Abstract:
In this paper, we obtain some new exponential inequalities for partial sums and their finite maximum of acceptable random variables by the results of Sung et al. (J. Korean Stat. Soc., 40, 109-114, 2011) and in different ways from theirs. The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal. In addition, we introduce some concepts and examples of widely acceptable random variables to extend our results mentioned above. Mathematics Subject Classification (2000) 60F15, 62G20

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.

Abstract:
We obtain a number of new general properties, related to the closedness of the class of long-tailed distributions under convolutions, that are of interest themselves and may be applied in many models that deal with "plus" and/or "max" operations on heavy-tailed random variables. We analyse the closedness property under convolution roots for these distributions. Namely, we introduce two classes of heavy-tailed distributions that are not long-tailed and study their properties. These examples help to provide further insights and, in particular, to show that the properties to be both long-tailed and so-called "generalised subexponential" are not preserved under the convolution roots. This leads to a negative answer to a conjecture of Embrechts and Goldie [10, 12] for the class of long-tailed and generalised subexponential distributions. In particular, our examples show that the following is possible: an infinitely divisible distribution belongs to both classes, while its Levy measure is neither long-tailed nor generalised subexponential.

Abstract:
In this paper, we show that the distribution class L(\gamma)\ OS for some \gamma>0 is not closed under convolution roots related to an infinitely divisible distribution. Precisely, we give two main conditions on Levy spectral distribution generated by Levy spectral measure of an infinitely divisible distribution, under each of which, there is an infinitely divisible distribution belonging to the class, however the corresponding spectral distribution is not. And we note that the two conditions can not be deduced from each other. For the distribution class (L(\gamma)\cap OS)\ S(\gamma), corresponding conclusion is also proved. In order to prove the results mentioned above, we explore some of the structural properties of the class, which include the closedness of the class under the convolution and the random convolution. In addition, we study some properties of a transformation between heavy-tailed and light-tailed distributions. The transformation is a key to finding the required Levy spectral distribution. On the other hand, we also give some conditions which guarantee the closeness of the class under convolution roots. The whole research in the paper is a deep and complete discussion on the famous conjecture due to Embrechts and Goldie [6, 7] (J. Austral. Math. Soc. (Ser. A) 29, 243-256, 1980; Stoch. Process. Appl. 13, 263-278, 1982).

Abstract:
In this paper, the local asymptotic estimation for the supremum of a random walk is presented. The summands of the random walk have common long-tailed and generalized strong subexponential distribution. The 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, which show that the intersection of the two above-mentioned distribution classes with long-tailed distribution class properly contain the strong subexponential distribution class and the locally subexponential distribution class, respectively.

Abstract:
In this paper, we present several heavy-tailed distributions belonging to the new class J of distributions obeying the principle of a single big jump introduced by Beck et al. [1]. We describe the structure of this class from different angles. First, we show that heavy-tailed distributions in the class J are automatically strongly heavy-tailed and thus have tails which are not too irregular. Second, we show that such distributions are not necessarily weakly tail equivalent to a subexponential distribution. We also show that the class of heavy-tailed distributions in J which are neither long-tailed nor dominatedly-varying-tailed is not only non-empty but even quite rich in the sense that it has a nonempty intersection with several other well-established classes. In addition, the integrated tail distribution of some particular of these distributions shows that the Pakes-Veraverbeke-Embrechts Theorem for the class J in [1] does not hold trivially.

Abstract:
In this paper, we discuss the laws of the iterated logarithm for nonstationarynegative associated sequences. From these results we can obtain a sort of strong convergencerates of variance estimate for negative associated errors.

Abstract:
In this paper,the law of iterated logarithm for product sums of positive as- sociated sequences with the strong stability and the strong law of large numbers for product sums of positive associated sequences with different distributions are proved.It is pointed out that the law of large numbers for product sums does not necessarily hold even though the law of large numbers for partial sums holds.Also the necessity of a condition in Theorem 2 is discussed.