$\{V_n\}$ and $\{W_n\}$ are two sequences of random variables, show that$V_n-W_n\overset{p}\to0$.

Question

Let $\{V_n\}$ and $\{W_n\}$ be two sequences of random variables satisfying the following conditions.

(i) For all $\delta>0$, there exists $\lambda$ s.t. $P(|W_n|>\lambda)<\delta$.

(ii) For all $k$ and all $\varepsilon>0$ $$\lim_{n\to\infty}P(V_n\le k,W_n\ge k+\varepsilon)=0$$ $$\lim_{n\to\infty}P(V_n\ge k+\varepsilon ,W_n\le k)=0.$$

Then $V_n-W_n\overset{p}\to0$,as $n\to\infty$.

How to understand and prove this proposition?

Answer 1

Thanks for all the comments. I find a proof in the book Order Statistics by David and Nagaraja (2003) on page 286.

Fix $\varepsilon>0, \delta>0$, by condtion (i) we can choose integers $m$ and $n_0$ s.t. $$P(|W_n|>m\varepsilon)<\delta,\quad for~~~ n\ge n_0.$$ Hence \begin{align*} P(|V_n&-W_n|>2\varepsilon)<\delta+P(|W_n|\le m\varepsilon,|V_n-W_n|>2\varepsilon)\\ &=\delta+\sum_{j=-m}^{m-1}P(j\varepsilon\le W_n\le (j+1)\varepsilon,|V_n-W_n|>2\varepsilon)\\ &\le\delta+\sum_{j=-m}^{m-1}\Big[P\Big(j\varepsilon\le W_n\le (j+1)\varepsilon,V_n>(j+2)\varepsilon\Big)\\ &+P\Big(j\varepsilon\le W_n\le (j+1)\varepsilon,V_n<(j-1)\varepsilon\Big)\Big] \end{align*} which tends to $0$ as $n\to\infty$ by condtion (ii).