Probability Theory – Problem 22: Prove that the random variables converge in distribution to

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

1. Checkpoints (max 7 pts total)

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

Chain A: Continuous Mapping Theorem Path (Official Solution)

• Conversion to convergence in probability [2 pts]: Explicitly state or prove 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 that the limit is a constant is not mentioned, resulting in 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., $(X_n, Y_n) \xrightarrow{d} (X, c)$.
• Application of the Continuous Mapping Theorem (CMT) [3 pts]:
• Construct the function $g(x,y) = xy$ and note its continuity (on $\mathbb{R}^2$ or on the support of the limit) [1 pt].
• Apply the CMT 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 the student merely states the conclusion $X_n Y_n \xrightarrow{d} cX$ without mentioning continuity or the mapping theorem, no credit is awarded for this step.*

Chain B: Direct Application of Slutsky's Theorem

• Verification of conditions [3 pts]: Explicitly state that Slutsky's theorem requires one of the variables to converge in probability to a constant, and derive or assert $Y_n \xrightarrow{p} c$ 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; the two cannot be treated as equivalent by default.*
• Application of the theorem [4 pts]: Correctly cite Slutsky's theorem (product form) to directly conclude $X_n Y_n \xrightarrow{d} cX$.

Chain C: Characteristic Functions or First-Principles Approach

• Conversion to convergence in probability [2 pts]: Establish $Y_n \xrightarrow{p} c$.
• Analytical proof [5 pts]: Use characteristic functions to decompose and estimate, or employ probability metric inequalities to rigorously prove convergence of the product.
• *If the argument contains a serious logical flaw (e.g., incorrectly interchanging limits), award 0 points for this part.*

Total (max 7)

2. Zero-credit items

• Merely copying the hypotheses (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 invoking any probabilistic limit theorem (such as Slutsky's theorem or the CMT).
• Claiming that $Y_n \to c$ holds almost surely without proof (the problem only gives convergence in distribution; this implication is incorrect).

3. Deductions

• Incorrect independence assumption: If the proof assumes that $X_n$ and $Y_n$ are independent (a condition not given in the problem), and this assumption is central to the argument (e.g., directly factoring the joint probability as $P(AB)=P(A)P(B)$): cap the score at 3/7 (only the first step receives credit).
• Logical gap: In Chain A/B, if the student directly uses $Y_n \xrightarrow{d} c$ as $Y_n \xrightarrow{p} c$ without mentioning that the limit is a constant: deduct 1 point.
• Notational error: Confusing random variables (uppercase $X$) with deterministic values (lowercase $x$) in a way that affects semantic clarity: deduct 1 point.