Scaling total life in a null-recurrent renewal process

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $S_0$ be an almost surely finite nonnegative delay, independent of independent identically distributed strictly positive recurrence times $X_1,X_2,\ldots$. Put $S_n=S_0+\sum_{j=1}^nX_j$, and assume that the recurrence law is nonlattice, meaning $\mathbb P(X_1\in h\mathbb Z)<1$ for every $h>0$, and null recurrent, meaning $\mathbb E X_1=\infty$. For $t\ge S_0$, let \[N_t=\inf\{n\ge0:S_n>t\},\quad A_t=t-S_{N_t-1},\quad B_t=S_{N_t}-t,\quad D_t=X_{N_t}=A_t+B_t,\quad U_t=A_t/D_t.\] For $t<S_0$, set $D_t=U_t=0$; this initial convention does not affect limits as $t\to\infty$. Write $\Rightarrow$ for convergence in distribution.

Statement recorded in the supplied source.

Does there exist a positive nondecreasing deterministic function $\phi$ for which $D_t/\phi(t)\Rightarrow D_\phi$ as $t\to\infty$, with $D_\phi$ a finite nondegenerate random variable? In particular, does the choice $\phi(t)=\mathbb E\min\{X_1,t\}$, $t>0$, have this property? [S2]

Short English statement

Can the length of the renewal interval containing a distant time be rescaled to a nonconstant limit in the infinite-mean case?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9900002 (AMR-098-0002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hermann Thorisson, Some Open Probability Problems, author preprint; original Universidad de Buenos Aires copy preserved in Internet Archive capture of 27 March 2024, Section 1, renewal definitions, p. 1, and Problem 1.2, p. 2. https://web.archive.org/web/20240327021031id_/http://cms.dm.uba.ar/depto/public/Some%20Open%20Problems-preprint.pdf.