1)^{1/p} sup_{i} |\frac{X_{i}(x)}{i}|, and for (X_{i}) iid with 𝔼(|X_{1}| log |X_{1}|) = ∞, the function sup_{i} |\frac{X_{i}(x)}{i}| is not integrable, as shown by D. Burkholder in [B]. So F^{∗}(x) = sup_{1<p<∞}G_{p}(x) is a supremum of nonintegrable func-

tions. This makes the handling of the function F^{∗}(x) somewhat delicate.

In the second part of this note we will focus on the ergodic stationary case. We will consider an ergodic dynamical system (X, F, μ, T) and a nonnegative measurable function f. Using (2) we will show first that

\frac{N_{n}(f)(x)}{n} = \frac{#\{k : \frac{f(T^{k}x)}{k} ≥ 1/n\}}{n}

converges in L^{1} norm to ∫ fdμ. Then using extrapolation methods we will show that

(5)  ∥∥(\frac{f(T^{k}x)}{k})∥_{1,∞}∥_{1} < ∞ forf ∈ L(Log L).

One of our interests in (5) lies in the following observation: If we denote by

\frac{f(T^{n^{∗}} x)}{n^{∗}} a decreasing rearrangement of the sequence \frac{f(T^{n}x)}{n}, then we have

(6)  ∥(\frac{f(T^{k}x)}{k})∥_{1,∞} = sup_{n} n\frac{f(T^{n^{∗}}x)}{n^{∗}}.

Hence for f ∈ L(Log L), (6) provides us with some information on the decreasing rate of the sequence \frac{f(T^{n}x)}{n}.

Using (5), we will prove that for f ∈ L log L, f ≥ 0,

(6′)  M_{1}^{∗}(x) = sup_{1<p<∞}(p − 1)^{1/p}(∑_{n=1}^{∞}(\frac{f(T^{n}x)}{n})^{p})^{1/p},

and

(7)  lim_{p→1^{+}}(p − 1)^{1/p}(∑_{n=1}^{∞}(\frac{f(T^{n}x)}{n})^{p})^{1/p} = ∫ fdμ a.e., (μ).

The integrability of M_{1}^{∗}(x) for f in LLogL extends the results on the integrability of the sup_{n} \frac{f(T^{n}x)}{n} in the ergodic case. We do not know at the present time if (7) holds for f ∈ L^{1} . Finally, in the third part of this paper we will study the connection between the maximal operators

M_{1}^{∗}(f)(x) = sup_{1<p<∞}(p − 1)^{1/p}(∑_{n=1}^{∞}(\frac{f(T^{n}x)}{n})^{p})^{1/p}, M_{2}^{∗}(f)(x) = sup_{N} \frac{1}{N} ∑_{n=1}^{N} f(T^{n}x)

and

∥(\frac{f(T^{n}x)}{n})∥_{1,∞} = N^{∗}(f)(x).

If there is no ambiguity we will simply denote these maximal functions by M_{1}^{∗}(x), M_{2}^{∗}(x) and N^{∗}(x).

2. Convergence of the p-series for iid sequences

2.1. The one dimensional case. The next elementary lemma will be useful for the convergence we are looking for.

Lemma 1. Let (x_{n})_{n} be a sequence of nonnegative numbers such that \frac{x_{k}}{k} →_{k} 0 and

\frac{#\{k : \frac{x_{k}}{k} ≥ 1/n\}}{n} ↦ x̄, then

(a) lim_{p→1^{+}}(p − 1)^{1/p} (∑_{n=1}^{∞}(\frac{x_{n}}{n})^{p})^{1/p} = x̄.

(b) If \frac{x_{k^{∗}}}{k^{∗}} is a decreasing rearrangement of the sequence (\frac{x_{k}}{k})_{k} then k · \frac{x_{k^{∗}}}{k^{∗}} converges to x̄.

Proof. We denote by R_{n} = \{k : \frac{x_{k}}{k} ≥ 1/n\} and N_{n} = #\{k : \frac{x_{k}}{k} ≥ 1/n\} = #R_{n}. To prove (a) it is enough to show that

lim_{p→1^{+}}(p − 1)(∑_{n=1}^{∞}(\frac{x_{n}}{n})^{p}) = x̄.

We can write the series (p − 1)(∑_{n=1}^{∞}(\frac{x_{n}}{n})^{p}) in the following way;

(p − 1)(∑_{n=1}^{∞}(\frac{x_{n}}{n})^{p}) = (p − 1)[ ∑_{n∈R_{1}}(\frac{x_{n}}{n})^{p} + ∑_{n∈ℕ^{∗}\R_{1}}(\frac{x_{n}}{n})^{p}]
= A_{p} + B_{p}.

