In the context of the proof of the first result of Theorem 4.11, prove that [Pleft(X_{n} leq
Question:
In the context of the proof of the first result of Theorem 4.11, prove that
\[P\left(X_{n} \leq x-\varepsilon-cight) \leq G_{n}(x)+P\left(\left|Y_{n}-cight|>\varepsilonight) .\]
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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+Y X + c as n o. 2. X,Y, dcX as n . 3. Xn/Yn + X/c as n as long as c +0.
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