We are interested in new proofs, connections with others theories and variations of the following theorem.
Th K (Karamata). If $f$ is monotone on some interval $[A,\infty)$,
$$ \lim{x\rightarrow \infty}(f(x+t)- f(x))= \psi(t) . \leqno (1)$$
for $t_1,t_2 \in \mathbb{R}$ such that $t_2/t_1$ is irrational and $\psi(t_1),\psi(t_2)$ finite, then (1) holds for all $ t \in \mathbb{R}$ and
$\psi(t)= c t$ for some real constant $c$.
For Th K (Karamata), see E. Seneta, Karamata's Characterization Theorem, Feller, and Regular Variation in Probability theory, Publ Inst Math 71(85) (2002), 79-89.
Credit for making the link between Karamata's regular variation and the probability limit theorems above explicit belongs to Sakovich (1956), writing - appropriately enough- in the first volume of the then new Soviet journal Theory of Probability and its Applications.
For some reason, Sakovich's work was overlooked, and had to be rediscovered later. In 1966, Feller published his sequel An introduction to probability theory and its
applications, Volume II, Feller (1966/71), to his earlier book Feller (1950/57/68).
See N. H. BINGHAM, REGULAR VARIATION AND PROBABILITY: THE EARLY YEARS, J. Computational and Applied Mathematics 200 (2007), 357-363 (J. L. Teugels Festschrift)
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