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Limit theorems for random Dirichlet series: boundary case

Alexander Iksanov, Ruslan Kostohryz

Abstract

Buraczewski et al (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series $\sum_{k\geq 2}(\log k)^αk^{-1/2-s}η_k$ as $s\to 0+$, where $α>-1/2$ and $η_1$, $η_2,\ldots$ are independent identically distributed random variables with zero mean and finite variance. We prove a FLT and a LIL in a boundary case $α=-1/2$. The boundary case is more demanding technically than the case $α>-1/2$.

Limit theorems for random Dirichlet series: boundary case

Abstract

Buraczewski et al (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series as , where and , are independent identically distributed random variables with zero mean and finite variance. We prove a FLT and a LIL in a boundary case . The boundary case is more demanding technically than the case .

Paper Structure

This paper contains 5 sections, 13 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['main']}
  4. Proof of Theorem \ref{['thm:flt']}
  5. A failure of approach based on a strong approximation

Key Result

Proposition 1.1

Assume that $\mathbb{E}[\eta]=0$, $\sigma^2:=\mathbb{E} [\eta^2]\in (0,\infty)$ and let $\alpha>-1/2$. Then on $C(0,\infty)$, where $(B(y))_{y\geq 0}$ is a standard Brownian motion.

Theorems & Definitions (22)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 12 more