Let \(\mathbf{X}_{1}, \ldots, \mathbf{X}_{n}\) be a set of two-dimensional independent and identically distributed random vectors from a distribution \(F\) with mean vector \(\boldsymbol{\mu}\). Let \(g(\mathbf{x})=x_{2}-x_{1}^{2}\) where \(\mathbf{x}^{\prime}=\left(x_{1}, x_{2}ight)\). Define \(\theta=g(\boldsymbol{\mu})\) with \(\hat{\theta}_{n}=g\left(\overline{\mathbf{X}}_{n}ight)\). Let \(R_{n}(\hat{\theta}, \theta)=n^{1 / 2}\left(\hat{\theta}_{n}-\thetaight)\) and \(H_{n}(t)=P\left[R_{n}\left(\hat{\theta}_{n}, \thetaight) \leq tight)\) with bootstrap estimate \(\hat{H}_{n}(t)=P^{*}\left[R_{n}\left(\hat{\theta}_{n}^{*}, \hat{\theta}_{n}ight) \leq tight]\). Using Theorem 11.16, under what conditions can we conclude that \(d_{\infty}\left(\hat{H}_{n}, H_{n}ight) \xrightarrow{\text { a.c. }} 0\) as \(n ightarrow \infty\) ? Explain how this result can be used to determine the conditions under which the bootstrap estimate of the sampling distribution of the sample variance is strongly consistent.