Showing that $(y-c)(1-F(y)) \underset{y \rightarrow \infty}{\rightarrow} 0$ for $F$ the cdf of an integrable random variable?

Question

I was reading the comments in this question and it was suggested that this limit is $0$ by dominated convergence theorem. I'm not able to see how. If anyone sees it and could give some hints?

Thank you.

score 1 · Accepted Answer · answered Apr 04 '20 at 17:38

It suffices to show that $yP(Y>y)\to 0$ as $y\to\infty$ as $cP(Y>y)\to 0$ is clear. To see this note that for $y>0$ $$ 0\leq yI(Y>y)\leq YI(Y>y). $$ Take expectations to yield that $$ yP(Y>y)\leq E(YI(Y>y))\to 0 $$ as $y\to \infty$ by the dominated convergence theorem.

Showing that $(y-c)(1-F(y)) \underset{y \rightarrow \infty}{\rightarrow} 0$ for $F$ the cdf of an integrable random variable?

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