2025/06/04 by Aaron Chong, Chong, Aaron, Konstantin Borovkov +1
Decision Sciences · Mathematics · #Probability and Risk Models #Statistical Distribution Estimation and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2506.03727
Let ξ1, ξ2,… be a sequence of independent and identically distributed random variables with zero mean, finite second moment and regularly varying right distribution tail. Motivated by a stop-loss insurance model, we consider a threshold sequence Mn≫ (nln n)1/2, n→ ∞, and establish the asymptotics of the probabilities of the large deviations of the form ∑j=1n(ξj \wedge Mn)>x in the whole spectrum of x-values in the region O(Mn). The asymptotic representations for these probabilities obey the "multiple large jumps principle" and have different forms in the vicinities of the multiples kMn of the censoring threshold values, on the one hand, and inside intervals of the form ((k-1)Mn, kMn), on the other. We show that there is a "smooth transition" of these representations from one to the other when the deviation x increases to a multiple of Mn, "crosses" it and then moves away from it.