Abstract
The precise large deviations asymptotics for the sums of independent identical random variables when the distribution of the summand belongs to the class S ∗ of heavy tailed distributions is studied. Under mild conditions, we extend the previous results from the paper Denisov et al. (2010) to asymptotics that are valid uniformly over some time interval. Finally, we apply the main result on the multi-risk model introduced by Wang and Wang (2007).
Highlights
In this paper, the precise large deviations for a random walk whose steps represent random variables with distribution F from a subclass S ∗ of the subexponential class S is studied
The most popular distributions with heavy tails belong to the class S ∗, among others Pareto, Burr, Cauchy, Lognormal and Weibull
The inclusion of the class S ∗ in the class of subexponential distribution is proved proper, namely have been found subexponential distributions that do not belong to S ∗ (see Denisov et al (2004))
Summary
The precise large deviations for a random walk whose steps represent random variables with distribution F from a subclass S ∗ of the subexponential class S is studied. The topic of large deviations of non-random sums has already been well studied. A review on large deviations for random sums is given in Mikosch and Nagaev (1998) and Mikosch and Nagaev (2001). Let us denote by Sn the sum of n independent identically distributed random variables X1 + · · · +. K be i.i.d. non-negative random variables with common distribution function Fi ( x ) and finite mean. ∑ Sni = ∑ ∑ Xij , i =1 i =1 j =1 found in Wang and Wang (2007), we formulate the following result: Let { Xi,j , j ≥ 1} be i.i.d. non-negative random variables with common distribution function Fi ( x ).
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