As lim_{p→1}A_{p} = 0 we just need to consider B_{p} = (p − 1) ∑_{n∈ℕ^{∗}\R_{1}}(\frac{X_{n}}{n})^{p}. But we have

(p − 1) ∑_{n=1}^{∞} \frac{N_{n+1} − N_{n}}{(n + 1)^{p}} ≤ B_{p} ≤ (p − 1) ∑_{n=1}^{∞} \frac{N_{n+1} − N_{n}}{n^{p}} .

It is then enough to prove that B_{p} is squeezed into two terms tending to the same limit x̄. We will only prove that the term (p − 1) ∑_{n=1}^{∞} \frac{N_{n+1}−N_{n}}{n^{p}} converges to x̄. The same argument shows the same conclusion for the second term (p − 1) ∑_{n=1}^{∞} \frac{N_{n+1}−N_{n}}{(n+1)^{p}}.

We have

(p − 1) ∑_{n=1}^{∞} \frac{N_{n+1} − N_{n}}{n^{p}} = (p − 1)(−\frac{N_{1}}{1^{p}} + ∑_{n=2}^{∞} \frac{N_{n}(n^{p} − (n − 1)^{p}))}{n^{p}(N − 1)^{p}})

= (p − 1)[−\frac{N_{1}}{1^{p}} + ∑_{n=2}^{∞} \frac{N_{n}(1 − (\frac{(n−1)}{n})^{p}))}{(n − 1)^{p}}]

∼ (p − 1)[−\frac{N_{1}}{1^{p}} + p ∑_{n=2}^{∞} \frac{N_{n}}{n} · \frac{1}{(n − 1)^{p}} ].

As \frac{N_{n}}{n} converges to x̄ and ∑_{n=2}^{∞} \frac{1}{(n−1)^{p}} ∼ \frac{1}{p−1} we conclude that

lim_{p→1^{+}}(p − 1) ∑_{n=1}^{∞} \frac{N_{n+1} − N_{n}}{n^{p}} = x̄.

(b) To obtain the convergence of the sequence k \frac{x_{k^{∗}}}{k^{∗}} to x̄ we can observe that

lim_{t→∞} \frac{#\{ℓ : \frac{x_{ℓ}}{ℓ} ≥ \frac{1}{t}\}}{t} = x̄.

If we take the increasing sequence t_{k} = \frac{k^{∗}}{x_{k^{∗}}} where \frac{x_{k^{∗}}}{k^{∗}} is the k^{th} term of the decreasing rearrangement of the sequence \frac{x_{k}}{k} we can see that

\frac{x_{k^{∗}}}{k^{∗}} · #\{ℓ : \frac{x_{ℓ}}{ℓ} ≥ \frac{x_{k^{∗}}}{k^{∗}} \} = k · \frac{x_{k^{∗}}}{k^{∗}} converges to x̄.

This ends the proof of this lemma. □

In this part we only consider sequences X_{n} of iid nonnegative random variables such that 𝔼(X_{1}) < ∞. This assumption can be made in view of the nature of our p series.

Theorem 2. Let (X_{n}) be a sequence of iid nonnegative random variables such that 𝔼(X_{1}) < ∞. Then we have

(a)  lim_{n→∞} \frac{N_{n}(x)}{n} = 𝔼(X_{1}) a.e. (with N_{n}(x) = #\{k : \frac{X_{k}(x)}{k} ≥ \frac{1}{n}\}),

lim_{p→1^{+}}(p − 1)^{1/p}(∑_{n=1}^{∞} (\frac{X_{n}(x)}{n})^{p})^{1/p} = 𝔼(X_{1}), a.e. (b)

Proof. By the previous lemma, (b) is an immediate consequence of (a), so we are left with proving (a).

In our proof of Lemma 1 in [A1], we showed that we have

∥\frac{X_{n}(x)}{n}∥_{1,∞} < ∞ a.e., because \overline{lim}_{n→∞} \frac{#\{k : \frac{X_{k}(x)}{k} ≥ \frac{1}{n}\}}{n} = 𝔼(X_{1}).

We proved this by noting that

N_{n}(x) = #\{k : \frac{X_{k}(x)}{k} ≥ \frac{1}{n}\} = ∑_{n=1}^{∞} 1\{x : \frac{X_{k}(x)}{k} ≥ \frac{1}{n}\}.

