Lower-moment norm equivalence for renewal partial sums

Let $(X_k)$ be independent copies of a nondegenerate nonnegative random variable, let $S_n=X_1+\cdots+X_n$ , and set $Z_n=\sum_{k=1}^n e^{\imath S_k}$ . For $p>0$ , write $\|Z_n\|_p=(\mathbb E|Z_n|^p)^{1/p}$ . Renewal-sum norm conjecture. One has

\|Z_n\|_p\asymp \|Z_n\|_2

for $p<2$ . The preceding argument establishes the analogous equivalence for $p\geq 2$ , while the lower-moment range is left as a conjectural extension.

References

Primary source

Jerzy Szulga, “Improper Poisson integral of the sinc function”, arXiv:2504.05574 (2025).

probability-inequalities renewal-processes

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A posted argument claims to prove the conjecture in all cases, but no independent source has checked it, so the problem is not verified as solved.

Szulga's 2025 paper studies renewal-point-process sums and indicates that broader questions are left as conjectures; it does not verify this norm equivalence.

Posted attempt

An unverified argument claims a complete proof: it interpolates between the known fourth-moment bound and the target lower moment when $|\mathbb{E}e^{\imath X_1}|<1$ , and treats the boundary case $|\mathbb{E}e^{\imath X_1}|=1$ as deterministic. The argument has not been independently verified.

Current status (as of August 2026): A complete proof has been posted but remains unverified; no corroborated resolution of the lower-moment conjecture is recorded.

Sources

arxiv.org

ProofThis solution needs a summarySee full solutionHide full solution

The conjecture follows from the fourth-moment estimate already proved in Lemma 2.1.1(3).

Let

S_k=X_1+\cdots+X_k, \qquad Z_n=\sum_{k=1}^n e^{iS_k},

and first assume the source hypothesis

r=\left|\mathbb E e^{iX_1}\right|<1.

The cited lemma supplies a constant $C_4\ge1$ , independent of $n$ , such that

\|Z_n\|_4\le C_4\|Z_n\|_2.

Fix any $0<p<2$ , and set

\theta=\frac{p}{4-p}, \qquad \frac12=\frac{\theta}{p}+\frac{1-\theta}{4}.

Log-convexity of moments gives

\|Z_n\|_2 \le\|Z_n\|_p^\theta\|Z_n\|_4^{1-\theta} \le C_4^{1-\theta} \|Z_n\|_p^\theta\|Z_n\|_2^{1-\theta}.

This interpolation inequality follows directly from Hölder's inequality and remains valid even when $0<p<1$ . If $\|Z_n\|_2>0$ , rearrangement yields

C_4^{-(4-2p)/p}\|Z_n\|_2 \le\|Z_n\|_p \le\|Z_n\|_2,

where the upper bound is monotonicity of moments on a probability space. If $\|Z_n\|_2=0$ , both sides vanish. Hence

\|Z_n\|_p\asymp\|Z_n\|_2 \qquad(0<p<2)

uniformly in $n$ .

The boundary case omitted by the source hypothesis is also immediate. If $r=1$ , equality for the expectation of the unit-modulus random variable $e^{iX_1}$ implies

e^{iX_1}=\lambda\quad\text{almost surely}

for some $|\lambda|=1$ . Thus

Z_n=\sum_{k=1}^n\lambda^k

is deterministic, and $\|Z_n\|_p=\|Z_n\|_2$ for every $p>0$ . Therefore the requested equivalence holds for every increment distribution, including the lattice boundary case.