Moving Ergodic Averages

• Moving ergodic averages are orbit averages where the averaging interval shifts, deforms, or reweights, generalizing the classical Birkhoff average.
• They incorporate methods such as shifted, bilateral, and weighted formulations to analyze pointwise convergence and the identification of invariant measures.
• This framework extends to higher dimensions and continuous time, impacting both theoretical convergence results and practical forecasting applications.

Moving ergodic averages are orbit averages in which the averaging set is allowed to shift, deform, or be reweighted, rather than remaining the initial interval $[0,N-1]$. In the literature, the term covers shifted window averages $M(v_n,L_n)^T$, centered bilateral averages, weighted Birkhoff averages, higher-dimensional box averages, and nonconventional multi-parameter averages built from polynomial or affine configurations. The main questions are pointwise almost-everywhere convergence, identification of the limiting factor or invariant measure, maximal and variational bounds, and the extent to which divergence or oscillation is forced when the averaging geometry is unfavorable (Adams et al., 2023, Cuny et al., 2020, Bou-Sakr-El-Tayar et al., 21 Nov 2025, Cheng et al., 21 Jul 2025).

1. Basic formulations

The classical reference point is the Birkhoff average

$A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$

for an ergodic measure-preserving transformation $T$. Moving averages replace the fixed initial interval by a variable window. Adams and Rosenblatt define

$$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$

where $v_n\in \mathbb Z$ and $L_n\in \mathbb Z_+$ is strictly increasing. This includes the classical ergodic averages when $v_n=0$ and $L_n=n$ (Adams et al., 2023).

Several adjacent notions are standard. Cuny and Derriennic study the centered bilateral average

$B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$

which is symmetric around the origin in time (Cuny et al., 2020). Weighted formulations replace the uniform kernel by nonnegative weights: $M(v_n,L_n)^T$0 When $M(v_n,L_n)^T$1, this is the classical Birkhoff average; when $M(v_n,L_n)^T$2 with a smooth bump vanishing at the endpoints, the orbit is tapered near the two ends (Bou-Sakr-El-Tayar et al., 21 Nov 2025).

Higher-dimensional versions average over boxes in $M(v_n,L_n)^T$3 or $M(v_n,L_n)^T$4. For a commuting $M(v_n,L_n)^T$5-action generated by $M(v_n,L_n)^T$6, one writes

$M(v_n,L_n)^T$7

with $M(v_n,L_n)^T$8 a translated box. This formulation brings geometric conditions on the family of boxes into the convergence problem (Cheng et al., 21 Jul 2025).

2. Universal convergence and geometric criteria

A central structural result is Adams–Rosenblatt’s equivalence between universal convergence of shifted windows and complete convergence of the standard ergodic averages. For ergodic $M(v_n,L_n)^T$9 and mean-zero $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$0, the condition

$A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$1

is equivalent to: for every choice of shifts $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$2 and every strictly increasing $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$3, the moving averages $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$4 almost everywhere (Adams et al., 2023). In the IID case, Hsu–Robbins and Erdős imply that finite second moment yields this complete-convergence property, and Adams–Rosenblatt also prove a Bernoulli-shift analogue for continuous observables on compact metric Bernoulli systems (Adams et al., 2023).

Bellow–Jones–Rosenblatt’s cone condition gives a geometric characterization for one-dimensional moving intervals. In the $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$5-plane, one associates to each pair $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$6 a right-angle cone with generating lines of slope $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$7; the condition $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$8 for all $A_n(f,x)=\frac1n\sum_{k=0}^{n-1} f(T^k x),$9 is necessary and sufficient for pointwise convergence of $T$0 for every ergodic $T$1 and every $T$2 (Adams et al., 2023).

The higher-dimensional extension is coordinatewise. For box families in $T$3 and $T$4, coordinate-wise cone-growth conditions $T$5 and $T$6 are necessary and sufficient for strong $T$7 boundedness of the associated maximal operators for every $T$8. When all coordinates satisfy the condition, the box averages converge almost everywhere to $T$9 for ergodic actions; failure of even one coordinate condition implies strong sweeping-out (Cheng et al., 21 Jul 2025).