Then we considered

V_{n}(x) = ∑_{n=1}^{∞} 1\{x : \frac{X_{k}(x)}{k} ≥ \frac{1}{n}\} − μ\{x : \frac{X_{k}(x)}{k} ≥ \frac{1}{n}\}.

Kolmogorov’s inequality for sums of independent random variables leads to the following inequality for each ϵ > 0.

∑_{n=1}^{∞} μ\{|\frac{N_{n^{2}}(x) − 𝔼(N_{n^{2}})}{n^{2}} | ≥ ϵ\} < ∞.

An application of the Borel-Cantelli lemma gave us

\overline{lim}\frac{N_{n^{2}}(x)}{n^{2}} = lim_{n} \frac{𝔼(N_{n^{2}})}{n^{2}} = 𝔼(X_{1}).

Then a simple interpolation allowed us to claim that

(8)  \overline{lim}_{n} \frac{N_{n}(x)}{n} = 𝔼(X_{1}).

But also in [A1], Theorem 3 shows that for each p, 1 < p ≤ ∞ we have

(9)  lim_{n→∞} \frac{#\{k : \frac{Y_{k}(x)}{k} ≥ 1/n\}}{n} = 𝔼(Y_{1})

for (Y_{n}) sequence of iid random variables where 𝔼(|Y_{1}|^{p}) < ∞ for some 1 < p ≤ ∞. We take M a positive constant; using (8) and (9) we get

𝔼(X_{1} ∧ M) = lim_{n} \frac{#\{k : \frac{X_{k}(x)∧M}{k} ≥ 1/n\}}{n}

≤ \underline{lim}\frac{#\{k : \frac{X_{k}(x)}{k} ≥ 1/n\}}{n}

= \overline{lim}\frac{#\{k : \frac{X_{k}(x)}{k} ≥ 1/n\}}{n}

= 𝔼(X_{1}).

As lim_{M} 𝔼(X_{1} ∧ M) = 𝔼(X_{1}) we have obtained a proof of (a) from which (b) now follows easily. □

2.2. The multidimensional case. The previous situation can be extended to a more general situation. In [A1] we proved the following:

Given H a positive integer and a nonnegative iid sequence (X_{1,n})_{n} on the probability measure space (Ω_{1}, F_{1}, μ_{1}) satisfying the condition 𝔼(X_{1,1}) < ∞, it is possible to find a set of full measure Ω̃_{1} such that if x_{1} ∈ Ω̃_{1} the following holds:

For all probability measure spaces (Ω_{2}, F_{2}, μ_{2}) and all nonnegative iid sequences (X_{2n})_{n} such that 𝔼(X_{2,1}) < ∞ it is possible to find a set of full measure Ω̃_{2} such that if x_{2} ∈ Ω̃_{2} the following holds:

For all probability measure spaces (Ω_{H}, F_{H}, μ_{H}) and all iid sequences (X_{H,n})_{n} of nonnegative random variables satisfying 𝔼(X_{H,1}) < ∞ we can find a set of full measure Ω̃_{H} for which if x_{H} ∈ Ω̃_{H} we have

(10)  \overline{lim}_{n} \frac{#\{k : ^{\frac{∏_{i=1}^{H} X_{i,k}(x_{i})}{k}} ≥ \frac{1}{n}\}}{n} = ∏_{i=1}^{H} 𝔼(X_{i,1}).

The difficulty resides in the way those sets of full measure Ω̃_{i} are obtained; they are independent of the incoming variables (X_{j,n}) for j > i.

We want to prove that in (10) we actually have convergence to ∏_{i=1}^{H} 𝔼(X_{i,1}). More precisely we have:

Theorem 3. Given H a positive integer and a nonnegative sequence of iid variables (X_{1n})_{n} on the probability measure space (Ω_{1}, F_{1}, μ_{1}) satisfying the condition 𝔼(X_{1,1}) < ∞, it is possible to find a set of full measure Ω̃_{1} such that if x_{1} ∈ Ω̃_{1} the following holds:

For all probability measure spaces (Ω_{2}, F_{2}, μ_{2}) and all nonnegative iid sequences (X_{2,n})_{n} such that 𝔼(X_{2,1}) < ∞, it is possible to find a set of full measure Ω̃_{2} such that if x_{2} ∈ Ω̃_{2} the following holds:

For all probability measure spaces (Ω_{H}, F_{H}, μ, H) and all iid sequences (X_{H,n})_{n} of nonnegative random variables satisfying 𝔼(X_{H,1}) < ∞ we can find a set of full measure Ω̃_{H} for which if x_{H} ∈ Ω̃_{H} we have

lim_{n} \frac{#\{k : ^{\frac{∏_{i=1}^{H} X_{i,k}(x_{i})}{k}} ≥ \frac{1}{n}\}}{n} = ∏_{i=1}^{H} 𝔼(X_{i,1}) (11)

and

lim_{p→1^{+}}((p − 1)(∑_{n=1}^{∞} (\frac{∏_{i=1}^{H} X_{i,n}(x_{i})}{n})^{p})^{1/p} = ∏_{i=1}^{H} 𝔼(X_{i,1}). (12)

Proof. As previously we just need to prove (11) to get (12). We use induction to prove (11). The result is true for H = 1, as shown in the previous theorem.

