Heavy-tailed Dependent Weighted Random Variables And The Results

This thesis deals with estimations for large deviations of randomly weighted sums∑i=1 nθiXi, where{θi, i≥1} is a sequence of nonnegative random variables, and{Xi, i≥1} is a sequence of dependent random variables from class D∩L; and also study some closure properties for dependent random variables from class S(γ). We have four chapters:In Chapter one, we first give some basic definitions (Section 1.1). Then we briefly discuss some properties (Section 1.2). Chapter 1 mainly serves the purpose of introducing definitions and properties, which relative to our problems.In Chapter two, we consider the uniform estimate and applications for large devi-ations of randomly weighted sums∑i=1 nθiXi and their maxima, where{Xi,i≥1} is a sequence of identically distributed and upper-tailed independent random variables from class D∩L, and{θi,i≥1} is a sequence of nonnegative random variables, which is independent of{Xi,i≥1} and satisfying certain moment and distributed conditions. Moreover, we assume{θi,i≥1} does not need any additional assumption on the depen-dent structure. In Section 2.1, we give briefly introductions for the problem in the present research. Some Lemmas for the problem study are given in Section 2.2. In Section 2.3, we show our main results and their proofs. Finally, we present applications for our main results.Chapter three studies some asymptotic results for both finite and ultimate ruin prob-abilities in a discrete time risk model with nonconstant interest rates, under assumptions that the individual net losses are pairwise upper-tail independent, identically distributed random variables in D∩L class. Additionally, it also establishes two-side bounds for ultimate ruin probability. In Section 3.1, we give briefly introductions for the problem in the present research. In Section 3.2, we state our main results, and their proofs are given in Section 3.3.In Chapter four, we let X and Y be two dependent random variables with distribu-tions belonging to the class S(γ). In this note we show that the distribution of min(X, Y) also belongs to the class S(γ) under certain assumptions. Especially, whenγ= 0, the result is true for the subexponential class, which was obtained in [1] under independent setting. In addition, we also obtain the results for n dependent random variables. In Section 4.1, we give briefly introductions for the problem in the present research. Section 4.2 contains some preliminaries for the problem, and we state and prove main results in Section4.3.