Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

#1 · Probability & Statistics › Limit Theorems and Markov Chains

“Convergence in probability ⇒ almost sure convergence” — false

Counterexample locked — unlock with Notes + PYQ

probabilityconvergence

#2 · Probability & Statistics › Limit Theorems and Markov Chains

“Convergence in probability implies almost sure convergence” — false

Counterexample: The typewriter sequence on [0,1]

$P (X_{n} \neq = 0) \to 0$ but every $ω$ is hit infinitely often, so there is no a.s. limit.

probabilityconvergence

#3 · Probability & Statistics › Limit Theorems and Markov Chains

“Xₙ → 0 almost surely implies E[Xₙ] → 0” — false

Counterexample locked — unlock with Notes + PYQ

probabilityconvergence