Let us assume that the result is true for H − 1. Hence if c_{k} = ∏_{i=1}^{H−1} X_{ik}(x_{i}) where x_{i} ∈ Ω̃_{i} we have

(13)  lim_{n→∞} \frac{#\{k : \frac{c_{k}}{k} ≥ \frac{1}{n}\}}{n} = ∏_{i=1}^{H−1} 𝔼(X_{i,1}).

The idea of the proof is the same as in Lemma 1 in [A1]. We have for x_{i} ∈ Ω̃_{i} , 1 ≤ i ≤ H − 1, (X_{H,n}) a sequence of nonnegative iid random variables and for all ϵ > 0

∑_{n=1}^{∞} μ\{x_{H} : |\frac{N_{n^{2}}(x_{H}) − 𝔼(N_{n^{2}})}{n^{2}} | ≥ ϵ\} < ∞ (14)

where

N_{n^{2}}(x_{H}) = \frac{#\{k : \frac{c_{k}X_{H,k}(x_{H})}{k} ≥ 1/n^{2}\}}{n^{2}}.

The inequality (14) is obtained by applying Kolmogorov’s inequality to the series of independent random variables

∑_{k=1}^{∞} 1_{\{x_{H}: \frac{c_{k}X_{H,k}(x_{H})}{k} ≥1/n\}} − μ\{x_{H} : \frac{c_{k}X_{H,k}(x_{H})}{k} ≥ 1/n\}.

The Borel-Cantelli lemma applied to (14) gives us

lim_{n→∞} \frac{N_{n^{2}}(x_{H}) − 𝔼(N_{n^{2}})}{n^{2}} = 0 a.e. (x_{H}).

As lim_{n→∞} \frac{𝔼(N_{n^{2}} )}{n^{2}} = ∏_{i=1}^{H} 𝔼(X_{i,1}) we have

lim_{n→∞} \frac{N_{n^{2}}(x_{H})}{n^{2}} = ∏_{i=1}^{H} 𝔼(X_{i,1}) a.e. (x_{H}).

The monotonicity of N_{n} gives us for p_{n}^{2} ≤ n ≤ (p_{n+1})^{2}

\frac{N_{p_{n}^{2}}(x_{H})}{p_{n}^{2}} ≤ \frac{N_{n}(x_{H})}{p_{n}^{2}} ≤ \frac{N_{(p_{n+1})^{2}}(x_{H})}{p_{n}^{2}} = \frac{N_{(p_{n+1})^{2}}(x_{H})}{(p_{n+1})^{2}} · \frac{(p_{n+1})^{2}}{(p_{n})^{2}} .

This last chain of inequalities implies that

lim_{n→∞} \frac{N_{n}(x_{H})}{n} = lim_{n→∞} \frac{N_{p_{n}^{2}}(x_{H})}{p_{n}^{2}} = ∏_{i=1}^{H} 𝔼(X_{i,1}) as \frac{p_{n}^{2}}{n} → 1.

□

3. Convergence of the p-series for ergodic stationary sequences

In this part the sequence X_{n} will be given by an ergodic dynamical system (X, F, μ, T) on a probability measure space (X, F, μ). The sequence is defined by the relation X_{n}(x) = f(T^{n}x) where f is a nonnegative integrable function.

