Pointwise bootstrap need not preserve pair dependence

The transition matrix is $\left(\begin{array}{cc}3\mathrm{/}4& 1\mathrm{/}4\\ 1\mathrm{/}4& 3\mathrm{/}4\end{array}\right)$, whose columns sum to one, so $(1\mathrm{/}2,1\mathrm{/}2)$ is invariant; multiplying this initial law by the transition probabilities gives the displayed four joint masses. Both marginals are Bernoulli with mean $1\mathrm{/}2$, while $\mathbb{P}(X_1=0,X_2=0)=3\mathrm{/}8\mathrm{\ne }1\mathrm{/}4=\mathbb{P}(X_1=0)\mathbb{P}(X_2=0)$, so the observations are dependent. Also $\mathbb{E}[X_1{X}_2]=3\mathrm{/}8$, and [F4] gives original covariance $3\mathrm{/}8-(1\mathrm{/}2)(1\mathrm{/}2)=1\mathrm{/}8$.

Fix any realized pair $x=(x_1,x_2)$ and put $m=(x_1+x_2)\mathrm{/}2$. By [F1]–[F3], each bootstrap coordinate has conditional mean $m$, and the two coordinates are conditionally independent; hence ${\mathbb{E}}^{\ast }[X_1^{\ast }{X}_2^{\ast }]=m^2$ and [F4] gives ${\mathrm{Cov}}^{\ast }(X_1^{\ast },X_2^{\ast })=0$.

Both the original pair and the bootstrap pair are bounded, so equality of their joint laws would force equality of their covariances. The original covariance is $1\mathrm{/}8$ but the conditional bootstrap covariance is $0$, so the laws differ and the statement is false. This finite counterexample concerns exact joint-law reproduction only; it does not decide asymptotic validity for any particular statistic.