Long-run central limit theorem for a dependent velocity sensor

Assume the Axiom of Choice. Let $H$ and the centered sensor observable $f_{\psi }:H\to \mathbb{R}$ be as in the Definition. The sequences in either branch below are defined on a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$. General stationary sensor branch. Let $(V_k{)}_{k\ge 0}$ be $H$-valued random variables with marginal law $\mu$ and put $Y_k=f_{\psi }(V_k)$. More generally, let $g:H\to \mathbb{R}$ be Borel measurable and satisfy ${\int }_{H}gd\mu =0$; under $V_0\sim \mu$, this is equivalent to $\mathbb{E}{Y}_0=0$ for $Y_0=g(V_0)$. Put $Y_k=g(V_k)$ and assume $\mathbb{E}{Y}_0^2<\mathrm{\infty }$. Assume $(Y_k)$ is strictly stationary: all its finite-dimensional laws are unchanged by common nonnegative integer shifts. For $n\ge 1$, define ${\alpha }_Y(n):={\sup }_{j\ge 0}{\sup }_{\begin{array}{c}A\in \sigma (Y_0,\dots ,Y_j)\\ B\in \sigma (Y_{j+n},Y_{j+n+1},\dots )\end{array}}\mathrm{\mid }\mathbb{P}(A\cap B)-\mathbb{P}(A)\mathbb{P}(B)\mathrm{\mid }\mathrm{.}$ Any one of the following is sufficient: $Y_0$ is essentially bounded and ${\sum }_n{\alpha }_Y(n)<\mathrm{\infty }$; for some $\delta >0$, $\mathbb{E}\mathrm{\mid }{Y}_0{\mathrm{\mid }}^{2+\delta }<\mathrm{\infty }$ and ${\sum }_n{\alpha }_Y(n{)}^{\delta \mathrm{/}(2+\delta )}<\mathrm{\infty }$; or ${\alpha }_Y(n)=O(r^n)$ for some $0<r<1$ and $\mathbb{E}[Y_0^2{\log }^{+}\mathrm{\mid }{Y}_0\mathrm{\mid }]<\mathrm{\infty }$. In each case ${\sum }_{k\ge 1}\mathrm{\mid }\mathrm{Cov}(Y_0,Y_k)\mathrm{\mid }<\mathrm{\infty }$ and $n^{-1\mathrm{/}2}{\sum }_{k=0}^{n-1}{Y}_k\Rightarrow N(0,{\sigma }_Y^2),{\sigma }_Y^2=\mathrm{Var}(Y_0)+2{\sum }_{k\ge 1}\mathrm{Cov}(Y_0,Y_k)\ge 0.$ Zero variance is included. No Markov, Harris, reversibility or state-space countable-generation assumption is needed for this branch. For a Reynolds product sensor, apply this clause to the centered product itself; its $(2+\delta )$ moment is an additional hypothesis. For two bounded linear sensors on $H$, a finite $(4+2\delta )$ moment of $\mathrm{\parallel }{V}_0{\mathrm{\parallel }}_H$ is a sufficient product-moment condition. No mixing property or rate is inferred from a fluid equation or the sampling interval.

Additional Markov-chain sufficient regimes. Let $\mathcal{B}(H)$ be countably generated, let $P$ be a probability kernel on $(H,\mathcal{B}(H))$ with invariant probability $\mu$, and let $(V_k{)}_{k\ge 0}$ be a stationary time-homogeneous $P$-chain whose canonical left shift preserves its path law and whose shift-invariant measurable events have probability zero or one. Assume $P$ is $\varphi$-irreducible, aperiodic, and Harris recurrent for a nonzero sigma-finite measure $\varphi$. Here $\varphi$-irreducibility means each $\varphi$-positive measurable set is reached with positive probability after some finite positive number of steps from every state; Harris recurrence means each such set is hit almost surely from every starting state; aperiodicity means there is no integer $q\ge 2$ and pairwise disjoint measurable sets $H_1,\dots ,H_q$ with $\mu (H_1)>0$ and $P(v,H_{i+1})=1$ for every $v\in H_i$ and $1\le i<q$, and $P(v,H_1)=1$ for every $v\in H_q$. Invariant means $\mu P=\mu$.