Proposition 4. Let (X, F, μ, T) be an ergodic dynamical system and f a nonnegative integrable function. We have

lim_{n→∞} ∥\frac{N_{n}(f)}{n} − ∫ fdμ∥_{1} = 0, where

\frac{N_{n}(f)(x)}{n} = \frac{#\{k : \frac{f(T^{k}x)}{k} ≥ 1/n\}}{n}

Proof. We know that lim_{n→∞} \frac{N_{n}(f)}{n} = ∫ fdμ a.e. for f ∈ L_{+}^{p} for some p, 1 < p ≤ ∞ (see Theorem 3 in [A1]). The difficulty at this level comes from the nature of the function of f, N_{n}(f); the map N_{n} is not linear nor positively homogeneous. But we have the following properties:

(A) ∥\frac{N_{n}(f)}{n}∥_{∞} ≤ ∥f∥_{∞}, (B) If f, g are nonnegative functions with disjoint support then we have

\frac{N_{n}(f+g)}{n} = \frac{N_{n}(f)}{n} + \frac{N_{n}(g)}{n} for all n ≥ 1. (C) For all f ≥ 0 integrable functions we have ∥\frac{N_{n}(f)}{n}∥_{1} ≤ ∥f∥_{1}.

(A) and (B) are easy to check.

To establish (C) we take f ∈ L^{1} for which we can find for each ϵ nonnegative numbers (α_{i})_{i} and sets (A_{i})_{i} such that f ≤ ∑ α_{i} 1_{A_{i}}, A_{i} ∩ A_{j} = ϕ if i ≠ j and ∫ ∑ α_{i}1_{A_{i}}dμ ≤ (1 + ϵ) ∫ fdμ. We have

\frac{N_{n}(f)}{n} ≤ \frac{N_{n}(∑_{i=1}^{∞} α_{i}1_{A_{i}})}{n} by monotonicity.

Thus

∥\frac{N_{n}(f)}{n}∥_{1} ≤ ∥\frac{N_{n}(∑_{i=1}^{∞} α_{i}1_{A_{i}})}{n}∥_{1}

= ∥∑_{i=1}^{∞} \frac{N_{n}(α_{i}1_{A_{i}})}{n}∥_{1} by (B)

= ∑_{i=1}^{∞} ∥\frac{N_{n}(α_{i}1_{A_{i}})}{n}∥_{1} .

As

\frac{N_{n}(α_{i}1_{A_{i}})}{n} = \frac{#\{k : \frac{1_{A_{i}} (t^{k}x)}{k} ≥ \frac{1}{nα_{i}} \}}{n}

= \frac{∑_{k=1}^{[nα_{i}]} 1_{A_{i}}(T^{k}x)}{n} we have

∥\frac{N_{n}(α_{i}1_{A_{i}})}{n}∥_{1} = ∑_{k=1}^{[nα_{i}]} \frac{μ(A_{i})}{n} ≤ \frac{(nα_{i})μ(A_{i})}{n} = α_{i}μ(A_{i}) .

So

∥\frac{N_{n}(f)}{n}∥ ≤ ∑_{i=1}^{∞} α_{i}μ(A_{i}) ≤ (1 + ϵ) ∫ fdμ.

As ϵ is arbitrary we have reached a proof of (C).

We are now in a position to prove Proposition 4.

For each positive real number M we can write f = f ∧ M + g_{M} with f ∧ M and g_{M} nonnegative functions with disjoint support.

We have

\frac{N_{n}(f)}{n} − ∫ fdμ = \frac{N_{n}(f ∧ M)}{n} − ∫ f ∧ Mdμ + \frac{N_{n}(g_{M})}{n} − ∫ g_{M}dμ.

Hence

\overline{lim_{n}} ∥\frac{N_{n}(f)}{n} − ∫ fdμ∥_{1} ≤ \overline{lim_{n}} ∥\frac{N_{n}(f ∧ M)}{n} − ∫ (f ∧ M)dμ∥_{1}

+ \overline{lim_{n}} ∥\frac{N_{n}(g_{M})}{n}∥_{1} + ∫ g_{M}dμ.

By the theorem mentioned at the beginning of this proof, associating the a.e. convergence of \frac{N_{n}(f)(x)}{n} to ∫ fdμ for functions in L^{p} for some p, we conclude that

\overline{lim_{n}} ∥\frac{N_{n}(f ∧ M)}{n} − ∫ f ∧ Mdμ∥_{1} = 0.

Hence

\overline{lim_{n}} ∥\frac{N_{n}(f)}{n} − ∫ fdμ∥_{1} ≤ 2 ∫ g_{M}dμ, by (C).

