Prove the second result of Theorem 4.11. That is, let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables that converge weakly to a random variable \(X\). Let \(\left\{Y_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables that converge in probability to a real constant \(c\). Prove that \(X_{n} Y_{n} \xrightarrow{d} c X\) as \(n ightarrow \infty\).

Transcribed Image Text:

Theorem 4.11 (Slutsky). Let {X} be a sequence of random variables that converge weakly to a random variable X. Let {Y}=1 be a sequence of random variables that converge in probability to a real constant c. Then, 1. Xn+YX+c as n o. 2. X,YcX as n o. 3. Xn/Yn X/c as n as long as c +0.