Variable-block coupling bounds for nonstationary statistics

Definitions and mathematical statement

The original question asks whether coupling can study statistical properties of a time-dependent nonstationary process. It specifies neither a process class, coupling, nor statistic, so it has no unique universal yes/no formulation. The aid considers the following time-inhomogeneous Markov model.

Let $(X_t)_{t\geq0}$ have transition kernels $P_t$ from time $t$ to $t+1$ on a standard Borel space. Choose integer block times $0=t_0<t_1<\cdots$. Suppose each full block kernel satisfies \[ P_{t_\ell,t_{\ell+1}}(x,\cdot)\geq\varepsilon_\ell\nu_\ell(\cdot) \quad\text{for every }x,\qquad0\leq\varepsilon_\ell\leq1, \] where $\nu_\ell$ is a probability measure and $P_{i,j}=P_i\cdots P_{j-1}$. For $i<j$, define \[ q(i,j)=\prod_{\ell:\ i\leq t_\ell<t_{\ell+1}\leq j}(1-\varepsilon_\ell), \] with empty product one. Total variation is $\|\mu-\nu\|_{\mathrm{TV}}=\sup_A|\mu(A)-\nu(A)|$.

An explicit version of the aid's reported transfer principle is that the same product controls a block-based coalescing coupling and loss of memory: \[ \Pr(\text{the coupling has not coalesced by }j)\leq q(i,j),\qquad \|\mu P_{i,j}-\eta P_{i,j}\|_{\mathrm{TV}}\leq q(i,j) \] for arbitrary starting laws $\mu,\eta$ at time $i$. After coalescence the two copies use the same future randomness. For bounded $U$ measurable with respect to the history through $i$, and bounded $V$ measurable with respect to the future from $j$, the corresponding covariance bound is \[ |\operatorname{Cov}(U,V)|\leq2\|U\|_\infty\|V\|_\infty q(i,j). \] For time-dependent centered observables $\phi_i(X_i)$, with ${\mathbb{E}}\phi_i(X_i)=0$ under the actual nonstationary law and $\|\phi_i\|_\infty\leq M_i$, put $S_n=\sum_{i=0}^{n-1}\phi_i(X_i)$. A compatible explicit variance bound is \[ \operatorname{Var}(S_n)\leq\sum_{i<n}M_i^2+ 4\sum_{0\leq i<j<n}M_iM_jq(i,j). \] In particular, if $M_i\leq M$ and $B=\sup_i\sum_{j>i}q(i,j)<\infty$, then \[ \operatorname{Var}(S_n)\leq M^2(1+4B)n,\qquad S_n/n\longrightarrow0\text{ in }L^2. \] No invariant probability measure is assumed. The record does not provide its own exact constants or block convention; these displayed conventions specify the transfer statement rather than claiming a verbatim theorem from the source.

The aid also reports two-state sharpness and failure examples. Their mechanisms can be expressed by reset kernels on $\{0,1\}$: \[ P_t(x,\cdot)=(1-\varepsilon_t)\delta_x+\varepsilon_t\nu. \] For a fixed common reset law $\nu$, the total-variation distance between the two distinct-state evolutions is exactly $\prod_{t=i}^{j-1}(1-\varepsilon_t)$, as is failure of the common-reset coupling. Thus $0<\varepsilon_t<1$ at every time does not imply memory loss when $\sum_t\varepsilon_t<\infty$.

For a separate obstruction, kernels $P_t(x,\cdot)=\nu_{t+1}$ have zero one-step Dobrushin coefficient and generate independent, nonidentically distributed states. Taking Bernoulli parameters alternating between two values with different variances on successive blocks dominating all previous lengths makes $n^{-1}\operatorname{Var}(\sum_{i<n}(X_i-{\mathbb{E}} X_i))$ fail to converge. The two values can be $1/4$ and $1/2$, with variances $3/16$ and $1/4$. Hence even perfect one-step coupling does not force a limiting diffusion rate. These examples distinguish memory bounds and a law of large numbers from convergence of normalized variance or a central limit theorem.

Short English statement

The aid gives a precise Markov-chain setting where block coupling bounds control memory, covariance and additive-functional variance without stationarity. It also records that repeated positive contraction need not erase memory, and even perfect coupling need not yield a limiting normalized variance.

Sources

• Local registry: permanent UnsolvedMath ID $20001390$, exact public code catalogue entry; aim-dynamical-systems-notes.json, notes[47].
• AIM workshop "Stochastic methods for non-equilibrium dynamical systems", Problem 5: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf. The supplied verification says the wording, including "time dependent", is exact but lacks a model and target statistic. The registry marks the statement reconstructed_unverified. No proposer or literature citation is supplied.
• Research attempt 1: partial_result, status at run partially_solved, rigor label proved, novelty confidence low, classification PARTIAL-PROGRESS. It reports variable-block minorization, the common coupling-failure product, uniform-summability consequences, an exact reset-chain memory example and a second two-state normalized-variance obstruction. No authoritative separate bibliography or full theorem text is supplied. The current registry status is partially_solved.