Family	Criterion	Consequence
$$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$0	complete convergence of $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$1	universal a.e. convergence for all $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$2, increasing $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$3
One-dimensional intervals	Cone Condition	a.e. convergence for every ergodic $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$4, every $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$5
Boxes in $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$6 or $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$7	all coordinate-wise $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$8 or $$M(v_n,L_n)^T f(x)=\frac1{L_n}\sum_{i=v_n+1}^{v_n+L_n} f(T^i x),$$9	strong $v_n\in \mathbb Z$0 maximal bounds, $v_n\in \mathbb Z$1, and a.e. convergence

These results distinguish universal theorems from system-specific or observable-specific convergence. A plausible implication is that the geometry of the averaging windows is as decisive as the dynamics once one asks for convergence uniformly over all ergodic systems.

3. Nonconventional and multiple moving averages

A major branch of the subject concerns averages built from several correlated orbit segments. Huang, Shao, and Ye study

$v_n\in \mathbb Z$2

for an ergodic invertible system and $v_n\in \mathbb Z$3. They construct a compact metric strictly ergodic model $v_n\in \mathbb Z$4 such that

$v_n\in \mathbb Z$5

is itself strictly ergodic under the commuting actions $v_n\in \mathbb Z$6 and $v_n\in \mathbb Z$7. From this they deduce that $v_n\in \mathbb Z$8 converges $v_n\in \mathbb Z$9-almost everywhere to the constant given by integrating $L_n\in \mathbb Z_+$0 against the unique $L_n\in \mathbb Z_+$1-invariant measure on $L_n\in \mathbb Z_+$2; the same conclusion holds for any tempered Følner sequence in $L_n\in \mathbb Z_+$3 (Huang et al., 2014).

The same paper proves that if $L_n\in \mathbb Z_+$4 is ergodic distal, then the one-parameter averages

$L_n\in \mathbb Z_+$5

converge almost everywhere. The proof uses Host–Kra seminorms and nilfactors, Weiss’s relative Jewett–Krieger theorem, a two-parameter Van der Corput estimate, Lindenstrauss’s theorem for tempered Følner sequences, and transfinite induction up the distal tower (Huang et al., 2014).

A related model-theoretic approach appears in "Strictly ergodic models and the convergence of non-conventional pointwise ergodic averages," where strengthened Jewett–Krieger models are built to be strictly ergodic for diagonal and skew-product actions, yielding almost-sure convergence for the same two-parameter averages and for cube averages along tempered Følner sequences in $L_n\in \mathbb Z_+$6 (Huang et al., 2013). For commuting transformations, Donoso and Sun prove pointwise convergence of a diagonal-moving average, pointwise convergence of cubic averages, and almost-everywhere convergence of the classical one-parameter averages $L_n\in \mathbb Z_+$7 in the distal case, using sated and magic extensions together with strictly ergodic cube models (Donoso et al., 2016).

4. Weighted, nonlinear, and continuous-time variants

Weighted Birkhoff averages retain the same ergodic limit under mild hypotheses. For ergodic $L_n\in \mathbb Z_+$8 and $L_n\in \mathbb Z_+$9,

$v_n=0$0

The point of weighting is quantitative rather than qualitative: for periodic dynamics the weighted error decays exponentially in $v_n=0$1; for quasiperiodic dynamics and $v_n=0$2, the error is superpolynomial, and for analytic $v_n=0$3 it is exponential; in chaotic or stochastic settings no faster rate is proved in general, although numerically the same $v_n=0$4 decay is observed with smaller constant and reduced oscillations (Bou-Sakr-El-Tayar et al., 21 Nov 2025).

Moving averages also arise along nonlinear integer sequences. O’Keeffe proves that for pairwise commuting measure-preserving transformations and Hardy field functions $v_n=0$5 that are non-polynomial and have distinct growth rates, the averages

$v_n=0$6

converge pointwise almost everywhere for every $v_n=0$7, $v_n=0$8. The proof proceeds through a long-variational inequality, which yields a maximal inequality; for monomial-type growth $v_n=0$9 with non-integer exponents, full $L_n=n$0-variation estimates are obtained (O'Keeffe, 2024).

Koutsogiannis treats multiple averages along tempered functions $L_n=n$1, with $L_n=n$2, and proves $L_n=n$3-convergence to the product $L_n=n$4 under strictly increasing growth and closure of nontrivial linear combinations inside the tempered class. PET induction, nilmanifold equidistribution, and Gowers–Host–Kra seminorms are the main inputs (Koutsogiannis, 2020).

