Lower-moment norm equivalence for renewal partial sums

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.