Logarithmic moving averages

Abstract

We introduce a moving average summability method, which is proved to be equivalent with the logarithmic ℓ-method. Several equivalence and Tauberian theorems are given. A strong law of large numbers is also proved.

Voronoi means, moving averages, and power series

“…(b) Here we closely follow [8]. From part (a) and the Abelian result of Theorem 9 (i), we have that (i) ⇒ (V, 1, q n , u n ) ⇒ (ii).…”

“…We now generalise the results of [8], which were established for the logarithmic mean ℓ. Let Λ denote the set of all functions u that are invertible and u(x) ∼ u([x]).…”

“…(δ) the deferred Cesàro mean (D, n/λ, n) of Agnew [1], which is the (V, 1, 1, n, λ) average; (µ) the logarithmic moving average L(λ) of [8], which is the (V, 1, 1/(1 + n), log n, λ) average. The next section states our results on the properties of the introduced methods, the relations between them, and a law of large numbers.…”

“…Proof of Theorem 5. Here we follow closely the approach of [8]. To prove (V, p n , q n , u n ) ⇒ (V, p n , q n , u n , λ), let t n → s (Ω).…”

Hardy, Littlewood and probability

“…The classical logarithmic method is a prime example of where one requires less (or deals with coarsenings of the Kolmogorov strong law). This is useful in a number of areas, particularly probability theory and analytic number theory (Bingham and Gashi [22] and [52, § § 4.16, 5.16, 21]).…”