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I have sequences of real-valued random variables $\{X_T\}, \{Y_T\}$ and a sequence of real numbers $\{a_T\}$. As $T\rightarrow\infty$, I know that $$ a_T \rightarrow \infty $$ and $$ X_T \overset{d}{\rightarrow} X $$ where $X$ is a real-valued random variable.

Furthermore, I have an inequality $$ X_T + Y_T \geq a_T $$ which holds for all $T$.

  1. How can I show that $Y_T \rightarrow \infty$ in probability?
  2. Does $Y_T\rightarrow\infty$ hold for other types of stochastic convergence?

(It all seems very obvious but I am struggling to come up with a formal argument)

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    $\begingroup$ Try proving the contrapositive: that is, if $Y_T\to\infty$ in probability is false, that means there exists some numbers $\epsilon\gt 0$ and $N\gt 0$ for which $\Pr(|Y_T|\le N)\ge\epsilon$ for infinitely many $T.$ Along with the constraint $X_T+Y_T\ge a_T$ and $a_T\to\infty,$ what does that imply about the distribution of $X_T$ for those infinitely many $T$? $\endgroup$
    – whuber ♦
    Commented Jul 9, 2020 at 13:09
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    $\begingroup$ Thanks, it was a little tedious but it worked! $\endgroup$
    – L D
    Commented Jul 9, 2020 at 19:03

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