Subexponential class group and unit group computation in large degree number fields

We describe how to compute the ideal class group and the unit group of an order in a number field in subexponential time. Our method relies on the generalized Riemann hypothesis and other usual heuristics concerning the smoothness of ideals. It applies to arbitrary classes of number fields, including those for which the degree goes to infinity.

ASYMPTOTIC RUIN PROBABILITIES IN FINITE HORIZON WITH SUBEXPONENTIAL LOSSES AND ASSOCIATED DISCOUNT FACTORS

Consider a discrete-time insurance risk model with risky investments. Under the assumption that the loss distribution belongs to a certain subclass of the subexponential class, Tang and Tsitsiashvili (Stochastic Processes and Their Applications 108(2): 299–325 (2003)) established a precise estimate for the finite time ruin probability. This article extends the result both to the whole subexponential class and to a nonstandard case with associated discount factors.

Some new results on the subexponential class

A distribution function F on (0,∞) belongs to the subexponential class if the ratio of 1 – F(2)(x) to 1 – F(x) converges to 2 as x →∞. A necessary condition for membership in is used to prove that a certain class of functions previously thought to be contained in has members outside of . Sufficient conditions on the tail of F are found which ensure F belongs to ; these conditions generalize previously published conditions.

Some new results on the subexponential class

A distribution function F on (0,∞) belongs to the subexponential class if the ratio of 1 – F (2)(x) to 1 – F(x) converges to 2 as x →∞. A necessary condition for membership in is used to prove that a certain class of functions previously thought to be contained in has members outside of . Sufficient conditions on the tail of F are found which ensure F belongs to ; these conditions generalize previously published conditions.

Transient renewal processes in the subexponential case

A transient renewal process based on a sequence of possibly infinite waiting times is defined. The process is studied when the (rescaled) distribution of the waiting times belongs to the subexponential class of distributions. In this case, even conditional on all waiting times observed by time t being finite, the distributions of the forward and backward delays at t are asymptotically degenerate. Also, the conditional moments of the number of events by time t converge to the same finite limits as the unconditional moments.

Transient renewal processes in the subexponential case

A transient renewal process based on a sequence of possibly infinite waiting times is defined. The process is studied when the (rescaled) distribution of the waiting times belongs to the subexponential class of distributions. In this case, even conditional on all waiting times observed by time t being finite, the distributions of the forward and backward delays at t are asymptotically degenerate. Also, the conditional moments of the number of events by time t converge to the same finite limits as the unconditional moments.

Subexponential distribution functions

A distribution function (F on [0,∞) belongs to the subexponential class if and only if 1−F(2) (x) ~ 2(1−F(x)), as x→ ∞. For an important class of distribution functions, a simple, necessary and sufficient condition for membership of is given. A comparison theorem for membership of and also some closure properties of are obtained.1980 Mathematics subject classification (Amer. Math. Soe.): primary 60 E 05; secondary 60 J 80.