As ∫ g_{M}dμ ⟶_{M} 0, the proof of this proposition is complete. □

Theorem 5. Let (X, F, μ, T) be an ergodic dynamical system and f ∈ L log L, f ≥ 0. Then we have

(a)  ∥∥(\frac{f(T^{k}x)}{k})∥_{1,∞}∥_{1} = ∥sup_{n} n · \frac{f(T^{n^{∗}}x)}{n^{∗}} ∥_{1} < ∞

where \frac{f(T^{n^{∗}} x)}{n^{∗}} is for μ a.e. x a decreasing rearrangement of the sequence \frac{f(T^{n}x)}{n}.

lim_{n→∞} \frac{N_{n}(f)(x)}{n} = ∫ fdμ, μ a.e. (b)

lim_{p→1+}((p − 1) ∑_{n=1}^{∞}(\frac{f(T^{n}x}{n})^{p})^{1/p} = ∫ fdμ, μ a.e. (c)

Proof. First we can make the following observations: For all measurable sets A we have

∥(\frac{1_{A}(T^{k}x)}{k})∥_{1,∞} = sup_{t>0} \frac{#\{k : \frac{1_{A}(T^{k}x)}{k} ≥ 1/t\}}{t} = sup_{n} \frac{#\{k : \frac{1_{A}(T^{k}x)}{k} ≥ 1/n\}}{n}

= sup_{n} \frac{N_{n}(1_{A})(x)}{n} (15)

= N^{∗}(1_{A})(x).

Because of the maximal inequality for the ergodic averages we have

(16)  μ\{x : N^{∗}(1_{A})(x) > λ\} ≤ \frac{1}{λ} · μ(A)  for all λ > 0.

(Note that N^{∗}(1_{A})(x) ≤ 1, hence for all p ≥ 1 we also have

(17)  μ\{x : N^{∗}(1_{A})(x) > λ\} ≤ \frac{1}{λ^{p}} · μ(A)).

For all positive real numbers y we have:

(18)  y^{\frac{i}{i+1}} = y^{\frac{i}{i+1}} · \frac{(i + 1)^{1/i+1}}{(i + 1)^{1/i+1}} ≤ \frac{y(i + 1)^{1/i}}{(i + 1)} i + \frac{1}{(i + 1)^{2}}

26 I. Assani (apply the inequality ab ≤ \frac{a^{p}}{p} + \frac{b^{q}}{q}, for a = y^{i/i+1} · (i + 1)^{1/i+1}, b = \frac{1}{(i+1)^{1/i+1}} , p = \frac{i+1}{i} and q = \frac{p}{p−1} = i + 1).

We proceed now with the proof of Theorem 5 (a).

We take f ∈ L log L and denote by A_{i} the set

A_{i} = {2^{i} ≤ f < 2^{i+1}}.

We have

N^{∗}(f) ≤ N^{∗}(∑_{i=1}^{∞} 2^{i+1}1_{A})) ≤ ∑_{i=1}^{∞} N^{∗}(2^{i+1}1_{A}))

= 2 ∑_{i=1}^{∞} 2^{i} · N^{∗}(1_{A}).

By taking the integral with respect to the measure μ we get

∥N^{∗}(f)∥_{1} ≤ 2 ∑_{i=1}^{∞} 2^{i}∥N^{∗}(1_{A_{i}})∥_{1}

Using (17) we get

∥N^{∗}(1_{A_{i}})∥_{1} ≤ \frac{p}{(p − 1)} sup_{t>0}[t · μ\{x : n^{∗}(1_{A_{i}})(x) > t\}]

≤ \frac{p}{(p − 1)} · (μ(A_{i}))^{1/p} for all p, 1 ≤ p < ∞.

((17) is combined with the inequality ∥g∥_{L^{1}} ≤ \frac{p}{(p−1)} sup_{t>0}[tμ\{x : |g(x)| > t\}^{1/p}].) Going back to the evaluation of ∥N^{∗}(f)∥_{1} we get

∥N^{∗}(f)∥_{1} ≤ 2 ∑_{i=1}^{∞} 2^{i} \frac{(i + 1/i)}{1/i}(μ(A_{i}))^{1/i+1}

= 2 ∑_{i=1}^{∞} 2^{i}(i + 1)(μ(A_{i}))^{i/i+1}

= 2 ∑_{i=1}^{∞}((2^{i}(i + 1))^{i+1/i}μ(A_{i}))^{i/i+1}.

Applying (18) to each term ((2^{i}(i + 1))^{i+1/i}μ(A_{i}))^{i/i+1} we get

∥N^{∗}(f)∥_{1} ≤ 2 ∑_{i=1}^{∞}[(2^{i}(i + 1))^{i+1/i} · (μ(A_{i}))\frac{(i + 1)^{1/i}}{i + 1} i + \frac{1}{(i + 1)^{2}} ]

≤ 4 ∑_{i=1}^{∞}[2^{i}iμ(A_{i}) · (1 + i)^{2/i} + \frac{1}{(i + 1)^{2}} ]

≤ 12 · ∑_{i=1}^{∞}[2^{i}iμ(A_{i}) + \frac{1}{(i + 1)^{2}} ]

≤ \frac{12}{ln 2}[∫ f log fdμ + 1].

