Total Variation Approximations for Multivariate Regularly Varying Random Walks Conditioned on Ruin (with J. C. Liu).

Summary:

There are several nice papers in the theory of large deviations for one dimensional heavy-tailed random walks and Lévy processes which have conditional limit theorems of the time to ruin, jointly with the overshoot, and the overshoot, and the path, etc. BUT, as far as I know this is the first result that derives these limiting results for multidimensional heavy-tailed random walks. AND it does so using completely different techniques, namely, change-of-measure techniques and Lyapunov inequalities. AND it not only provides conditional functional central limit theorems but also provides total variation approximations of the whole conditional path. To give a sense of a qualitative difference between the multidimensional and the one-dimensional case. It is known that asymptotic (normalized) time to ruin is Pareto in the one-dimensional case. This is no longer true in the multidimensional case. The (normalized) time to ruin is asymptotically still regularly varying, but no longer purely Pareto.

Bibtex:

@Article{BlaLiu_TV14,

author = { J. Blanchet and J. C. Liu},

title = {Total Variation Approximations for Multivariate Regularly Varying Random Walks Conditioned on Ruin},

journal = {Bernoulli},

year = {2014},

volume = {20},

pages = {416-456}

}