Probability Theory – Problem 51: Prove that the random variable converges in distribution to

The following is the detailed grading rubric for this problem (maximum 7 points).

1. Checkpoints (max 7 pts total)

Select exactly one of the following three paths that fully matches the student's approach; do not combine points across paths.

Chain A: Continuous Mapping Theorem Path (Official Solution)

• Convergence in probability conversion [2 pts]: Explicitly stating or proving that since $Y_n$ converges in distribution to the 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 for this item.*
• Joint distribution convergence [2 pts]: Using $X_n \xrightarrow{d} X$ and $Y_n \xrightarrow{p} c$ to conclude that the two-dimensional 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]:
• Constructing the function $g(x,y) = xy$ and noting its continuity (on $\mathbb{R}^2$ or on the support of the limit) [1 pt].
• Applying the Continuous Mapping Theorem to obtain $g(X_n, Y_n) \xrightarrow{d} g(X, c)$, i.e., $X_n Y_n \xrightarrow{d} cX$ [2 pts].
• *Note: If only the conclusion $X_n Y_n \xrightarrow{d} cX$ is stated without mentioning continuity or the mapping theorem, no credit for this step.*

Chain B: Direct Citation of Slutsky's Theorem Path

• Condition verification [3 pts]: Explicitly stating that Slutsky's theorem requires one variable to converge in probability to a constant, and deriving or asserting $Y_n \xrightarrow{p} c$ 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 treat the two as equivalent by default.*
• Theorem application [4 pts]: Accurately citing Slutsky's theorem (product form) to directly conclude $X_n Y_n \xrightarrow{d} cX$.

Chain C: Characteristic Functions or First-Principles Approach

• Convergence in probability conversion [2 pts]: Obtaining $Y_n \xrightarrow{p} c$.
• Analytical proof [5 pts]: Using characteristic functions to decompose and estimate, or rigorously proving the convergence of the product via 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 implication is incorrect).

3. Deductions

• Incorrectly assuming independence: If the proof assumes $X_n$ and $Y_n$ are mutually independent (a condition not given in the problem), and this assumption is central to the proof (e.g., directly factoring the joint distribution probability $P(AB)=P(A)P(B)$): score capped at 3/7 (only the first step receives credit).
• Logical gap: In Chain A/B, if $Y_n \xrightarrow{d} c$ is directly used as $Y_n \xrightarrow{p} c$ without any mention of "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.