Let be a branching process with and . Define if . Set , and let . Prove that a.s., and also in the sense. Finally, prove that .

Stochastic Processes Solution – Problem 39: Prove that a.s., and also in the sense

Question

Let $\{\xi_{i,m}\}$ be a branching process with $\mu = \mathbb{E}\xi_{i,m} > 1$ and $\sigma^2 = \mathrm{Var}\,\xi_{i,m} < \infty$ . Define $Z_n = \sum_{k=1}^{Z_{n-1}} \xi_{n,k}$ if $Z_{n-1} \neq 0$ . Set $Z_0 = 1$ , and let $X_n = \frac{Z_n}{\mu^n}$ . Prove that $X_n \to W$ a.s., and also in the $L^2$ sense. Finally, prove that $\mathbb{E}W = 1$ .

Step-by-step solution

A branching process starting with $Z_0 = 0$ would have $Z_n = 0$ for all $n$ , making convergence trivial but yielding $W = 0$ and $\mathbb{E}W = 0$ , contradicting the requirement $\mathbb{E}W = 1$ . Therefore the standard setting is $Z_0 = 1$ . All proofs below assume $Z_0 = 1$ .

Step 1. Prove that $\{X_n\}$ is a martingale with respect to $\mathcal{F}_n = \sigma(Z_0, Z_1, \dots, Z_n)$ . Integrability: Since $Z_n \ge 0$ and $\mu > 0$ , $X_n \ge 0$ , so $\mathbb{E}|X_n| = \mathbb{E}[X_n]$ . By the branching property: $\mathbb{E}[Z_n | \mathcal{F}_{n-1}] = Z_{n-1}\mu$ . Hence $\mathbb{E}[Z_n] = \mu \mathbb{E}[Z_{n-1}]$ , giving $\mathbb{E}[Z_n] = \mu^n$ and $\mathbb{E}[X_n] = 1 < \infty$ . Martingale property: $\mathbb{E}[X_n | \mathcal{F}_{n-1}] = \frac{1}{\mu^n}\mathbb{E}[Z_n | \mathcal{F}_{n-1}] = \frac{Z_{n-1}\mu}{\mu^n} = \frac{Z_{n-1}}{\mu^{n-1}} = X_{n-1}$ .

Step 2. Prove $X_n \to W$ a.s. Since $\{X_n\}$ is a nonneg martingale, by the martingale convergence theorem, there exists $W$ such that $X_n \to W$ a.s.

Step 3. Prove $L^2$ convergence. By the $L^p$ martingale convergence theorem, an $L^2$ -bounded martingale converges in $L^2$ . We need $\sup_n \mathbb{E}[X_n^2] < \infty$ . Using the variance decomposition: $\mathrm{Var}(Z_n) = \sigma^2 \mu^{n-1} + \mu^2 \mathrm{Var}(Z_{n-1})$ . Setting $v_n = \mathrm{Var}(Z_n)/\mu^{2n}$ : $v_n = v_{n-1} + \sigma^2/\mu^{n+1}$ , with $v_0 = 0$ . So $v_n = \frac{\sigma^2}{\mu^2} \sum_{k=1}^{n} (1/\mu)^{k-1}$ . Since $\mu > 1$ , this geometric series converges: $v_n \to \frac{\sigma^2}{\mu(\mu-1)}$ . Therefore $\mathbb{E}[X_n^2] = v_n + 1 \le \frac{\sigma^2}{\mu(\mu-1)} + 1 < \infty$ . Hence $X_n \to W$ in $L^2$ .

Step 4. Prove $\mathbb{E}W = 1$ . $L^2$ convergence implies $L^1$ convergence, so $\mathbb{E}[W] = \lim_{n \to \infty} \mathbb{E}[X_n] = 1$ .

Final answer

QED.

Marking scheme

This rubric is based on the official solution, total 7 points.

1. Checkpoints (max 7 pts total)

Part 1: Martingale Property and Almost Sure Convergence (2 pts)

Prove the martingale property [1 pt]
Correctly compute $\mathbb{E}[Z_n | \mathcal{F}_{n-1}] = \mu Z_{n-1}$ and derive $\mathbb{E}[X_n | \mathcal{F}_{n-1}] = X_{n-1}$ .
*Note: If only $\mathbb{E}X_n = \mathbb{E}X_{n-1}$ is verified without using $\mathcal{F}_{n-1}$ , award 0 pts.*
Conclude a.s. convergence [1 pt]
Explicitly cite the nonneg martingale convergence theorem.

Part 2: $L^2$ Boundedness and Convergence (4 pts)

Establish the variance/second moment recursion [1 pt]
Correctly use the total variance formula to obtain $\mathrm{Var}(Z_n) = \sigma^2 \mu^{n-1} + \mu^2 \mathrm{Var}(Z_{n-1})$ .
Series summation/solve the recursion [1 pt]
Express $\mathbb{E}[X_n^2]$ or $\mathrm{Var}(Z_n)/\mu^{2n}$ as a geometric series sum.
Prove $L^2$ boundedness [1 pt]
Use $\mu > 1$ to show the series converges, hence $\sup_n \mathbb{E}[X_n^2] < \infty$ .
Conclude $L^2$ convergence [1 pt]
State that an $L^2$ -bounded martingale converges in $L^2$ .

Part 3: Expected Value (1 pt)

Prove $\mathbb{E}W=1$ [1 pt]
Use $L^2$ (or $L^1$ ) convergence to justify exchanging limit and expectation.

Total (max 7)

2. Zero-credit items

Only copying definitions or restating $Z_n$ 's formula.
Claiming $X_n$ is a constant sequence.
Asserting $L^2$ convergence without proving $L^2$ boundedness.
If the student assumes $Z_0=0$ leading to all-zero terms without discussing the nontrivial case ( $Z_0=1$ ), the entire solution gets 0 pts.

3. Deductions

[-2 pts] Logical error in variance computation: Treating $Z_{n-1}$ as a constant instead of a random variable.
[-1 pt] Missing convergence justification: Not mentioning convergence guarantees when computing $\mathbb{E}W$ .
[-1 pt] Series convergence error: Not explicitly using $\mu > 1$ when judging convergence.

Stochastic Processes – Problem 39: Prove that a.s., and also in the sense