Convergence of the p-Series for Stationary Sequences 27

Thus we have proved the following inequality

(19) ∥N^{∗}(f)∥_{1} ≤ \frac{12}{ln 2}[∫ f log fdμ + 1] for all f ≥ 0, f ∈ L log L.

This clearly ends the proof of Theorem 5 (a).

It remains to show (b). Our goal is to prove that for f ≥ 0, f ∈ L log L

(20)  \frac{lim}{n}N^{∗}(f − f ∧ n) = 0 a.e.

Using (19) we have for all t > 0,

∥N^{∗}(t(f − f ∧ n))∥_{1} ≤ \frac{12}{ln 2}[∫ (t(f − f ∧ n)) log(t(f − f ∧ n))dμ + 1]

for all f ≥ 0, f ∈ L log L.

This last inequality gives us

∥N^{∗}(f − f ∧ n)∥_{1} ≤ \frac{12}{ln 2}[∫ (f − f ∧ n) log(t(f − f ∧ n)]dμ + \frac{1}{t}].

At the expense of taking a subsequence, we derive from it

lim_{k} ∥N^{∗}(f − f ∧ n_{k})∥_{1} ≤ \frac{12}{ln 2} · \frac{1}{t}.

Then we easily get \frac{lim}{n}N^{∗}(f − f ∧ n) = 0 a.e. This proves (20).

As lim_{n} \frac{N_{k}(f∧n)}{k} = ∫ f ∧ ndμ, because f ∧ n is clearly bounded we have

\frac{N_{k}(f ∧ n)}{k} ≤ \frac{N_{k}(f)}{k} = \frac{N_{k}(f ∧ n)}{k} + \frac{N_{k}(f − f ∧ n)}{k}

and after taking the limits we obtain

∫ f ∧ ndμ ≤ \frac{lim}{k} \frac{N_{k}(f)}{k} ≤ \overline{lim}\frac{N_{k}(f)}{k} ≤ \overline{lim}\frac{N_{k}(f ∧ n)}{k} + N^{∗}[f − f ∧ n]

= ∫ f ∧ ndμ + N^{∗}[f − f ∧ n].

Finally, by taking the lim inf with respect to n we can conclude that

lim_{k} \frac{N_{k}(f)}{k} = ∫ fdμ a.e.

This proves Theorem 5 (b). Theorem 5 (c) now follows easily from Lemma 1. This ends the proof of Theorem 5. □

Corollary 6. Let (X, F, μ, T) be an ergodic dynamical system and f ∈ L log L, f ≥ 0. Then there exists an absolute constant C such that

∥ sup_{1<p<∞}(p − 1)^{1/p}(∑_{n=1}^{∞}(\frac{f(T^{n}x)}{n})^{p})^{1/p}∥_{1} ≤ C[∫ f log fdμ + 1]

Proof. By Theorem 5 we know that

∥sup_{n} n · \frac{f(T^{n^{∗}}x)}{n^{∗}} ∥_{1} < ∞.

As for μ a.e. x , for each p, we have

(p − 1)^{1/p}(∑_{n=1}^{∞}(\frac{f(T^{n}x)}{n})^{p})^{1/p} ≤ (sup_{n} n · \frac{f(T^{n^{∗}}x)}{n^{∗}}) · ((p − 1) ∑_{n=1}^{∞} 1/n^{p})^{1/p},

the corollary follows easily. □

Remark.

1) One can see that the limit when p tends to ∞ of the p-Series is equals to sup_{n} \frac{f(T^{n}x)}{n} . This is the reason why we only focus on the existence of the limit when p tends to 1+.

2) We proved in [A2] that if N^{∗}(f)(x) is a.e finite for all functions f ∈ L_{+}^{1} then M_{2}^{∗}(f)(x) is also a.e finite for all functions f ∈ L^{1}.

3) The results obtained in this note can be extended to increasing sequences of integers (p_{n})_{n}. The corresponding maximal function to consider is simply

∥(\frac{f(T^{p_{n}})(x)}{n})∥_{1,∞}.

To illustrate this we have the following Proposition.

Proposition 7. Let (X, F, μ, T) be an ergodic dynamical system , p a fixed positive real number 1 < p < ∞ and (p_{n})_{n} an increasing sequence of positive integers. Consider the following statements

(a)  ∥(\frac{f(T^{p_{n}}x)}{n})∥_{1,∞} < ∞ a.e. for all f ∈ L_{+}^{p}.

