A gap-optimized Poisson approximation bound with a clustering obstruction

Definitions and mathematical statement

The source asks for rates of convergence to a Poisson law in slowly mixing systems. Its example "from a decay rate of 1/n 4 to 1/n" does not unambiguously specify an exponent, the mixing estimate, or the probability metric. The aid instead states a rare-target total-variation criterion.

Let $(\Omega,\mu,T)$ be a probability-preserving system, let $A$ be measurable with $p=\mu(A)>0$, fix $t>0$, and set \[ N=\lfloor t/p\rfloor,\qquad X_i=\mathbf1_A\circ T^i, \qquad W_A=\sum_{i=1}^NX_i. \] For integer gap $g\ge1$, define the positive short-return excess \[ Q_A(g)=\sum_{j=1}^g\max\{\mu(A\cap T^{-j}A)-p^2,0\}. \] For the entire far-field sigma algebra $\mathcal B_{i,N,g}=\sigma(X_j:1\le j\le N,\ |j-i|>g)$, use the normalized event-mixing coefficient \[ M_A(g)=\sup_{N\ge1}\sup_{1\le i\le N} \frac1p\sup_{B\in\mathcal B_{i,N,g}} |{\mathbb{E}}[(X_i-p)\mathbf1_B]|. \] The supremum over all horizons makes this a whole-far-field condition, not just pairwise correlation decay. Assume $M_A(g)\le C_Mg^{-\beta}$ with $\beta>0$. With total variation defined by supremum over events, the aid reports \[ d_{\mathrm{TV}}(\mathcal L(W_A),\operatorname{Pois}(t)) \le t(4g+1)p+\frac{2tQ_A(g)}p+2C_Mt g^{-\beta}+p. \] The last term accounts for $|Np-t|\le p$. The first term retains the independent $p^2$ neighborhood contribution rather than assigning all short returns to clustering. The factor $p$ in normalized far-field dependence is essential before summing $N$ trials. Exact dependency-neighborhood and total-variation constants require verification from the missing proof artifact.

For a family of shrinking targets with $p\to0$, constants uniform in the family, and fixed $t,\beta$, choose \[ g_p=\lceil p^{-1/(\beta+1)}\rceil. \] If additionally $Q_A(g_p)\le C_Rp^{1+\gamma}$ with $\gamma>0$, the aid obtains \[ d_{\mathrm{TV}}(\mathcal L(W_A),\operatorname{Pois}(t)) =O\left(p^{\beta/(\beta+1)}+p^\gamma\right). \] This is conditional gap optimization, not a consequence of an unspecified polynomial mixing rate alone. In particular decay for a restricted class of observables need not give the displayed target-uniform far-field bound.

The aid gives a fair Bernoulli-shift obstruction using periodic cylinders, for example $A_\ell$ requiring $\ell$ consecutive zero coordinates. Then $p=2^{-\ell}$ and $\mu(A_\ell\cap T^{-1}A_\ell)=p/2$, so overlapping visits cluster despite independent underlying coordinates and perfect separation beyond the cylinder length. It reports that this family fails the ordinary Poisson limit. Thus strong long-range mixing does not replace short-return control. The record does not identify a concrete slowly mixing system satisfying all estimates, determine a sharp universal rate, or recover the source's ambiguous numerical example.

Short English statement

The source asks for Poisson approximation rates in slowly mixing systems. The supplied aid gives a conditional total-variation bound separating neighborhood size, short-return clustering, and far-field dependence, with an optimized exponent and a Bernoulli-cylinder example showing why mixing alone is insufficient.

Sources

• Local UnsolvedMath registry, permanent ID $20001394$, source code catalogue entry; aim-dynamical-systems-notes.json, notes[51], tag problem.
• AIM workshop "Stochastic methods for non-equilibrium dynamical systems," Problem 9: https://aimath.org/pastworkshops/noneqdynsysproblems.pdf. Statement status exact; the ambiguous "1/n 4 to 1/n" example is retained as unresolved source notation. No PDF was accessed for this draft.
• Arratia--Goldstein--Gordon dependency-neighborhood method, named in the aid without a complete title, date, theorem number, or URL. No original remarks, literature field, or authoritative proof artifact for the candidate bound is supplied.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor label proved, novelty confidence low. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.