"Recall that a distribution F belongs to the subexponential distribution class introduced by Chistyakov , denoted by F ∈ S, if F * 2 (x) ∼ 2F (x), where the notation g(x) ∼ h(x) means that ratio g(x)(h(x)) −1 → 1 for two positive functions g and h supported on [0, ∞). The class S is properly included in the following larger distribution class. "
[Show abstract][Hide abstract] ABSTRACT: Beck et al. (2013) introduced a new distribution class J which contains many
heavy-tailed and light-tailed distributions obeying the principle of a single
big jump. Using a simple transformation which maps heavy-tailed distributions
to light-tailed ones, we find some light-tailed distributions, which belong to
the class J but do not belong to the convolution equivalent distribution class
and which are not even weakly tail equivalent to any convolution equivalent
distribution. This fact helps to understand the structure of the light-tailed
distributions in the class J and leads to a negative answer to an open question
raised by the above paper.
Journal of Mathematical Analysis and Applications 11/2014; 430(2). DOI:10.1016/j.jmaa.2015.05.011 · 1.12 Impact Factor
"It is known that the subexponential distribution class was introduced by Chistyakov (1964) in the study of the branching process, where it was proved that the subexponential distribution class belonged to the following heavy-tailed distribution subclass. We say that a distribution F belongs to the long-tailed distribution class, denoted by F ∈ L, if for any y ∈ (−∞, ∞), F (x + y) ∼ F (x). "
[Show abstract][Hide abstract] ABSTRACT: In this paper, the local asymptotic estimation for the supremum of a random
walk is presented, where the summands of the random walk has common long-tailed
and generalized strong subexponential distribution. The generalized strong
subexponential distribution class and corresponding generalized local
subexponential distribution class are two new distribution classes with some
good properties. Further, some long-tailed distributions with intuitive and
concrete forms are found, showing that the generalized strong subexponential
distribution class and the generalized locally subexponential distribution
class properly contain the strong subexponential distribution class and the
locally subexponential distribution class, respectively.
[Show abstract][Hide abstract] ABSTRACT: In this paper, we present two heavy-tailed distributions related to the new
class J of distributions obeying the principle of a single big jump introduced
by Beck et.al. . The first example belongs to the class J, but is neither
long-tailed nor dominatedly-varying-tailed, and its integrated tail
distribution is strongly subexponential. The other one does not belong to the
class J,but its integrated tail distribution does but is neither long-tailed
nor dominatedly-varying-tailed.These two examples of heavy-tailed distributions
help to illustrate properties of the class J and its relationship to
well-established other distribution classes. In particular, the second example
shows that the Pakes-Veraverbeke-Embrechts Theorem for the class J in  does
not hold trivially.
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