Problem
Let ( X_1, X_2, \dots ) be i.i.d. with ( \mathbbE[X_1] = 0 ) and ( \mathbbE[X_1^2] = 1 ). Define ( S_n = X_1 + \dots + X_n ). Prove that
[ \fracS_n\sqrtn \quad \textdoes NOT converge almost surely. ]
Solution outline
Use Kolmogorov’s 0-1 law: the event ( \limsup S_n/\sqrtn \le c ) is a tail event, so its probability is 0 or 1. If almost sure convergence occurred, the limit would be constant a.s., but CLT gives non-degenerate distribution, contradiction. Hence no a.s. convergence. advanced probability problems and solutions pdf
Here are the hidden gems I recommend to my graduate students. Problem Let ( X_1, X_2, \dots ) be i
A well-constructed advanced probability problems PDF will span several interconnected domains: Solution outline Use Kolmogorov’s 0-1 law: the event
A good solutions PDF complements these problems with rigorous, step-by-step solutions, often highlighting measure-theoretic justifications (e.g., “by Fubini’s theorem” or “by the monotone class lemma”).