On the Dependence of the Convergence Rate in the Strong Law of Large Numbers for Stationary Processes on the Rate of Decay of the Correlation Function

An effective Ratner equidistribution result for SL(2,R)⋉R2

“…y), which is clearly subsumed by the error term in (12). Hence from now on we focus on the terms with c = 0.…”

“…On the other hand, using the fact that the flow {U t } is mixing on smooth vectors in L 2 (X) with a rate t ε−1 as t → ∞ (as follows from [4] combined with an argument as in [37,Lemma 2.3] 2 ), one can prove that for sufficiently nice test functions f on Γ\G, and for µ-almost all Γg ∈ X, the deviation of the ergodic average in the left hand side of (9) decays like T ε− 1 2 as T → ∞; cf. [12]. In this last statement the µ-null set of exceptional points Γg is non-explicit and depends on f ; furthermore the implied constant in the bound depends on both f and Γg in a non-explicit way; the strength of Theorem 1.6 lies of course in the fact that it gives a bound where all these dependencies are explicit.…”

“…Hence (23) accounts for the leading term in (12) in Theorem 3.1. We also note that the error term in (23) is subsumed by the error term in (12)…”

Extensions of the Menchoff-Rademacher theorem with applications to ergodic theory

“…The case b n ≡ 1 was treated in Gaposhkin [18,Theorem 3] when p = 2 and T is unitary on L 2 , in Derriennic and Lin [11,Corollary 3.7] when T is a Dunford-Schwartz operator, and in Weber [52, Proposition 1.6] in the general case treated here. Applying Kronecker's lemma to the series in (i) (with b n ≡ 1) yields the same "strong law with rate" as Weber for β > (p − 1)/p, but our rate obtained directly in (i) is better; the rate in the "strong law of large numbers" obtained from (ii) by Kronecker's lemma is the same as Weber's when β = (p − 1)/p, but worse than Weber's (in the power of the logarithm) when β < (p − 1)/p.…”

Uniform bounds under increment conditions

“…For T unitary on L 2 and f ∈ L 2 satisfying (1.14), the rates obtained by Gaposhkin [Ga,Theorem 3,cases (vii), (iv), and (iii)] are better than those of Proposition 1.8.…”