Pointwise Ergodic Theorem Research Articles

Ergodic theorems and the basis of science

New results in ergodic theory show that averages of repeated measurements will typically diverge with probability one if there are random errors in the measurement of time. Since mean-square convergence of the averages is not so susceptible to these anomalies, we are led again to compare the mean and pointwise ergodic theorems and to reconsider efforts to determine properties of a stochastic process from the study of a generic sample path. There are also implications for models of time and the interaction between observer and observable.

Weighted $L_p$ spaces and pointwise ergodic theorem

In this paper we give an operator theoretic version of a recent result of F. J. Mart¹n-Reyes and A. de la Torre concerning the problem of finding necessary and sufficient conditions for a nonsingular point transformation to satisfy the Pointwise Ergodic Theorem in $L_p$. We consider a positive conservative contraction $T$ on $L_1$ of a $\sigma$-finite measure space $(X,\Cal F,\mu)$, a fixed function $e$ in $L_1$ with $e>0$ on $X$, and two positive measurable functions $V$ and $W$ on $X$. We then characterize the pairs $(V,W)$ such that for any $f$ in $L_p(V\,d\mu)$ the averages $$ R_0^n(f,e)=\left.\left(\sum^n_{k=0}T^kf\right)\right/\left(\sum^n_{k=0}T^ke\right) $$ converge almost everywhere to a function in $L_p(W\,d\mu)$. The characterizations are given for all $p$, $1\le p<\infty$.

Pointwise ergodic theorems for radial averages on simple Lie groups I

Pointwise ergodic theorems for radial averages on simple Lie groups I

Harmonic Analysis and Pointwise Ergodic Theorems for Noncommuting Transformations

Let ${F_k}$ denote the free group on $k$ generators, $1 < k < \infty$, and let $S$ denote a set of free generators and their inverses. Define ${\sigma _n} \stackrel {d}{=} \frac {1}{{\# {S_n}}}{\Sigma _{w \in {S_n}}}w$, where ${S_n} = \{ w:|w| = n\}$, and $| \cdot |$ denotes the word length on ${F_k}$ induced by $S$. Let $(X, \mathcal {B}, m)$ be a probability space on which ${F_k}$ acts ergodically by measure preserving transformations. We prove a pointwise ergodic theorem for the sequence of operators $\sigma _n^\prime = \frac {1}{2}({\sigma _n} + {\sigma _{n + 1}})$ acting on ${L^2}(X)$, namely: $\sigma _n^\prime f(x) \to \int _X {f dm}$ almost everywhere, for each $f$ in ${L^2}(X)$. We also show that the sequence ${\sigma _{2n}}$ converges to a conditional expectation operator with respect to a $\sigma$-algebra which is invariant under ${F_k}$. The proof is based on the spectral theory of the (commutative) convolution subalgebra of ${\ell ^1}({F_k})$ generated by the elements ${\sigma _n}, \;n \geq 0$. We then generalize the discussion to algebras arising as a Gelfand pair associated with the group of automorphisms $G({r_1},\;{r_2})$ of a semi-homogeneous tree $T({r_1},\;{r_2})$, where ${r_1} \geq 2,\;{r_2} \geq 2,\;{r_1} + {r_2} > 4$. (The case of ${F_k}$ corresponds to that of a homogeneous tree of valency $2k$.) We prove similar pointwise ergodic theorems for two classes of subgroups of $G({r_1},\;{r_2})$. One is the class of closed noncompact boundary-transitive subgroups, including any simple algebraic group of split rank one over a local field, for example, $PS{L_2}({\mathbb {Q}_p})$. The second class is that of lattices complementing a maximal compact subgroup. We also prove a strong maximal inequality in ${L^2}(X)$ for the groups listed above, as well as a mean ergodic theorem for unitary representations of the groups (due to ${\text {Y}}$. Guivarc’h for ${F_k}$). Finally, we describe the structure and spectral theory of a noncommutative algebra which arises naturally in the present context, namely the double coset algebra associated with the subgroup of $G({r_1},\;{r_2})$ stabilizing a geometric edge. The results are applied to prove mean ergodic theorems for a family of lattices in $G({r_1},\;{r_2})$, which includes, for example, $PS{L_2}(\mathbb {Z})$.

Multiparameter Weighted Ergodic Theorems

AbstractIn this paper we show that multi-dimensional bounded Besicovitch weights are good weights for the pointwise ergodic theorem for Dunford-Schwartz operators and positively dominated contractions of LP. This in particular implies new weighted results for multi-parameter measure preserving point transformations. The proofs show that Besicovitch weights are a very natural class when considered from the operator point of view. We also show that for 1 ≤ r &lt; ∞, the r-bounded Besicovitch classes are all the same, generalizing a result of Bellow and Losert.

