In both the past literature and industrial practice, it was often implicitly used without any justification that the classical strong law of large numbers applies to the modeling of equitylinked insurance. However, as all policyholders’ benefits are linked to common equity indices or funds, the classical assumption of independent claims is clearly inappropriate for equity-linked insurance. In o...

In this article we prove a strong law of large numbers for Borel measurable nonseparably valued random elements in the case of generalized random sets.

Abstract: Limit theorems for functionals of classical (homogeneous) Markov renewal and semi-Markov processes have been known for a long time, since the pioneering work of R. Pyke and R. Schaufele (1964). Since then, these processes, as well as their time-inhomogeneous generalizations, have found many applications, for example in finance and insurance. Unfortunately, no limit theorems have been ...

Let {Xn,n 1} be a sequence of φ -mixing random variables with non-identical distribution and {ani;1 i n,n 1} be an array of real constants. In this paper, we study the strong law of large numbers for the maximal weighted sums of φ -mixing random variables. The results obtained generalize and improve the previous known result of Bai and Cheng (Z.D. Bai and P.E. Cheng, 2000. Marcinkiewicz strong ...

Let { Xij } be a double sequence of pairwise independent random variables. If P {|Xmn| ≥ t}≤ P{|X| ≥ t} for all nonnegative real numbers t and E|X|p( log+ |X|)3 <∞, for 1 <p < 2, then we prove that ∑m i=1 ∑n j=1 ( Xij−EXij ) (mn)1/p → 0 a.s. as m∨n →∞. (0.1) Under the weak condition of E|X|p log+ |X| <∞, it converges to 0 in L1. And the results can be generalized to an r -dimensional array of r...

In the paper we define the convergence of compact fuzzy sets as a convergence of α-cuts in the topology of compact subsets of a metric space. Furthermore we define typical convergences of fuzzy variables and show relations with convergence of their fuzzy distributions. In this context we prove a general formulation of the Strong Law of Large Numbers for fuzzy sets and fuzzy variables with Archi...

For a sequence of pairwise negative quadrant dependent random variables {Xn, n ≥ 1}, conditions are given under which normed and centered partial sums converge to 0 almost certainly. As special cases, new results are obtained for weighted sums { ∑n j=1 ajXj , n ≥ 1} where {an, n ≥ 1} is a sequence of positive constants and the {Xn, n ≥ 1} are also identically distributed. A result of Matu la [1...