Almost sure bounds and tail phases for critical non-Gaussian holomorphic chaos

Abstract

Gu and Zhang asked for almost sure upper and lower bounds on the Taylor coefficients $a_n$ of $\exp(\sum_{k\ge1}X_kz^k/\sqrt{k})$ in the unit disc, and for the dependence of these bounds on the input tail. For independent, identically distributed, rotationally invariant complex inputs with $\mathbb E|X_1|^2=1$, we prove the universal almost sure bound $|a_n|=O((\log n)^{1/4})$. Almost surely, $\limsup_{n\to\infty}|a_n|/(\log n)^{1/4}$ is positive and finite exactly when $\mathbb E[|X_1|^2\log^+|X_1|^2]<\infty$, and is zero otherwise. For each input law, we define a deterministic scale such that, almost surely, the coefficients are bounded exactly when the scale is bounded, and tend to zero exactly when it vanishes. We also determine the almost sure logarithmic growth exponent as the maximum of contributions from collective fluctuations and individual large inputs. Explicit tails exhibit distinct thresholds for boundedness and for upper bounds at the scale in the phase criteria. Examples show that neither the leading tail nor the set of finite logarithmic moments determines the almost sure phase.