(b)  sup_{k} k^{p−1} ∑_{n=k}^{∞}(\frac{f(T^{p_{n}}x)}{n})^{p} < ∞ a.e. for all f ∈ L_{+}^{p}.

(c)  sup_{N} \frac{1}{N} ∑_{n=1}^{N} f(T^{p_{n}}x) < ∞ a.e. for all f ∈ L_{+}^{p}.

(d)  sup_{N} \frac{1}{N} ∑_{n=1}^{N} f(T^{p_{n}}x) < ∞ a.e. for all f ∈ L(p, 1).

Then we have the following implications: (a) implies (d), (b) is equivalent to (c),

(b) implies (a) and (c) implies (d).

Proof. The implications (c) implies (b) and (b) implies (a) can be proved the same way we did in [A1] for the usual Cesaro averages. (See the proof of Theorem 3 part b) in [A1]). The implication (c) implies (d) is a direct consequence of the structure of L(p, q) spaces as shown in [SW].

It remains to prove the implications (a) implies (d) and (b) implies (c). For (a) implies (d), we can notice that (a) implies the existence of a finite constant C_{p} such that for all f ∈ L_{+}^{p}

(21)  μ\{x : ∥(\frac{f(T^{p_{n}}x}{n})∥_{1,∞} > λ\} ≤ \frac{C_{p}}{λ^{p}} ∫ |f|^{p}dμ  for all λ > 0.

In the particular case of f = 1_{A}, (21) will give us the following

(22)  μ\{x : sup_{N} \frac{1}{N} ∑_{n=1}^{N} 1_{A}(T^{p_{n}}x) > λ\} ≤ \frac{C_{p}}{λ^{p}} μ(A)  for all λ > 0

because

∥(\frac{1_{A}(T^{p_{n}}x)}{n})∥_{1,∞} = sup_{N} \frac{1}{N} ∑_{n=1}^{N} 1_{A}(T^{p_{n}}x).

As this inequality is valid for all measurable sets A, we can conclude that the maximal operator

M^{∗}(f)(x) = sup_{N} \frac{1}{N} ∑_{n=1}^{N} f(T^{p_{n}}x)

is of restricted weak type (p,p) (see [SW]). In other words, the maximal operator M^{∗} maps the characteristic function of any measurable set A from L(p, 1) into L(p, ∞). It is shown in [SW] that the nature of the maximal operator M^{∗} and the existence of an equivalent norm on L(p, ∞), making it a Banach space, M^{∗} maps continuously all functions f ∈ L(p, 1) into L(p, ∞). This means the existence of a finite constant C_{p} such that for all f ∈ L(p, 1),

(23)  μ\{x : sup_{N} \frac{1}{N} ∑_{n=1}^{N} f(T^{p_{n}}x) > λ\} ≤ \frac{C_{p}}{λ^{p}} ∥f∥_{p,1}^{p}  for all λ > 0.

From this we can clearly derive (d).

The implication (b) implies (c) can be obtained by summation. For f ∈ L_{+}^{p} let us denote by C_{x} the finite constant which dominates the sup on k. Then for each k, we have

k^{p−1} ∑_{n=k}^{2k}(\frac{f(T^{p_{n}}x)}{2k})^{p} < C_{x}.

This implies the inequality

sup_{k} \frac{1}{k} ∑_{n=k}^{2k} f(T^{p_{n}}x)^{p} < 2^{p}C_{x}.

From this we can derive by convexity the uniform boundedness of the averages \frac{1}{k} ∑_{n=k}^{2k} f(T^{p_{n}}x). This property allows us to obtain (c) without difficulty. □

Remark. It would be interesting to know if (a) implies (c) for any increasing sequence p_{n} and any p, 1 ≤ p < ∞ . For p = 1, p_{n} = n we already mentioned that (a) implies (c) (see [A2]).

30 I. Assani

References

[A1] I. Assani, Strong laws for weighted sums of iid random variables, Duke Math. J. 88 (1997), 217–246.

[A2] I. Assani, Return times and Birkhoff theorems in L1 of product measures, Preprint.

[B] D. Burkholder, Successive conditional expectations of an integrable function, AM St 33 (1962), 887–893.

[JOP] B. Jamison, S. Orey, and W. Pruitt, Convergence of weighted averages of independent random variables, Z. Wahrheinlichkeitstheorie Verw. Gebiete. 4 (1965), 40–44.

[SW] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971.

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599

assani@math.unc.edu Typeset by AMS-TEX