We use the properties of the Matuszewska indices to show asymptotic inequalities for hazard rates. We discuss the relation between membership in the classes of dominatedly or extended rapidly varying tail distributions and corresponding hazard rate conditions. Convolution closure is established for the class of distributions with extended rapidly varying tails.

The asymptotic behaviour of the tail expectation ?E(Sn?)??{Sn?>x} is investigated, where exponent ? a nonnegative real number and Sn?=?1+…+?n sum dominatedly varying not necessarily identically distributed random summands, following specific dependence structure. It turns out that such can be asymptotically bounded from above below by sums expectations ?E?i??{?i>x} with correcting constan...

This paper investigates the tail asymptotic behavior of the severity of ruin (the deficit at ruin) in the renewal model. Under the assumption that the tail probability of the claimsize is dominatedly varying, a uniform asymptotic formula for the tail probability of the deficit at ruin is obtained. 1. Model and main result Throughout this paper, for any 0 ≤ a < b < ∞ the integral symbol ∫ b a is...

Suppose that ξ+ is the positive part of a random variable defined on probability space (Ω,F,P) with distribution function Fξ. When moment Eξ+p order p&gt;0 finite, then truncated F¯ξ,p(x)=min1,Eξp1I{ξ&gt;x}, for all x⩾0, survival or, in other words, tail Fξ,p. In this paper, we examine which regularity properties transfer from Fξ to Fξ,p and The construction describes transformation initial Our...

 This paper mainly considers a nonstandard risk model with a constant interest rate‎, ‎where both the claim sizes and the inter-arrival times follow some certain dependence structures‎. ‎When the claim sizes are dominatedly varying-tailed‎, ‎asymptotics for the infinite time ruin probability of the above dependent risk model have been given‎.

This paper considers the real-valued random variables X1, . . . , Xn with corresponding distributions F1, . . . , Fn, such that X1, . . . , Xn admit some dependence structure and n(F1 + · · · + Fn) belongs to the class dominatedly varying-tailed distributions. The weak equivalence relations between the quantities P(Sn > x), P(max{X1, . . . , Xn} > x), P(max{S1, . . . , Sn} > x) and ∑n k=1 Fk(x)...

this paper mainly considers a nonstandard risk model with a constant interest rate‎, ‎where both the claim sizes and the inter-arrival times follow some certain dependence structures‎. ‎when the claim sizes are dominatedly varying-tailed‎, ‎asymptotics for the infinite time ruin probability of the above dependent risk model have been given‎.