Question: (Slutskys lemma) Let {Xn, n = 0, 1, 2 . . . } and {Yn, n = 0, 1, 2 . . . } be

(Slutsky’s lemma) Let {Xn, n = 0, 1, 2 . . . } and {Yn, n = 0, 1, 2 . . . } be sequences of random variables. If Xn d−

→ X0 and Yn p−

→ c for a constant c as n → ∞, then show that

(1) Xn + Yn d−

→ X0 + c.

(2) XnYn d−

→ cX0.

(3) Xn/Yn d−

→ X0/c, (c ̸= 0).

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