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Negative Moments of Steinhaus Sums

Martin Rapaport, Tomasz Tkocz, Isabella Wu

TL;DR

This work resolves sharp negative-moment bounds for sums of Steinhaus random variables by recasting the problem through a Fourier-inversion formula involving $J_0$ and introducing the key decoupling function $\Psi_p$. A Nazarov–Podkorytov–style inductive framework, augmented by a carefully crafted concave envelope $\Phi_p$, yields a complete proof that for $-1<p<0$, the optimal constant is $A_p=\|(\xi_1+\xi_2)/\sqrt{2}\|_p$. The authors also correct a gap in Ko's earlier argument and derive a sharp Rényi entropy bound for Steinhaus sums, illustrating the broad impact on both moment inequalities and information-theoretic quantities. Altogether, the paper closes the sharp $L_p-L_2$ Khintchin-type programme for Steinhaus sums and furnishes precise constants and methods with potential applications to entropy and related inequalities.

Abstract

We prove a sharp upper bound on negative moments of sums of independent Steinhaus random variables (that is uniform on circles in the plane). Together with the series of earlier works: König-Kwapień (2001), Baernstein II-Culverhouse (2002), and König (2014), this closes the investigation of sharp $L_p-L_2$ Khinchin-type inequalities for the Steinhaus sums. Incidentally, we fix a mistake in an earlier paper, as well as provide an application to sharp bounds on Rényi entropy.

Negative Moments of Steinhaus Sums

TL;DR

This work resolves sharp negative-moment bounds for sums of Steinhaus random variables by recasting the problem through a Fourier-inversion formula involving and introducing the key decoupling function . A Nazarov–Podkorytov–style inductive framework, augmented by a carefully crafted concave envelope , yields a complete proof that for , the optimal constant is . The authors also correct a gap in Ko's earlier argument and derive a sharp Rényi entropy bound for Steinhaus sums, illustrating the broad impact on both moment inequalities and information-theoretic quantities. Altogether, the paper closes the sharp Khintchin-type programme for Steinhaus sums and furnishes precise constants and methods with potential applications to entropy and related inequalities.

Abstract

We prove a sharp upper bound on negative moments of sums of independent Steinhaus random variables (that is uniform on circles in the plane). Together with the series of earlier works: König-Kwapień (2001), Baernstein II-Culverhouse (2002), and König (2014), this closes the investigation of sharp Khinchin-type inequalities for the Steinhaus sums. Incidentally, we fix a mistake in an earlier paper, as well as provide an application to sharp bounds on Rényi entropy.

Paper Structure

This paper contains 9 sections, 10 theorems, 78 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and the main result
  2. Proof of Theorem \ref{['thm:cp']}
  3. Overview.
  4. Specifics
  5. Induction
  6. The technical part: Proof of Lemma \ref{['lm:Psi<Psi(2)']}
  7. The inductive argument: Proof of Theorem \ref{['thm:main-ind']}
  8. A comment on Ko
  9. Sharp Rényi entropy comparison

Key Result

Theorem 1

With notation eq:Ap-Bp-sharp, for $-1 < p < 0$, we have $A_p = \|(\xi_1+\xi_2)/\sqrt{2}\|_p$.

Figures (1)

  • Figure 1: An ingenious construction of $\Phi_p$: the plots show $\phi_p$, $\Phi_p$ and their common tangent at $x=1$.

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 2
  • Corollary 3
  • Remark 4
  • Remark 5
  • Lemma 6: Lemma 33 in CST, Lemma 20 in CKT
  • Theorem 7
  • Lemma 8
  • proof
  • Lemma 9
  • ...and 9 more