Conditional weak convergence in probability

Let $E$ be a separable complete metric space, let ${\mathcal{D}}_n$ be the sigma-algebra generated by the observed data at sample size $n$, and let $Z_n^{\ast }$ be a bootstrap random element whose remaining randomness is the resampling randomization. Write ${\mathcal{L}}^{\ast }(Z_n^{\ast })$ for its conditional Borel probability law given ${\mathcal{D}}_n$. Supply as part of the data a metric $d_w$ on Borel probability measures on $E$ such that $d_w({\mu }_n,\mu )\to 0$ exactly when ${\mu }_n\Rightarrow \mu$ in the sense of Weak convergence of borel probability measures. Assume the displayed metric distance is a measurable real random variable. For a fixed Borel probability measure $\mu$ on $E$, we say that $Z_n^{\ast }$ converges conditionally weakly in probability to $\mu$ when

$d_w\text{\hspace{0.17em}⁣}({\mathcal{L}}^{\ast }(Z_n^{\ast }),\mu )⟶0\text{in probability}\mathrm{.}$

The metric is supplied as a hypothesis; this definition makes no metric-existence assertion. It applies to separable complete metric spaces, including separable Banach spaces, their closed linear subspaces and finite products.

The conditioning fixes the observations and averages only over bootstrap randomization. A conditional limit in probability is not an assertion of almost-sure conditional convergence.