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Kernel conditional expectations for square-integrable Markov observables

Statement

Assume AC. Let $(X_{k})_{k \geq 0}$ be a stationary time-homogeneous Markov chain with probability kernel $P$ , invariant law $π$ , and natural filtration $F_{k} = σ (X_{0}, \dots, X_{k})$ . Let $g \in L^{2} (π)$ (The space $L^{p} (μ)$ as the quotient by null functions) and fix a finite real measurable representative, also denoted by $g$ . Define $P g (x) = \int g (y) P (x, d y)$ and $P (g^{2}) (x) = \int g (y)^{2} P (x, d y)$ on the $π$ -full measurable set where the respective integrals are finite, and set these functions to zero on the exceptional set. Then, for each $k \geq 0$ , $E [g (X_{k + 1}) ∣ F_{k}] = P g (X_{k}) .$ $E [g (X_{k + 1})^{2} ∣ F_{k}] = P (g^{2}) (X_{k}) a.s.$ Consequently, if $D_{k + 1} = g (X_{k + 1}) - P g (X_{k})$ , then $(D_{k + 1})$ is a square-integrable martingale-difference sequence and $E [D_{k + 1}^{2} ∣ F_{k}] = P (g^{2}) (X_{k}) - (P g (X_{k}))^{2} a.s.$

Facts & Assumptions

Given: AC, a stationary chain $X$ with kernel $P$ and invariant law $π$ , its natural filtration, and a real measurable representative $g$ with $π (g^{2}) < \infty$ .

[A1]

AC is the choice principle used by the library's conditional-expectation class convention. (The Axiom of Choice)

[F1]

For a bounded measurable $h$ and a $P$ -chain, $E [h (X_{m + n}) ∣ F_{m}] = P^{n} h (X_{m})$ ; in particular, at $n = 1$ the conditional mean is the kernel integral. (Chapman-Kolmogorov equations)

[F2]

Increasing nonnegative inputs have increasing conditional-expectation versions converging almost surely to the conditional expectation of their limit. (Conditional monotone convergence)

[F3]

Invariance is the event identity $\int P (x, A) π (d x) = π (A)$ for every measurable $A$ . (Invariant and stationary distribution for a Markov kernel)

[F4]

Nonnegative increasing functions pass through an integral by monotone convergence. (Monotone convergence for the integral)

[F5]

Kernel integration of a nonnegative product-measurable function is measurable, with infinity permitted. (Measurability of integration against a kernel)

[F6]

Conditional expectation is linear and preserves order. (Basic algebra and order properties of conditional expectation)

[F7]

Conditional Jensen gives $∣ E [Z ∣ G] ∣^{2} \leq E [∣ Z ∣^{2} ∣ G]$ for square-integrable $Z$ . (Conditional jensen inequality)

[F8]

On a probability space, Hölder gives $π (∣ g ∣) \leq π (g^{2})^{1 / 2} < \infty$ . (Holder's inequality for integrals, including the endpoint cases)

[F9]

An integrable $F_{k}$ -measurable factor can be taken out of a conditional expectation when the products are integrable. (Taking out what is known)

[F10]

Products of square-integrable random variables are integrable by Hölder's inequality. (Holder's inequality for integrals, including the endpoint cases)

Proof

1.1A1F1F5given

For bounded measurable $h$ , [F1] with $n = 1$ gives $E [h (X_{k + 1}) ∣ F_{k}] = P h (X_{k})$ . Kernel measurability [F5] makes the right side measurable.

2.1F2F3F4F6F8step 1.1

Write $g = g^{+} - g^{-}$ . By [F8], both parts are integrable. For $N \geq 1$ let $g_{N}^{\pm} = g^{\pm} \land N$ . These functions are bounded and increase to $g^{\pm}$ . Step 1.1 gives the conditional kernel identity for each $g_{N}^{\pm}$ ; [F2] makes the conditional expectations increase to $E [g^{\pm} (X_{k + 1}) ∣ F_{k}]$ , while [F4] makes $P g_{N}^{\pm} (X_{k}) ↑ P g^{\pm} (X_{k})$ . Invariance [F3], first for nonnegative simple functions and then by [F4], gives $E [P g^{\pm} (X_{k})] = π (g^{\pm}) < \infty$ ; hence both limits are finite almost surely. Subtracting the two finite identities using [F6] proves $E [g (X_{k + 1}) ∣ F_{k}] = P g (X_{k})$ .

2.2F2F3F4step 1.1

Apply the same argument to the nonnegative integrable function $g^{2}$ : the truncations $g^{2} \land N$ increase to $g^{2}$ , and invariance gives $E [P (g^{2}) (X_{k})] = π (g^{2}) < \infty$ . Thus $E [g (X_{k + 1})^{2} ∣ F_{k}] = P (g^{2}) (X_{k})$ . These identities do not depend on the chosen representative: if two representatives differ on a measurable $π$ -null set $A$ , invariance gives $π (P 1_{A}) = 0$ , hence $P (x, A) = 0$ for $π$ -almost every $x$ ; their kernel integrals agree there, and their evaluations at each $X_{k}$ agree almost surely.

3.1F3F6F7F9F10step 2.1step 2.2∎

By [F7] and step 2.2, $(P g (X_{k}))^{2} \leq P (g^{2}) (X_{k})$ almost surely, so stationarity and [F3] imply $E [(P g (X_{k}))^{2}] \leq π (g^{2})$ ; therefore $D_{k + 1} \in L^{2}$ . Step 2.1, the preceding square-integrability bound, conditional linearity and [F9] for the $F_{k}$ -measurable factor $P g (X_{k})$ give $E [D_{k + 1} ∣ F_{k}] = 0$ . The product $g (X_{k + 1}) P g (X_{k})$ is integrable by the two $L^{2}$ bounds and [F10], so [F9] also gives its conditional expectation as $(P g (X_{k}))^{2}$ . Expanding the square and using step 2.2 now yields $E [D_{k + 1}^{2} ∣ F_{k}] = P (g^{2}) (X_{k}) - (P g (X_{k}))^{2}$ . Thus the sequence is a square-integrable martingale-difference sequence.

Source notes

Jones §2 defines the kernel operator $P g$ and states the conditional one-step kernel convention. The bounded-to-square-integrable extension and the exceptional-set version are proved here; no pointwise identity is asserted outside a $π$ -full set.

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Sources

Galin L. Jones, On the Markov Chain Central Limit Theorem, §2 (standard reference, not scraped)
Charles J. Geyer, Markov Chain Monte Carlo Lecture Notes, §§2.2 and 4.8 (standard reference, not scraped)