# Conclude convergence in probability from uniform convergence on a set of limiting probability 1

Let

• $$\kappa_d$$ be a Markov kernel on $$(\mathbb R^d,\mathcal B(\mathbb R^d))$$ for $$d\in\mathbb N$$
• $$f_d:\mathbb R^d\times\mathbb R^d\to\mathbb R$$ be Borel measurable for $$d\in\mathbb N$$
• $$B_d\in\mathcal B(\mathbb R^d)$$ for $$d\in\mathbb N$$ with $$\sup_{x\in B_d}\int\kappa_d(x,{\rm d}y)|f(x,y)|\xrightarrow{d\to\infty}0\tag1$$
• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space
• $$X^{(d)}:\Omega\to\mathbb R^d$$ be $$(\mathcal A,\mathcal B(\mathbb R^d))$$-measurable with $$\operatorname P\left[X^{(d)}\in B_d\right]\xrightarrow{d\to\infty}1\tag2$$ for $$d\in\mathbb N$$
• $$Y^{(d)}:\Omega\to\mathbb R^d$$ be $$(\mathcal A,\mathcal B(\mathbb R^d))$$-measurable with$$^1$$ $$Y^{(d)}_\ast\operatorname P=\left(X^{(d)}_\ast\operatorname P\right)\kappa_d\tag3$$ for $$d\in\mathbb N$$

Are we able to conclude $$\left|f\left(X^{(d)},Y^{(d)}\right)\right|\xrightarrow{d\to\infty}0\tag4$$ in probability (or even almost surely)?

We may note that $$\operatorname E\left[\left|f\left(X^{(d)},Y^{(d)}\right)\right|\right]=\int\operatorname P\left[X^{(d)}\in{\rm d}x\right]\int\kappa_d(x,{\rm d}y)|f(x,y)|\tag5$$ for all $$d\in\mathbb N$$. We may split the outer integral and clearly $$\int_{B_d}\operatorname P\left[X^{(d)}\in{\rm d}x\right]\int\kappa_d(x,{\rm d}y)|f(x,y)|\le\operatorname P\left[X^{(d)}\in B_d\right]\sup_{x\in B_d}\int\kappa_d(x,{\rm d}y)|f(x,y)|\xrightarrow{d\to\infty}0\tag6.$$ So, a possible way would to show $$\int_{\mathbb R^d\setminus B_d}\operatorname P\left[X^{(d)}\in{\rm d}x\right]\int\kappa_d(x,{\rm d}y)|f(x,y)|\xrightarrow{d\to\infty}0\tag7.$$ If the inner integral wouldn't depend on $$d$$, this would be a simple application of the dominated convergence theorem. However, since it depends on $$d$$, this might be problematic.

$$^1$$ $$\left(X^{(d)}_\ast\operatorname P\right)\kappa_d$$ denotes the composition of the distribution $$X^{(d)}_\ast\operatorname P$$ of $$X^{(d)}$$ under $$\operatorname P$$ and $$\kappa_d$$.