A Note on the Conditions for COS Convergence

TL;DR

This paper establishes simple, verifiable conditions under which the COS method converges, extending applicability to heavy-tailed distributions and higher dimensions by analyzing truncation errors and moment conditions.

Contribution

It provides new, easy-to-check criteria for COS convergence that include heavy-tailed distributions and multi-dimensional cases, broadening previous theoretical results.

Findings

COS converges when density is in L1 and L2 with finite weighted L2 moment

Extension to multiple dimensions requires moments to exceed the dimension

Includes heavy-tailed distributions like Student t with small degrees of freedom

Abstract

We study the truncation error of the COS method and give simple, verifiable conditions that guarantee convergence. In one dimension, COS is admissible when the density belongs to both L1 and L2 and has a finite weighted L2 moment of order strictly greater than one. We extend the result to multiple dimensions by requiring the moment order to exceed the dimension. These conditions enlarge the class of densities covered by previous analyses and include heavy-tailed distributions such as Student t with small degrees of freedom.