Stochastic Processes – Problem 6: Prove that the sequence of random variables converges in the mean-square sense

This rubric is based on the official solution and is designed to assess the student's mastery of the proof of $L^2$ convergence (mean-square convergence) of martingales.

1. Checkpoints (Total: 7 pts)

Score exactly one chain (Chain A or Chain B). If a student mixes methods, grade the most complete chain.

Chain A: Based on Cauchy Sequence [Official Solution Path]

• Key identity and orthogonality (3 pts): Derive or use the martingale property to prove the $L^2$ orthogonality relation: for $m \le n$, prove $E[Y_n Y_m] = E[Y_m^2]$ OR directly derive $E[(Y_n - Y_m)^2] = E[Y_n^2] - E[Y_m^2]$. (Note: If using the orthogonal increment method $Y_n = \sum D_k$ and noting $E[D_i D_j]=0$ for $i \ne j$, thereby obtaining $E[Y_n^2] = \sum E[D_k^2]$, this item receives full marks.) [additive]
• Monotonicity and convergence of the norm (2 pts): State that $E[Y_n^2]$ is a nondecreasing sequence in $n$ (or that $Y_n^2$ is a submartingale), combined with the upper bound $K$ from the hypothesis, conclude that the sequence $\{E[Y_n^2]\}$ converges. [additive]
• Cauchy sequence determination (1 pt): Combine the conclusions from the previous two steps to state that $\lim_{n,m \to \infty} E[(Y_n - Y_m)^2] = 0$, thereby asserting that $\{Y_n\}$ is a Cauchy sequence in $L^2$. [additive]
• Completeness and conclusion (1 pt): Invoke the completeness of $L^2$ (or Hilbert space property) to conclude that the Cauchy sequence must converge to some random variable $Y$. (Note: If the student only writes "since it is a Cauchy sequence it converges," this implicitly uses completeness and should receive credit.) [additive]

Chain B: Based on Maximal Inequality and Dominated Convergence (Doob's Maximal Inequality & DCT)

• Almost sure convergence (1 pt): Cite Doob's Martingale Convergence Theorem to state that $Y_n \to Y_\infty$ almost surely. [additive]
• Maximal inequality and integrable domination (3 pts): Use Doob's $L^2$ maximal inequality to prove $\sup_{n} |Y_n| \in L^2$ (i.e., $E[(\sup |Y_n|)^2] \le 4 \sup E[Y_n^2] \le 4K$). [additive]
• Application of the Dominated Convergence Theorem (3 pts): Use the above integrable dominating function $(\sup |Y_n|)^2$ and apply the Dominated Convergence Theorem to prove $E[(Y_n - Y_\infty)^2] \to 0$. [additive]

2. Zero-credit items

• Only copying the problem conditions: e.g., only writing that $\{Y_n\}$ is a martingale or $E Y_n^2 \le K$, with no subsequent derivation.
• Circular reasoning: Assuming mean-square convergence and then deriving properties from it.
• Incorrect logic: Asserting convergence solely from "boundedness" (e.g., $|Y_n|$ bounded $\implies$ convergence), ignoring that this does not hold in infinite-dimensional spaces (Cauchy property or monotonicity is needed).
• Only citing the theorem name: For a proof problem, if only writing "by Doob's $L^2$ convergence theorem the conclusion holds" with no derivation, this is treated as an incomplete proof (unless the exam explicitly allows direct citation of this theorem; otherwise typically receives 0 or very low marks).

3. Deductions

• Logic gap (Max -2): In Chain A, if the specific expression relating $E[(Y_n - Y_m)^2]$ to $E[Y_n^2]$ is not established, and convergence of the norm is directly asserted to imply sequence convergence (confusing convergence of real sequences with convergence of random variable sequences).
• Notational confusion (Max -1): Confusing $E[Y_n^2]$ as a random variable with a constant, or failing to specify whether the limit symbol $\lim$ is in the $L^2$ sense or the real number sense, causing logical ambiguity.
• Limit object undefined (No penalty): If the Cauchy sequence property is proved but the convergence target "$Y$" is not explicitly written, as long as completeness is implicitly used, no deduction.