Let be i.i.d. random variables with , regularly varying with exponent and slowly varying as . We give the limit distribution of in the threshold case which separates the Brownian phase corresponding to where the limit of is , with , is the γ-Hölder norm of a standard Brownian motion and the Fréchet phase corresponding to where the limit of is with Fréchet distribution , . We prove that , converges in distribution to some random variable Z if and only if L has a limit at infinity. In such case, there are , such that in distribution, where for , with and independent and , . When , a possible choice for the normalization is and , with . We also build an example where L has no limit at infinity and has for each a subsequence converging after normalization to .
"On the asymptotic of the maximal weighted increment of a random walk with regularly varying jumps: the boundary case." Electron. J. Probab. 26 1 - 31, 2021. https://doi.org/10.1214/21-EJP691