On the UMD constant of the space $ℓ₁^{N}$

• Autorzy: Adam Osękowski
• Języki publikacji: EN

Abstrakty

EN

Let N ≥ 2 be a given integer. Suppose that $df = (dfₙ)_{n≥0}$ is a martingale difference sequence with values in $ℓ₁^{N}$ and let $(εₙ)_{n≥0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate
$ℙ(sup_{n≥0} ||∑_{k=0}^{n} ε_{k} df_{k}||_{ℓ₁^{N}} ≥ 1) ≤ (ln N + ln(3ln N))/(1 - (2ln N)^{-1}) sup_{n≥0} 𝔼||∑_{k=0}^{n} df_{k}||_{ℓ₁^{N}}$.
It is shown that this result is asymptotically sharp in the sense that the least constant $C_{N}$ in the above estimate satisfies $lim_{N→ ∞} C_{N}/ln N = 1$. The novelty in the proof is the explicit verification of the ζ-convexity of the space $ℓ₁^{N}$.

Kategorie tematyczne

• 60G42: Martingales with discrete parameter
• 60G44: Martingales with continuous parameter

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2016
• Tom: 142
• Numer: 1
• Strony: 135-147

Daty

wydano

2016

Twórcy

autor

Adam Osękowski

• Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Identyfikatory

DOI

10.4064/cm142-1-7