Suppose is a random variable with the following density function: Given , the random variable is uniformly distributed on . Prove that is stationary and ergodic.

Stochastic Processes Solution – Problem 23: Prove that is stationary and ergodic

Question

Suppose $X_{0}$ is a random variable with the following density function:

$f(\alpha) = \begin{cases} 2c, & 0 < \alpha \leq 1 \\ 0, & \text{elsewhere} \end{cases}$

Given $X_{0},X_{1},\cdots,X_{n}$ , the random variable $X_{n+1}$ is uniformly distributed on $(1-X_{n},1]$ . Prove that $\{X_{n}\}$ is stationary and ergodic.

Step-by-step solution

Step 1. Determine the density of $X_0$ and verify normalization. From $\int_0^1 2c\,d\alpha = 2c = 1$ , we get $c = 1/2$ , so $X_0 \sim \text{Uniform}(0,1]$ .

Step 2. Describe the Markov property and transition kernel. $\{X_n\}$ is a Markov chain: given $X_n = x$ , $X_{n+1} \sim \text{Uniform}(1-x, 1]$ . The transition density is $p(y|x) = 1/x$ for $1-x < y \le 1$ , and 0 elsewhere.

Step 3. Prove irreducibility. If $y \in (1-x, 1]$ (i.e., $x > 1-y$ ), then $P(X_1 \in U(y) | X_0 = x) > 0$ in one step. If $y \le 1-x$ , take $n=2$ : $X_1 \sim (1-x,1]$ , and for $z > 1-y$ we have $y \in (1-z,1]$ , so $P(X_2 \in U(y)|X_0=x) > 0$ . Hence $\{X_n\}$ is irreducible.

Step 4. Prove aperiodicity. For continuous state spaces with a continuous transition kernel, there is no non-trivial periodic structure, so $\{X_n\}$ is aperiodic (period 1).

Step 5. Verify the Feller property. For a closed set $A$ , $P(x, A) = \text{length}(A \cap (1-x,1])/x$ . As $x \to x_0$ , the interval endpoint $1-x \to 1-x_0$ continuously, so $P(x,A) \to P(x_0,A)$ . The Feller property holds.

Step 6. Prove positive recurrence. Since the transition kernel is continuous and nonzero on $(1-x,1]$ , and the state space $S$ is bounded, return times have finite expectation, so $\{X_n\}$ is positive recurrent.

Step 7. Apply the Markov chain ergodic theorem. For a continuous-state Markov chain satisfying irreducibility, aperiodicity, the Feller property, and positive recurrence, the chain is ergodic: there exists a unique stationary distribution $\pi$ such that for any initial distribution, $X_n$ converges weakly to $\pi$ . - Stationarity: with $X_0 \sim \pi$ , the finite-dimensional distributions are shift-invariant. - Ergodicity: time averages equal space averages a.s., and all invariant sets have probability 0 or 1.

Step 8. In summary, the Markov chain $\{X_n\}$ is stationary and ergodic.

Final answer

QED.

Marking scheme

1. Checkpoints (max 7 pts total)

Model parameters and kernel (1 pt): Compute $c=1/2$ via normalization and write the correct transition density $p(y|x) = (1/x)\mathbb{I}_{(1-x,1]}(y)$ . If only $c$ is found without the kernel, award 0.
Irreducibility proof (2 pts)
[additive] Argue one-step reachability ( $n=1$ ) for $y > 1-x$ , or define irreducibility. [1 pt]
[additive] Argue multi-step reachability (e.g., $n=2$ ) for $y \le 1-x$ , proving irreducibility on all of $(0,1]$ . [1 pt]
Aperiodicity (1 pt): Note that continuous state space with a density (or no discrete cyclic structure) implies period 1.
Feller property / continuity (1 pt): Verify $P(x,A)$ is continuous in $x$ , or state the Feller property holds.
Positive recurrence (1 pt): Use boundedness of the state space to argue finite mean return time, or cite Krylov-Bogolyubov for existence of an invariant measure.
Ergodic theorem and conclusion (1 pt): Cite the Markov chain ergodic theorem (irreducible + aperiodic + positive recurrent/Feller $\Rightarrow$ ergodic with unique stationary distribution $\pi$ ). Conclude the process is stationary and ergodic.

Total (max 7)

2. Zero-credit items

Merely copying the density or distribution definition from the problem.
Listing terms like "stationary" or "ergodic" without verifying any conditions.
Asserting irreducibility or positive recurrence without computing the transition kernel.
Only computing $c=1/2$ with no further discussion of the stochastic process.

3. Deductions

Cap at 5/7: Completely ignoring the integration domain restriction (not handling $1-x < y$ ), causing a serious gap in the irreducibility proof.
-1: Writing the transition density without the indicator function or boundary conditions.
-1: Citing the ergodic theorem but omitting a key prerequisite (e.g., not mentioning irreducibility or positive recurrence).

Stochastic Processes – Problem 23: Prove that is stationary and ergodic