For $k\ge 0$, put ${\gamma }_k:={\mathrm{Cov}}_{\mu }(f_{\psi }(V_0),f_{\psi }(V_k))$. Suppose at least one of the following six regimes holds:

1. For some $m>1$ there is a measurable $M\ge 0$ with ${\int }_{H}Md\mu <\mathrm{\infty }$ and $\mathrm{\parallel }{P}^n(v,\cdot )-\mu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le M(v)n^{-m}$ for every $v$ and $n\ge 1;$f_\psi$isessentiallyboundedunder$\mu$.
2. For some $m>0$ there is a measurable $M\ge 0$ with ${\int }_{H}Md\mu <\mathrm{\infty }$ and $\mathrm{\parallel }{P}^n(v,\cdot )-\mu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le M(v)n^{-m}$ for every $v$ and $n\ge 1;forsome$\delta>0$,$f_\psi\in L^{2+\delta}(\mu)$and$m\delta>2+\delta$.
3. $P$ is geometrically ergodic, meaning $\mathrm{\parallel }{P}^n(v,\cdot )-\mu {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le M(v)r^n$ for a finite measurable $M$ and some $0<r<1$, and $f_{\psi }\in L^{2+\delta }(\mu )$ for some $\delta >0$.
4. $P$ is geometrically ergodic in the same total-variation sense and ${\int }_H{f}_{\psi }^2{\log }^{+}\mathrm{\mid }{f}_{\psi }\mathrm{\mid }d\mu <\mathrm{\infty }$, with $0{\log }^{+}0:=0$.
5. $P$ is geometrically ergodic in the same total-variation sense, is reversible in the symmetric-flow sense $\mu (dv)P(v,dw)=\mu (dw)P(w,dv)$, and $f_{\psi }\in L^2(\mu )$.
6. $P$ is uniformly ergodic, meaning the geometric total-variation bound holds with $M$ bounded on $H$, and $f_{\psi }\in L^2(\mu )$.

Then ${\sum }_{k\ge 1}\mathrm{\mid }{\gamma }_k\mathrm{\mid }<\mathrm{\infty }$ and $\frac{1}{\sqrt{n}}{\sum }_{k=0}^{n-1}{f}_{\psi }(V_k)\Rightarrow N(0,{\sigma }_{\mathrm{L}\mathrm{R}}^2(f_{\psi })),$ where ${\sigma }_{\mathrm{L}\mathrm{R}}^2(f_{\psi }):={\gamma }_0+2{\sum }_{k=1}^{\mathrm{\infty }}{\gamma }_k$ is the covariance long-run variance. Here $\mathrm{\parallel }\alpha -\beta {\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}:={\sup }_{A\in \mathcal{B}(H)}\mathrm{\mid }\alpha (A)-\beta (A)\mathrm{\mid }$, and $f\in L^p(\mu )$ for finite $p$ means ${\int }_H\mathrm{\mid }f{\mathrm{\mid }}^{p}d\mu <\mathrm{\infty }$. Also set ${\log }^{+}t:=\max (0,\log t)$ for $t>0$ and $0{\log }^{+}0:=0$. The zero-variance case is included. This is an asymptotic model deduction; it does not make finite samples normally distributed or establish a physical sensor's stationarity, Markov property, or mixing assumptions. A central limit theorem alone is not a consistency result for an estimated Monte Carlo standard error.

Strong-mixing CLTs under bounded, moment, and logarithmic-moment conditions

Strong, uniform, and maximal-correlation mixing coefficients

Finite-second-moment random velocity sensor law

The L2 velocity sensor is well-defined

The Axiom of Choice

Sufficient MCMC CLT regimes

Long-run variance for a stationary Markov chain

Measure kernel and probability kernel

Time-homogeneous Markov chain with transition kernel

Invariant and stationary distribution for a Markov kernel

Phi-irreducibility for a general-state Markov kernel

Aperiodicity for a general-state Markov kernel

Harris recurrence relative to an irreducibility measure

Geometric and uniform ergodicity for MCMC CLTs

Polynomial ergodicity with an integrable rate envelope

General-state reversibility by symmetric flow measure

The space $L^p(\mu )$ as the quotient by null functions

Total variation distance for probability laws

Ergodicity relative to an invariant measure

Measure-preserving transformations and systems

Jones, On the Markov Chain Central Limit Theorem, Section 4.1

Geyer, Markov Chain Monte Carlo Lecture Notes, Sections 4.8.1–4.9