[Paper Review] A note on directly Riemann integrable functions

Francesco Caravenna|arXiv (Cornell University)|Oct 8, 2012

Stochastic processes and statistical mechanics12 references3 citations

TL;DR

This paper establishes that for any Lebesgue-integrable function $ f $ satisfying mild moment and $ L^p $-boundedness conditions, some $ k $-fold convolution $ f^{*k} $ becomes directly Riemann integrable (d.R.i.). The key result shows that $ (1+|x|^{\varepsilon})f^{*k}(x) $ is bounded, continuous, and d.R.i. for sufficiently large $ k $, enabling applications in renewal theory and local limit theorems.

ABSTRACT

A non-negative function f, defined on the real line or on a half-line, is said to be directly Riemann integrable (d.R.i.) if the upper and lower Riemann sums of f over the whole (unbounded) domain converge to the same finite limit, as the mesh of the partition vanishes. In this note we show that, for a Lebesgue-integrable function f, very mild conditions are enough to ensure that some n-fold convolution of f with itself is d.R.i.. Applications to renewal theory and to local limit theorems are discussed.

Motivation & Objective

• To identify minimal conditions under which repeated convolutions of an $ L^1 $ function become directly Riemann integrable.
• To bridge the gap between general $ L^1 $ functions and the class of directly Riemann integrable functions, which are crucial in renewal theory and local limit theorems.
• To provide a practical criterion ensuring that $ f^{*k} $ is d.R.i. for large $ k $, even when $ f $ itself is not.
• To demonstrate that the combination of $ L^p $-boundedness of some convolution and finite $ \varepsilon $-moment ensures d.R.i. behavior for higher-order convolutions.
• To offer a theoretical foundation for applying d.R.i. techniques to heavy-tailed and non-lattice renewal processes.

Proposed method

• Uses the definition of direct Riemann integrability via convergence of upper and lower Riemann sums over unbounded partitions.
• Applies Hölder's inequality and iterative convolution estimates to show that $ f_{2^k} \in L^1 \cap L^{p^{2^k}} $, leading to $ \widehat{f} \in L^{2^{k+1}} $ via Fourier transform properties.
• Employs a bootstrapping argument on the upper Riemann sum $ S^{g_k}_1(0) $, where $ g_k(x) = (1+|x|^\varepsilon)|f_k(x)| $, to show finiteness propagates across convolutions.
• Relies on the inequality $ (a+b)^\varepsilon \leq 2^\varepsilon(a^\varepsilon + b^\varepsilon) $ to control growth in the convolution estimate.
• Uses the relation $ S^g_\delta(x) \leq (1 + 2\delta/\delta') S^g_{\delta'}(x') $ to compare Riemann sums across different mesh sizes.
• Establishes that $ f_k $ is bounded and continuous for large $ k $, and that $ (1+|x|^\varepsilon)f_k(x) $ is d.R.i. under the stated assumptions.

Experimental results

Research questions

• RQ1 Under what conditions on an $ L^1 $ function $ f $ does some convolution $ f^{*k} $ become directly Riemann integrable?
• RQ2 How do moment conditions $ \int |x|^\varepsilon |f(x)| dx < \infty $ and $ L^p $-boundedness of $ f^{*k_0} $ interact to ensure d.R.i. behavior?
• RQ3 Can the d.R.i. property be inherited through convolution even when $ f $ itself is not d.R.i.?
• RQ4 What is the minimal regularity required for a function to be d.R.i. in the context of renewal processes and local limit theorems?
• RQ5 How can the Fourier transform be used to characterize the $ L^p $-boundedness of convolutions, which is a key assumption in the main result?

Key findings

• For any $ f \in L^1(\mathbb{R}) $ satisfying $ f^{*k_0} \in L^\infty $ and $ \int |x|^\varepsilon |f(x)| dx < \infty $ for some $ \varepsilon > 0 $, there exists $ k_1 \in \mathbb{N} $ such that $ f^{*k} $ is bounded, continuous, and directly Riemann integrable for all $ k \geq k_1 $.
• The function $ x \mapsto (1+|x|^\varepsilon)f_k(x) $ is bounded and continuous for all $ k \geq k_1 $, ensuring integrability and regularity.
• The upper and lower Riemann sums of $ f_k $ converge to the same finite limit as the mesh size $ \delta \to 0 $, satisfying the definition of d.R.i. for $ k \geq k_1 $.
• The condition $ f \in L^1 \cap L^p $ for $ p > 1 $ implies $ f^{*k_0} \in L^\infty $, making the $ L^p $-boundedness assumption practically verifiable in most cases.
• The result applies even when $ f $ is not d.R.i. itself, as shown by the example $ f(x) = \frac{1}{x(\log x)^2} \mathbf{1}_{[e,\infty)}(x) $, which is in $ L^1 $ but not satisfying the moment condition.
• The proof uses a bootstrapping argument on the upper Riemann sum $ S^{g_k}_1(0) $, showing that finiteness at one $ k $ implies finiteness for all larger $ k $, via a linear recurrence inequality.