bibkey: “stone1967local” authors: “Charles Stone” year: 1967 title: “On local and ratio limit theorems” doi: “10.1525/9780520325340-017” url: “https://digitalassets.lib.berkeley.edu/math/ucb/text/math_s5_v2_p2_article-15.pdf” claim: “A normalized nondegenerate probability law with finite covariance satisfies a uniform local central limit theorem, in lattice, nonlattice and mixed cases; the nonlattice specialization concerns fixed-width interval probabilities and does not require a density.” strata_touched: [] license: “citation-only” triage: “anchor”

On local and ratio limit theorems

The primary source is Stone’s eight-page paper in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, volume II, part 2, Contributions to Probability Theory, University of California Press, 1967, pp. 217–224. Berkeley’s scanned copy supplies the text. Crossref identifies the chapter DOI above and the page range 217–224.

Theorem 1 and Corollary 1, printed p. 218, give the local limit theorem for a normalized law. The definitions on p. 217 distinguish its lattice and nonlattice coordinates through the set where the characteristic function has modulus one. In the one-dimensional nonlattice case, the normalization imposes no lattice rescaling. Corollary 1 specializes as follows: if independent, identically distributed real variables have mean $m$, variance $v\in (0,\infty )$ and a nonlattice law, then for every fixed $h>0$,

\sup_{x\in\mathbb R}
\left|
\sqrt n\,\mathbb P\{Y_1+\cdots+Y_n\in[x,x+h)\}
-\frac{h}{\sqrt{2\pi v}}
 \exp\left(-\frac{(x-nm)^2}{2nv}\right)
\right|\longrightarrow0.

The original formulation uses centered cubes; translating the interval gives the displayed version, with a uniformly negligible change in the Gaussian term. Endpoint conventions have the same limit: place any endpoint atom in an interval of fixed width, apply the uniform bound and then let that width shrink. The same argument shows that the largest atom is $o(n^{-1/2})$. Nonlattice does not mean absolutely continuous. A discrete law can satisfy this theorem.

Corollary 1 requires finite covariance. The paper introduces its “Cramér’s condition” only afterward, in (2.9), as existence of an exponential moment. Theorem 2 on p. 219 adds uniformity over compact sets of exponential tilts. Neither the corollary used here nor that terminology means the stronger condition that the characteristic function stay uniformly below one at infinity. No such condition is assumed for the two-jump compound-Poisson law. Stone attributes the nonlattice case of Theorem 1 to his 1965 paper, A local limit theorem for nonlattice multi-dimensional distribution functions, Annals of Mathematical Statistics 36, 546–551, DOI 10.1214/aoms/1177700165. The inspected primary text for the present use is the 1967 paper.

For the sparse parity model, exponential tilting at the fixed critical parameter gives a compound-Poisson process with fixed jump sizes $\log (1+r)$ and $\log (1-r)$. Its unit-time increment has positive mass at zero and at both jumps. Irrationality of their ratio makes its law nonlattice, and its variance is positive and finite. Decomposing a real time into its integer part and an independent remainder of length less than one reduces its interval estimate to the corollary. Uniformity in the location allows this bounded-time remainder to be integrated out.

The local limit theorem and exponential-tilt method are literature-attested. The compensated parity model’s actual-path transport, mixed rare-row counts, logarithmic correction and exact minimax recovery curve require the additional model-specific arguments in PARITY_HIDDEN_ARROW. Stone’s paper does not supply those statistical conclusions, a growing-row comparison, or an equivalence of complete experiments. Its nonlattice specialization does not justify using continuous tail prefactors at an arithmetic jump ratio.

Nonlattice capacity localization

The parity fluctuation volume’s Chapter 33 applies the same fixed-width local limit to both sides of a critical compensated-score threshold. Actual one-row and two-row comparisons put order q/sqrt(lambda) positions in either window for each fixed positive width. This exceeds the order sqrt(q) capacity correction. The width is sent to zero only after the sample limit. The argument therefore does not require a density or a relative local approximation in a window shrinking at the rank-fluctuation scale.

The resulting actual pair/path theorem has a linear Gaussian coarse loss but retains a smaller posterior-centered Gaussian variance mixture. Stone’s local limit and the conditional-Bernoulli tools used for that mixture are classical. The model-specific localization, exact finite minimax mean transfer and joint fixed-support coupling are repo-derived ordinary mathematics; the source does not state them. This comparison neither establishes global originality nor a finer universal expansion of the capacity correction’s expectation.

Chapter 33 also gives an actual-model obstruction to a shrinking-window relative formula. At fixed amplitude one half, a central integer Poisson-count pair has probability of order inverse lambda. The support-size and sample-size sequence keeps its compensated score inside a window of width sqrt(lambda/q); actual one-row comparison transfers the atom to both stationary experiments. Its mass exceeds the density-times-width scale q^(-1/2) by an unbounded factor. Stirling’s formula and the discrete nonlattice obstruction are classical; this construction verifies the obstruction with the model’s compensation, legal integer parameters and actual observation laws. It does not refute the fixed-width theorem or establish a finer mean-cost expansion.