Stochastic integral convergence: A white noise calculus approach

Abstract

By virtue of long-memory time series, it is illustrated in this paper that white noise calculus can be used to handle subtle issues of stochastic integral convergence that often arise in the asymptotic theory of time series. A main difficulty of such an issue is that the limiting stochastic integral cannot be defined path-wise in general. As a result, continuous mapping theorem cannot be directly applied to deduce the convergence of stochastic integrals $\int^{1}_{0}H_{n}(s)\,dZ_{n}(s)$ to $\int^{1}_{0}H(s)\,dZ(s)$ based on the convergence of $(H_{n},Z_{n})$ to $(H,Z)$ in distribution. The white noise calculus, in particular the technique of $\mathcal{S}$-transform, allows one to establish the asymptotic results directly.