A Note on the Conditions for COS Convergence

TL;DR

The paper addresses convergence conditions for the COS method by deriving weaker, moment-based criteria for COS-admissibility. It shows that in 1D, membership in $L^1 \cap L^2$ plus a finite $p$-th squared moment for some $p>1$ suffices, and extends this to higher dimensions with $p>d$, broadening applicability to heavy-tailed densities such as Student-t. It provides explicit bounds on the tail energy $B(L)$ that decay as $O(L^{-p})$ and develops a multidimensional bound with a constructive constant, enabling rectangular truncations. The results significantly expand the class of densities for which COS converges and furnish practical, verifiable criteria for practitioners working with heavy-tailed and multivariate models.

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.

Theorems & Definitions (11)

• Definition 1
• Theorem 1
• Proposition 1: Moment-based COS bound
• proof
• Corollary 1: Bounded density with finite $(1+\varepsilon)$ moment
• proof : Proof of Corollary
• Remark 1
• Proposition 2: Weighted $d$-dimensional COS bound
• proof
• Remark 2: Rectangular truncation domains