Article
A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes
Theory of Probability and Its Applications (Impact Factor: 0.52). 01/1964; 9(4). DOI: 10.1137/1109088

 "Recall that a distribution F belongs to the subexponential distribution class introduced by Chistyakov [2], 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. "
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ABSTRACT: Beck et al. (2013) introduced a new distribution class J which contains many heavytailed and lighttailed distributions obeying the principle of a single big jump. Using a simple transformation which maps heavytailed distributions to lighttailed ones, we find some lighttailed 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 lighttailed distributions in the class J and leads to a negative answer to an open question raised by the above paper. 
 "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 heavytailed distribution subclass. We say that a distribution F belongs to the longtailed distribution class, denoted by F ∈ L, if for any y ∈ (−∞, ∞), F (x + y) ∼ F (x). "
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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 longtailed 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 longtailed 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. 
 "The classes S and D were introduced in [6] and [8], respectively. Further, we say that a distribution F is generalized subexponential, denoted by F ∈ OS, if "
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ABSTRACT: In this paper, we present two heavytailed distributions related to the new class J of distributions obeying the principle of a single big jump introduced by Beck et.al. [2]. The first example belongs to the class J, but is neither longtailed nor dominatedlyvaryingtailed, 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 longtailed nor dominatedlyvaryingtailed.These two examples of heavytailed distributions help to illustrate properties of the class J and its relationship to wellestablished other distribution classes. In particular, the second example shows that the PakesVeraverbekeEmbrechts Theorem for the class J in [2] does not hold trivially.
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