数据科学基础 (Fall 2024)/Problem Set 6

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Assumption throughout Problem Set 6

Without further notice, we are working on probability space [math]\displaystyle{ (\Omega,\mathcal{F},\Pr) }[/math].

Without further notice, we assume that the expectation of random variables are well-defined.

Problem 1 (Convergence) (Bonus)

• [Convergence in [math]\displaystyle{ r }[/math]-th mean] Suppose [math]\displaystyle{ X_n{\xrightarrow {r}} X }[/math], where [math]\displaystyle{ r\ge 1 }[/math]. Prove or disprove that [math]\displaystyle{ \mathbb E[X_n^r]\to\mathbb E[X^r] }[/math].
• [Dominated convergence] Suppose [math]\displaystyle{ |X_n|\le Z }[/math] for all [math]\displaystyle{ n\in\mathbb N }[/math], where [math]\displaystyle{ \mathbb E(Z)\lt \infty }[/math]. Prove that if [math]\displaystyle{ X_n \xrightarrow P X }[/math] then [math]\displaystyle{ X_n \xrightarrow 1 X }[/math].
• [Slutsky’s theorem] Let [math]\displaystyle{ (X_n)_{n \ge 1}, (Y_n)_{n \ge 1}, X, Y }[/math] be random variables and [math]\displaystyle{ c\in\mathbb{R} }[/math] be a real number.
  1. Suppose [math]\displaystyle{ X_n \overset{D}{\to} X }[/math] and [math]\displaystyle{ Y_n \overset{D}{\to} c }[/math]. Prove that [math]\displaystyle{ X_nY_n \overset{D}{\to} cX }[/math].
  2. Construct an example such that [math]\displaystyle{ X_n \overset{D}{\to} X }[/math] and [math]\displaystyle{ Y_n \overset{D}{\to} Y }[/math] but [math]\displaystyle{ X_nY_n }[/math] does not converge to [math]\displaystyle{ XY }[/math] in distribution.

Problem 2 (LLN & CLT)

• [Proportional betting] In each of a sequence of independent bets, a gambler either wins 30%, or loses 25% of her current fortune, each with probability [math]\displaystyle{ 1/2 }[/math]. Denoting her fortune after [math]\displaystyle{ n }[/math] bets by [math]\displaystyle{ F_n }[/math], show that [math]\displaystyle{ \mathbb E(F_n)\to\infty }[/math] as [math]\displaystyle{ n \to\infty }[/math], while [math]\displaystyle{ F_n \to 0 }[/math] almost surely.
• [Transience] Let [math]\displaystyle{ X_1,X_2,\dots }[/math] be independent identically distributed random variables taking values in the integers [math]\displaystyle{ \mathbb Z }[/math] and having a finite mean. Show that the Markov chain [math]\displaystyle{ S = \{S_n\} }[/math] given by [math]\displaystyle{ S_n = \sum^n_{i=1} X_i }[/math] is transient, i.e. [math]\displaystyle{ \forall n\in\mathbb N,\Pr(\exists n'\gt n, S_{n'}=S_n)\lt 1 }[/math], if [math]\displaystyle{ \mathbb E(X_1)\ne 0 }[/math].
• [Controlling a Fair Voting] In a society of [math]\displaystyle{ n }[/math] isolated (independent) and neutral (uniform) peoples, how many peoples are there enough to manipulate the result of a majority vote with [math]\displaystyle{ 1-\delta }[/math] certainty? You have to use the Berry–Esseen theorem to solve this problem.