Functional-specific effective sample size

Let $(X_n{)}_{n\ge 0}$ be random elements of a measurable space $(E,\mathcal{E})$ on a common probability space, and let $h:E\to \mathbb{R}$ be measurable. Suppose $Y_n:=h(X_n)$ is a strictly stationary real process in the sense of Stationary process and canonical path shift, and suppose $0<{\sigma }_h^2:=\mathrm{Var}(Y_0)<\mathrm{\infty }$ using Moments, variance, and covariance on a probability space. For $k\ge 1$, put

${\rho }_h(k):=\frac{\mathrm{Cov}(Y_0,Y_k)}{{\sigma }_h^2}\mathrm{.}$

If ${\sum }_{k\ge 1}\mathrm{\mid }{\rho }_h(k)\mathrm{\mid }<\mathrm{\infty }$ and ${\tau }_h:=1+2{\sum }_{k\ge 1}{\rho }_h(k)>0$, define the theoretical functional-specific effective sample size for an integer $n\ge 1$ draws by ${\mathrm{ESS}}_h(n):=n\mathrm{/}{\tau }_h$. Here $Y_n$ is the scalar functional whose Monte Carlo average is being assessed. When a central limit theorem for that average has asymptotic variance ${\sigma }_h^2{\tau }_h$, this definition is the iid sample size with the same asymptotic variance. It does not assert that such a central limit theorem holds. If the covariance series is not absolutely summable or ${\tau }_h\le 0$, this definition does not assign a finite positive effective sample size.