In continuous time, Christ, Durcik, Kovač, and Roos establish almost-everywhere convergence of the quadratic average

$L_n=n$5

for commuting $L_n=n$6-actions and exponents $L_n=n$7 satisfying $L_n=n$8. The analytic core is a curved multilinear singular integral estimate related to the triangular Hilbert transform (Christ et al., 2020).

5. Oscillation, divergence, and generic behavior

Convergence does not remove oscillation. For the bilateral averages $L_n=n$9, Cuny and Derriennic prove a sharp dichotomy: $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$0 converges almost everywhere if and only if both the forward averages $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$1 and backward averages $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$2 converge almost everywhere to the same limit. If $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$3 fails to converge, then

$B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$4

If $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$5, then $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$6 changes sign infinitely often for almost every $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$7, via a reduction to coboundaries and Furstenberg’s double-recurrence theorem (Cuny et al., 2020).

Mondal, Rosenblatt, and Wierdl show that for a fixed ergodic transformation, generic non-constant $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$8 has classical ergodic averages fluctuating around the mean infinitely often almost everywhere. They extend the same perspective to subsequence averages, weighted averages, convolution operators, uniform distribution, and martingales. In this sense, infinite up-down fluctuation is generic even when convergence is known (Mondal et al., 2024).

Adams–Rosenblatt sharpen the distinction between universal and existential convergence. They prove that given $B_n f(x)=\frac1{2n+1}\sum_{i=-n}^n f(T^i x),$9 and ergodic $M(v_n,L_n)^T$00, one can choose a moving average with $M(v_n,L_n)^T$01 strictly increasing such that $M(v_n,L_n)^T$02 does not satisfy the cone condition, yet pointwise convergence still holds almost everywhere. Conversely, for any nonzero $M(v_n,L_n)^T$03, there is a generic class of ergodic maps $M(v_n,L_n)^T$04 such that each map admits an associated moving average $M(v_n,L_n)^T$05 which does not converge pointwise. They also show that coboundary-based universal convergence holds in $M(v_n,L_n)^T$06 but does not extend to $M(v_n,L_n)^T$07 for $M(v_n,L_n)^T$08 (Adams et al., 2023).

6. Extensions, applications, and open directions

Moving ergodic averages extend beyond the invariant-measure setting. Blank introduces natural measures, observable measures, and weak ergodicity for systems in which the transfer operator may be discontinuous or invariant measures may not exist. Under Cesàro attraction of all $M(v_n,L_n)^T$09-smooth initial measures to a limit measure $M(v_n,L_n)^T$10, weak ergodicity of $M(v_n,L_n)^T$11, and absence of wandering measures, the set

$M(v_n,L_n)^T$12

has full reference measure. This shows that almost-everywhere convergence of moving averages can survive without genuine invariance (Blank, 2017).

A distinct line of work treats weighted averages as computational primitives. Weighted Birkhoff averages have been incorporated into weighted Dynamic Mode Decomposition, weighted Extended DMD, weighted SINDy, weighted spectral-measure estimation, and weighted diffusion forecasting. The additional cost is a one-time $M(v_n,L_n)^T$13 multiplication by diagonal weights, while the examples reported include orders-of-magnitude error reduction in quasiperiodic or periodic regimes and measurable gains in El Niño forecasting skill (Bou-Sakr-El-Tayar et al., 21 Nov 2025).

Several open problems remain explicit in the literature. Huang–Shao–Ye note that, via Furstenberg’s structure theorem, the general problem of pointwise convergence for the classical multiple averages is reduced to the remaining weakly mixing extension step (Huang et al., 2014). In higher dimensions, weak type $M(v_n,L_n)^T$14 estimates for moving box averages and convergence along curved shapes or non-flat submanifolds remain unresolved directions; locally flat submanifolds already fail $M(v_n,L_n)^T$15-genericity (Cheng et al., 21 Jul 2025). In continuous time, the purely linear average $M(v_n,L_n)^T$16 for commuting flows is identified as open in the quadratic-curve framework (Christ et al., 2020).

Taken together, these results show that moving ergodic averages form a broad framework rather than a single theorem. Their behavior is controlled by a combination of averaging geometry, characteristic factors, unique ergodicity or distal structure, maximal or variational inequalities, and, in many regimes, an irreducible oscillatory component.