1
$\begingroup$

Let $X_1, X_2, ...$ and $Y_1, Y_2, ...$ be two sequences of nonnegative random variables. Assume that each $n$ random variable $Y_n$ is uniform in the interval $[0, X_n]$.

  1. Show that if $X_n$ converges to 0 in probability, then $Y_n$ converges to 0 in probability as well.

  2. Show that if $Y_n$ converges to 0 in probability, then $X_n$ converges to 0 in probability as well.

These are my answers, but I'm not sure if I'm right.

  1. Given $\epsilon >0$, $$P(|Y_n| \geq \epsilon) = P(Y_n \geq \epsilon , X_n \geq \epsilon) = P(Y_n \geq \epsilon | X_n \geq \epsilon) * P(X_n \geq \epsilon)$$ So $P(X_n \geq \epsilon) \rightarrow 0$ implies $P(Y_n \geq \epsilon) \rightarrow 0$.

  2. From the above equation, we get $$P(|X_n| \geq \epsilon) = P(Y_n \geq \epsilon)/P(Y_n \geq \epsilon | X_n \geq \epsilon)$$ So $P(Y_n \geq \epsilon) \rightarrow 0$ implies $P(X_n \geq \epsilon) \rightarrow 0$.

Am I right? Am I at least headed in the right direction?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.