On the Pointwise Ergodic Theorem in $L_{p}$

On the Pointwise Ergodic Theorem in $L_{p}$

A generalization of Birkhoff's pointwise ergodic theorem

A generalization of Birkhoff's pointwise ergodic theorem

Pointwise ergodic theorems for functions in Lorentz spaces $L_{pq}$ with p ≠ ∞

Let τ be a null preserving point transformation on a finite measure space. Assuming τ is invertible, P. Ortega Salvador has recently obtained sufficient conditions for the almost everywhere convergence of the ergodic averages in $L_{pq}$ with 1 < p < ∞, 1

Harmonic analysis and pointwise ergodic theorems for noncommuting transformations

Let F k {F_k} denote the free group on k k generators, 1 &gt; k &gt; ∞ 1 &gt; k &gt; \infty , and let S S denote a set of free generators and their inverses. Define σ n = d 1 # S n Σ w ∈ S n w {\sigma _n} \stackrel {d}{=} \frac {1}{{\# {S_n}}}{\Sigma _{w \in {S_n}}}w , where S n = { w : | w | = n } {S_n} = \{ w:|w| = n\} , and | ⋅ | | \cdot | denotes the word length on F k {F_k} induced by S S . Let ( X , B , m ) (X, \mathcal {B}, m) be a probability space on which F k {F_k} acts ergodically by measure preserving transformations. We prove a pointwise ergodic theorem for the sequence of operators σ n ′ = 1 2 ( σ n + σ n + 1 ) \sigma _n^\prime = \frac {1}{2}({\sigma _n} + {\sigma _{n + 1}}) acting on L 2 ( X ) {L^2}(X) , namely: σ n ′ f ( x ) → ∫ X f d m \sigma _n^\prime f(x) \to \int _X {f\,dm} almost everywhere, for each f f in L 2 ( X ) {L^2}(X) . We also show that the sequence σ 2 n {\sigma _{2n}} converges to a conditional expectation operator with respect to a σ \sigma -algebra which is invariant under F k {F_k} . The proof is based on the spectral theory of the (commutative) convolution subalgebra of ℓ 1 ( F k ) {\ell ^1}({F_k}) generated by the elements σ n , n ≥ 0 {\sigma _n},\,\;n \geq 0 . We then generalize the discussion to algebras arising as a Gelfand pair associated with the group of automorphisms G ( r 1 , r 2 ) G({r_1},\;{r_2}) of a semi-homogeneous tree T ( r 1 , r 2 ) T({r_1},\;{r_2}) , where r 1 ≥ 2 , r 2 ≥ 2 , r 1 + r 2 &gt; 4 {r_1} \geq 2,\;{r_2} \geq 2,\;{r_1} + {r_2} &gt; 4 . (The case of F k {F_k} corresponds to that of a homogeneous tree of valency 2 k 2k .) We prove similar pointwise ergodic theorems for two classes of subgroups of G ( r 1 , r 2 ) G({r_1},\;{r_2}) . One is the class of closed noncompact boundary-transitive subgroups, including any simple algebraic group of split rank one over a local field, for example, P S L 2 ( Q p ) PS{L_2}({\mathbb {Q}_p}) . The second class is that of lattices complementing a maximal compact subgroup. We also prove a strong maximal inequality in L 2 ( X ) {L^2}(X) for the groups listed above, as well as a mean ergodic theorem for unitary representations of the groups (due to Y {\text {Y}} . Guivarc’h for F k {F_k} ). Finally, we describe the structure and spectral theory of a noncommutative algebra which arises naturally in the present context, namely the double coset algebra associated with the subgroup of G ( r 1 , r 2 ) G({r_1},\;{r_2}) stabilizing a geometric edge. The results are applied to prove mean ergodic theorems for a family of lattices in G ( r 1 , r 2 ) G({r_1},\;{r_2}) , which includes, for example, P S L 2 ( Z ) PS{L_2}(\mathbb {Z}) .

On the simulation of expectations of random variables depending on a stopping time

Birkhoff's pointwise ergodic theorem with the shift operator over [0,1]11yields a new practical method to compute expectations of functionals of stochastic process. Indeed converges to as Nconverges toward infinity, almost surely. By numerical simulations we will explain the efficiency of this method especially when compared to the classic Monte-Carlo one. It will furthermore be proven that under suitable assumptions a. central limit theorem holds. These assumptions are satisfied in most encountered practical problems. It will precisely be fulfilled when with a stopping time T having a moment of order p, p > 2.Moreover, under this assumptions a “weak” law of iterated logarithm applies. Such that: Numerical simulations were processed.

