Randomized quasi-Monte Carlo point sets and estimators

Let $d,n\ge 1$ and let $X_1,\dots ,X_n$ be measurable random elements of $[0,1{)}^d$. The array is marginally uniform if every $X_i$ has normalized Lebesgue law on the cube; the points need not be independent. An RQMC design consists of marginal uniformity together with a separately stated low-discrepancy condition. For a family indexed by $n$, one such condition is an almost-sure bound $D_n^{\ast }(X_1,\dots ,X_n)\le {\delta }_n$ with deterministic ${\delta }_n\to 0$, where $D_n^{\ast }$ is the star discrepancy of Star discrepancy, digital nets, and digital sequences. For a fixed $n$, an exact net-count property may instead be declared; no star-discrepancy rate follows merely from marginal uniformity.

For a Borel function $f$ integrable on the cube, set $I(f)={\int }_{[0,1{)}^d}f(x)dx,{\widehat{I}}_n(f)=\frac{1}{n}{\sum }_{i=1}^{n}f(X_i)\mathrm{.}$ An independent full-set randomization is an independent copy of the whole vector $(X_1,\dots ,X_n)$. Independence is across copies, not among points within one copy. The point count $n$ and number of independent randomizations $R$ are separate parameters.