Probability Theory – Problem 77: Prove that the random variables converge in distribution to

The following is the detailed marking scheme for this problem (total: 7 points).

1. Checkpoints (max 7 pts total)

Choose one of the following three paths that fully matches the student's approach; do not accumulate points across paths.

Chain A: Continuous Mapping Theorem Path (Official Solution)

• Convergence in probability conversion [2 pts]: Explicitly state or prove: since $Y_n$ converges in distribution to a constant $c$, it follows that $Y_n$ converges in probability to $c$ ($Y_n \xrightarrow{p} c$).
• *Note: If the key condition "constant" is not mentioned, causing a logical gap, no credit is awarded for this item.*
• Joint distributional convergence [2 pts]: Using $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{p} c$, conclude that the bivariate random vector converges in distribution to $(X, c)$, i.e., $(X_n, Y_n) \xrightarrow{d} (X, c)$.
• Continuous Mapping Theorem (CMT) application [3 pts]:
• Construct the function $g(x,y) = xy$ and state its continuity (on $\mathbb{R}^2$ or on the support of the limit) [1 pt].
• Apply the Continuous Mapping Theorem to conclude $g(X_n, Y_n) \xrightarrow{d} g(X, c)$, i.e., $X_n Y_n \xrightarrow{d} cX$ [2 pts].
• *Note: Simply writing the conclusion $X_n Y_n \xrightarrow{d} cX$ without mentioning continuity or the mapping theorem earns no credit for this step.*

Chain B: Direct Slutsky's Theorem Path

• Condition verification [3 pts]: Explicitly state that Slutsky's theorem requires one variable to converge in probability to a constant, and derive or declare $Y_n \xrightarrow{p} c$ from the hypothesis $Y_n \xrightarrow{d} c$.
• *Key point: The student must demonstrate awareness of the distinction between "convergence in distribution" and "convergence in probability"; they cannot assume the two are equivalent by default.*
• Theorem application [4 pts]: Accurately cite Slutsky's theorem (product form) to directly conclude $X_n Y_n \xrightarrow{d} cX$.

Chain C: Characteristic Function or First-Principles Approach

• Convergence in probability conversion [2 pts]: Obtain $Y_n \xrightarrow{p} c$.
• Analytic proof [5 pts]: Use characteristic functions to decompose and estimate, or rigorously prove convergence of the product using probability metric inequalities.
• *If the argument contains serious logical flaws (e.g., incorrectly interchanging limits), this part receives 0 points.*

Total (max 7)

2. Zero-credit items

• Merely copying the problem conditions (e.g., "$X_n \to X, Y_n \to c$").
• Asserting the conclusion based solely on deterministic calculus limit rules ("the limit of a product equals the product of the limits") without citing any probabilistic limit theorem (such as Slutsky or CMT).
• Claiming $Y_n \to c$ is "almost sure convergence" without proof (the problem only gives convergence in distribution; this is an incorrect implication).

3. Deductions

• Incorrectly assuming independence: If the proof assumes $X_n$ and $Y_n$ are mutually independent (the problem does not state this), and this assumption is central to the proof (e.g., directly factoring the joint probability $P(AB)=P(A)P(B)$): score capped at 3/7 (only the first step receives credit).
• Logical gap: In Chain A/B, directly using $Y_n \xrightarrow{d} c$ as $Y_n \xrightarrow{p} c$ without mentioning "because the limit is a constant": deduct 1 point.
• Notation error: Confusing random variables (uppercase $X$) with specific values (lowercase $x$) in a way that affects semantic understanding: deduct 1 point.