  Motivated mainly by applications to partial differential equations with
random coefficients, we introduce a new class of Monte Carlo estimators, called
Toeplitz Monte Carlo (TMC) estimator for approximating the integral of a
multivariate function with respect to the direct product of an identical
univariate probability measure. The TMC estimator generates a sequence
$x_1,x_2,\ldots$ of i.i.d. samples for one random variable, and then uses
$(x_{n+s-1},x_{n+s-2}\ldots,x_n)$ with $n=1,2,\ldots$ as quadrature points,
where $s$ denotes the dimension. Although consecutive points have some
dependency, the concatenation of all quadrature nodes is represented by a
Toeplitz matrix, which allows for a fast matrix-vector multiplication. In this
paper we study the variance of the TMC estimator and its dependence on the
dimension $s$. Numerical experiments confirm the considerable efficiency
improvement over the standard Monte Carlo estimator for applications to partial
differential equations with random coefficients, particularly when the
dimension $s$ is large.