On the Spectral SLLN and Pointwise Ergodic Theorem in $L^\alpha$

We obtain criteria for the SLLN to hold for processes which are Fourier transforms of random measures. With this spectral approach, we also give criteria for the pointwise ergodic theorem to hold, for some classes of operators between $L^\alpha$-spaces, $1 \leq \alpha < + \infty$. These results apply in particular to contractions on $L^2$. Some random fields extensions are also studied.

Random sets for the pointwise ergodic theorem

AbstractWe generalize a result of Bourgain and devise more general criteria which guarantee that the corresponding random set in Z+ almost surely satisfies a pointwise ergodic theorem on Lp for p &gt; 1. Several large classes of examples are constructed. We also show that under a simple condition the corresponding random set in Z+ almost surely satisfies a pointwise ergodic theorem not only on Lp for p &gt; 1 but also on L1. On the other hand, we establish a criterion to conclude that a certain class of random sets have Banach density zero. In particular, all of the examples mentioned have Banach density zero.

Subsequence pointwise ergodic theorems for operators inL p

In this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations are extended to Dunford-Schwartz operators, positive isometries, and power bounded Lamperti operators.

On numerical integration by the shift and application to Wiener space

Since the advantages of quasi-Monte Carlo methods vanish when the dimension of the basic space increases, the question arises whether there are better methods than the classical Monte Carlo in large or infinite-dimensional basic spaces. We study here the use of the shift operator with the pointwise ergodic theorem whose implementation is particularly interesting. After recalling the theoretical results on the speed of convergence in a form useful for applications, we give sufficient criteria for the law of iterated logarithm in several cases and, in particular, in situations involving the Wiener space.

The symmetric simple exclusion process, II: Applications

We consider the one dimensional nearest neighbour symmetric simple exclusion process. We use the probability estimates obtained in a companion paper, Ferrari et al. (1991), to study some ‘collective’ properties of the particle system. In particular we give another proof of a pointwise ergodic theorem.

Pseudospectral Operators and the Pointwise Ergodic Theorem

We show that for a class of operators which properly contains the normal operators on ${L_2}$, \[ \frac {1}{n}\sum \limits _{i = 0}^{n - 1} {{T^i}f \to a.e.} {\text {iff}}\frac {1}{{{2^n}}}\sum \limits _{i = 0}^{{2^n} - 1} {{T^i}f \to a.e.} \] This theorem is used to give an alternate form of a theorem of Gaposhkin concerning the pointwise ergodic theorem for normal operators.

Pseudospectral operators and the pointwise ergodic theorem

We show that for a class of operators which properly contains the normal operators on L 2 {L_2} , \[ 1 n ∑ i = 0 n − 1 T i f → a . e . iff 1 2 n ∑ i = 0 2 n − 1 T i f → a . e . \frac {1}{n}\sum \limits _{i = 0}^{n - 1} {{T^i}f \to a.e.} {\text {iff}}\frac {1}{{{2^n}}}\sum \limits _{i = 0}^{{2^n} - 1} {{T^i}f \to a.e.} \] This theorem is used to give an alternate form of a theorem of Gaposhkin concerning the pointwise ergodic theorem for normal operators.

On pointwise ergodic theorems for positive operators

On pointwise ergodic theorems for positive operators

Pointwise ergodic theorems for arithmetic sets

converge almost surely for N -+ co, assuming f a function of class L~(~, ~). Here and in the sequel, one denotcs by ~ a probability measure and by T a measure-preserving automorphism. The natural problem of developing the L~-thcory for p < 2 was studied in [B~] and a partial result was obtained. We continue this line of investigation here. The approach used in [Ba] , [Ba] relies on a method which may be summarized as follows: a) Reduction of the general problem to statements about the shift S on Z, which are of a finite and quantitative nature (in the sense of inequalities involving finitely many iterates of the transformation). b) Proof of certain maximal function inequalities, relative to the shift, by Fourier Analysis methods. c) Use of the major arc description of the relevant exponential sums, similar to that in the Hardy-Littlewood circle method.

The spectral measure and Hilbert transform of a measure-preserving transformation

V. F. Gaposhkin gave a condition on the spectral measure of a normal contraction on L 2 {L^2} sufficient to imply that the operator satisfies the pointwise ergodic theorem. We prove that unitary operators which come from measure-preserving transformations satisfy a stronger version of this condition. This follows from the fact that the rotated ergodic Hubert transform is a continuous function of its parameter. The maximal inequality on which the proof depends follows from an analytic inequality related to the Carleson-Hunt Theorem on the a.e. convergence of Fourier series.