Open Access
2015 Stochastic integral convergence: A white noise calculus approach
Chi Tim Ng, Ngai Hang Chan
Electron. J. Statist. 9(2): 2035-2057 (2015). DOI: 10.1214/15-EJS1070


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.


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Chi Tim Ng. Ngai Hang Chan. "Stochastic integral convergence: A white noise calculus approach." Electron. J. Statist. 9 (2) 2035 - 2057, 2015.


Received: 1 February 2015; Published: 2015
First available in Project Euclid: 16 September 2015

zbMATH: 1327.62472
MathSciNet: MR3397400
Digital Object Identifier: 10.1214/15-EJS1070

Primary: 62M10
Secondary: 62P20

Keywords: $\mathcal{S}$-transform , convergence , fractional Dickey-Fuller statistic , stochastic integral , white noise calculus

Rights: Copyright © 2015 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.9 • No. 2 • 2015
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