Large deviations for random walks under subexponentiality: The big-jump domain

By Denis Denisov, Antonius B. Dieker, V. Shneer. 2007

182 citations recorded by the source corpus.

For a given one-dimensional random walk {Sn} with a subexponential step-size distribution, we present a unifying theory to study the sequences {xn} for which as n→∞ uniformly for x≥xn. We also investigate the stronger “local” analogue, . Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory. When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results.

Subfields: Management Science and Operations Research, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/df493edc3a7668de0b38f8615fff7a66290aa299

Citations in the corpus

• Asymptotics of randomly stopped sums in the presence of heavy tails (2008, unknown): “The advantage of the result in Theorem 5 is its simplicity and universality since it is vali d for all distributions fromS ∗ without any further moment or regularity assumptions, compare with a series of results in [2, 3, 6] where the hazard rate is assumed to be sufficiently smooth.”
• Conditional distribution of heavy tailed random variables on large deviations of their sum (2009, unknown): “y provided in [2]. The asymptotics for the large deviation probabilities are now given by lim n!1 sup xd n n P S n2x+ x+ 1 = 0; (2.6) and sucient conditions on d nfor (2.6) to hold can be found in [5]. We are interested in the conditional distribution of the variables fX ngsubject to a large deviati”
• Scaling limits of random planar maps with a unique large face (2012, unknown): “13) = n(1 + o(1))P(ξ1 = ⌊n(1 − κ) − 1⌋) as n → ∞, see [23] for more general statements.”