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.
Highlights
Stochastic integrals are widely used in the asymptotic theories of time series problems
Functionals of Brownian motion are employed in [5] to derive the asymptotic distributions of the least squares estimates of the autoregressive processes in the presence of unit roots
To test the long-memoryness of a non-stationary time series, [9] extend the results of [7] and develop a fractional Dickey-Fuller test that is based on stochastic integrals involving both Brownian motions and fractional Brownian motions
Summary
Stochastic integrals are widely used in the asymptotic theories of time series problems. To test the long-memoryness of a non-stationary time series, [9] extend the results of [7] and develop a fractional Dickey-Fuller test that is based on stochastic integrals involving both Brownian motions and fractional Brownian motions. Going beyond the construction problem of stochastic integrals, it is unclear if such convergence theories can be applied directly to establish asymptotic results in statistics. The asymptotic results of the fractional Dickey-Fuller statistic can be generalized to test the fractional cointegration of bivariate time series.
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