Asymptotics of Multivariate Sequences

Asymptotics of Multivariate Sequences , Part I : Smooth Points Ofthe Singular

Asymptotics of Multivariate Sequences, Part I: Smooth Points of the Singular Variety

Given a multivariate generating function F (z1, . . . , zd) = ∑ ar1,... ,rdz r1 1 · · · z rd d , we determine asymptotics for the coefficients. Our approach is to use Cauchy’s integral formula near singular points of F , resulting in a tractable oscillating integral. This paper treats the case where the singular point of F is a smooth point of a surface of poles. Companion papers will treat sin...

Asymptotics of multivariate sequences, part III: quadratic points

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deforma...

ec 2 00 8 Asymptotics of multivariate sequences , part III : quadratic points

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deforma...

Asymptotics for the infinite time ruin probability of a dependent risk model with a constant interest rate and dominatedly varying-tailed claim sizes

 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.

A new method for computing asymptotics of diagonal coefficients of multivariate generating functions

Let P n∈Nd Fnx n be a multivariate generating function that converges in a neighborhood of the origin of C. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients Fa1n,...,adn and show its superiority over the standard, univariate diagonal method. Several examples are given in detail.