Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A renewal theorem for relatively stable variables   Let $F\{dx\}$ be a relatively stable probability distribution on the whole
real line and $S_n$ the random walk started at the origin with step
distribution $F$. We obtain an exact asymptotic form of the Green measure
$U\{x+dy\}= \sum_{n=0}^\infty P[S_n-x \in dy]$ as $x\to \infty$ when $S_n$ is
transient and $S_n\to \infty$ in probability. If $F$ is concentrated on
$[0,\infty)$, it is relatively stable if and only if $\ell(x) :=\int_0^x
F\{(t,\infty)\}dt$ is slowly varying at infinity; our result entails that if
$F$ is non-arithmetic and relatively stable, then $\lim_{x\to\infty}\,
\ell(x)U\{[x, x+h)\} = h$ for each $h>0$. This surpasses the known result due
to Erickson \cite{Ec}, the latter assuming the stronger condition that
$xF\{(x,\infty)\}$ is slowly varying. An obvious analog also holds for
arithmetic variables.

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The categories of this paper are: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex