Weak convergence of dependent empirical measures with application to subsampling in function spaces

Consider the problem of inference for a parameter of a stationary time series, where the parameter takes values in a metric space (such as a function space). In this paper, we develop asymptotic theory based on subsampling to approximate the distribution of estimators for such parameters. The reason for this level of abstraction is to be able to consider parameters that take values in a function space. For example, we consider the estimation of the distribution of the empirical process and the spectral process. In order to accomplish this, we provide a general result based on simple arguments. The main technical result relies on the weak convergence of triangular arrays of dependent empirical measures, where the variables making up the arrays can take values in a (possibly nonseparable) metric space. This approach based on subsampling is quite powerful in that it leads to straightforward arguments where corresponding results based on the moving blocks bootstrap are much harder to obtain.

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