Discussion
Suppose \(X_{n} \to X\) almost surely. Suppose \(Y_{n}\) and Y are defined as \(Y_{n} = X_{n}^{3} + 2X_{n}^{2} + 3X_{n} + 2\) and \(Y = X^{3} + 2X^{2} + 3X + 2\). The following are three statements :
(I) \(Y_{n} \to Y\) almost surely
(II) \(Y_{n} \to Y\) in probability
(III) \(Y_{n} \to Y\) in law.
Which of the following is true ?
Answer: D
All three are true.
\(g(x) = x^{3} + 2x^{2} + 3x + 2\) is continuous, and \(Y_{n} = g(X_{n})\), \(Y = g(X)\). By the continuous mapping theorem, almost sure convergence of \(X_{n}\) gives almost sure convergence of \(g(X_{n})\): wherever \(X_{n}(\omega) \to X(\omega)\), \(g(X_{n}(\omega)) \to g(X(\omega))\). So (I) holds.
Almost sure convergence implies convergence in probability, which in turn implies convergence in law, so (II) and (III) follow from (I).
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