Statistical Inference for the Logistic Limit Law of Random Sums and Statistics Constructed from Samples with Random Sizes

Korolev (2024) proved necessary and sufficient conditions under which sums of a random number of independent random variables, and asymptotically normal statistics computed from samples whose size is itself random, converge to the logistic distribution, or to a newly introduced asymmetric logistic family defined as normal variance-mean mixtures whose mixing law is the distribution of $4D^2$ with $D$ Kolmogorov-distributed; the mixing is transferred from the asymptotic distribution of the random sample size. The paper observes in Remark 1 that the mixing law is in principle recoverable from data, since normal scale mixtures are identifiable, but develops no estimator, consistency theory, inference procedure, robustness analysis, or model-selection methodology. This problem asks for that missing inferential layer: consistent estimation of the mixing law (equivalently, of the limit law of the normalized random sample size) and of the parameters of the asymmetric logistic family, with convergence rates and asymptotic distributions, in both the idealized model of exact draws from the limit mixture and the pre-limit random-sample-size model; a characterization of what replaces the logistic limit when the Lindeberg condition, independence of summands, or divergence of the random index in probability fails; and goodness-of-fit and discrimination tools separating logistic limits from the normal law and other normal-mixture limits.

Background

In many data-collection settings the number of observations available in a given period is not fixed in advance: insurance claim counts, trade arrivals in high-frequency markets, or events registered until a stopping time. When a statistic is computed from such a sample, its size $N_n$ is itself random, and classical asymptotic theory, which treats the sample size as a deterministic parameter tending to infinity, no longer applies directly. A random sum is $S_{N_n}=X_1+\dots+X_{N_n}$, where the number of summed terms is random; more generally one considers $T_{N_n}$, a statistic built from the first $N_n$ observations of a sequence.

A convenient way to describe the asymptotics of such objects is via mixing. If $N_n\to\infty$ in probability, the summands $X_i$ are independent (not necessarily identically distributed), have zero means and finite variances, and satisfy the Lindeberg condition (the standard central-limit-theorem hypothesis excluding dominant heavy tails: no single summand contributes a non-negligible share of the total variance $\sigma_1^2+\dots+\sigma_n^2$), and the index is independent of the summands, then the normalized random sum converges if and only if the squared total variance $B_{N_n}^2$ (random through $N_n$), normalized by a deterministic scale $d_n^2$, converges in distribution to some nonnegative random variable $U$; the limit law is then the normal scale mixture $\mathsf{E}\,\Phi(xU^{-1/2})$, obtained by averaging the standard normal distribution function $\Phi$ over the random variance $U$. Normal scale mixtures are identifiable: distinct mixing laws give distinct mixtures (Teicher, 1961). These transfer theorems for random sums and statistics with random indices go back to Korolev (1994–1995); a published modern account is Korolev and Zeifman (2016). The same mechanism applies to asymptotically normal statistics: if $\delta\sqrt{n}(T_n-\theta)$ is asymptotically standard normal for deterministic $n$, then $\delta\sqrt{n}(T_{N_n}-\theta)$ converges if and only if the normalized index $N_n/n$ converges, and the limit is again a normal scale mixture, with the index limit acting as the reciprocal mixing variance.

The logistic distribution, with distribution function $F(x)=1/(1+e^{-x})$, enters through a representation found by Stefanski (1991):

$$F(x)=\int_0^\infty \Phi\!\left(\frac{x}{2z}\right)\,dK(z),$$

where $K$ is the Kolmogorov distribution function — the classical limit law of the supremum of the empirical process, written in the source's parametrization as $K(x)=1-2\sum_{j\ge1}(-1)^{j+1}e^{-j^2x^2}$ for $x\ge0$. In words, a logistic random variable is distributed as $2D\cdot X$, the product of an independent standard normal $X$ and twice a Kolmogorov-distributed factor $D$. Earlier, Gnedenko and Gnedenko (1982) had shown that the logistic law arises as a limit for extreme order statistics in samples whose size has a geometric distribution. Building on both inputs, Korolev (2024) proved sharp necessary-and-sufficient versions for additive statistics: under the Lindeberg and $N_n\to\infty$ hypotheses, the normalized random sum $S_{N_n}/(\sigma\sqrt{n})$ converges to the logistic law exactly when the normalized index converges to $4D^2$; an asymptotically normal statistic computed from a random-size sample converges to the logistic law exactly when $N_n/n$ converges to $\tfrac14 D^{-2}$; and, in a double-array scheme, random sums converge to a new asymmetric generalization

$$F_{a,b}(x)=\int_0^\infty \Phi\!\left(\frac{x-az}{b\sqrt{z}}\right)\,dK^\ast(z),\qquad K^\ast(z)=\mathsf{P}(4D^2<z),$$

exactly when the normalized index converges to $K^\ast$. Here $F_{a,b}$ is a normal variance–mean mixture (the distribution of $2bD\cdot X+4aD^2$; the general framework of variance–mean mixtures is due to Barndorff-Nielsen, Kent and Sørensen, 1982). A natural index sequence realizing the logistic case is $N_n=P(4nD^2)$, a Cox process — a Poisson process whose intensity is itself random — which models counting with a stochastic, environment-driven rate, as is common in insurance and finance.

What the source paper explicitly leaves out is the statistical side. Its Remark 1 observes that, because normal scale mixtures are identifiable, the mixing law $U$ can in principle be reconstructed from the mixture distribution by solving an integral equation, or approximately from a homogeneous sample using techniques of statistical separation of mixtures — but no estimator, consistency theorem, rate result, or inference procedure is developed; there is no analysis of what happens when the Lindeberg condition fails (heavy-tailed summands), when the summands are dependent, or when the index does not tend to infinity in probability; and no goodness-of-fit or model-comparison methodology is provided for the logistic family against the normal law or other normal mixtures. The surrounding statistical literature exists only in fragments. Nonparametric maximum likelihood estimation of mixing distributions dates to Kiefer and Wolfowitz (1956), with modern convex-optimization implementations by Koenker and Mizera (2014). Deconvolution theory shows that recovering a law mixed with normal noise is statistically slow: Fan (1991) proved that optimal rates are only logarithmic in the sample size. For random sums with mixed Poisson and fractional-Poisson indices, parametric estimation of the resulting mixture limits has been developed (Oliveira, Barreto-Souza and Silva, 2020 and 2021), and limit distributions of parametric estimators under random sample sizes have been derived (Kudryavtsev and Shestakov, 2023). None of this addresses the logistic model specifically — the Kolmogorov mixing law $K^\ast$, the asymmetric family $F_{a,b}$, nonparametric recovery of an arbitrary mixing law $U$, or the pre-limit setting in which each observation is a finite-$n$ statistic computed from a sample whose size is random.

Problem Statement

Develop the statistical inference theory for the logistic limit model of random sums and statistics with random sample sizes established by Korolev (2024), with one coherent objective: make the mixing law — equivalently, the limit law of the normalized random sample size — an estimable and testable object from data. Throughout, $\Phi$ is the standard normal distribution function; $D$ denotes a random variable with the Kolmogorov distribution, that is, distribution function $K(x)=1-2\sum_{j\ge1}(-1)^{j+1}e^{-j^2x^2}$ for $x\ge0$, the classical limit law of the supremum of the empirical process; $K^\ast(z)=\mathsf{P}(4D^2<z)=K(\tfrac12\sqrt{z}\,)$ is the distribution function of $4D^2$; and $F_{a,b}(x)=\int_0^\infty \Phi\!\left(\frac{x-az}{b\sqrt{z}}\right)\,dK^\ast(z)$ with $a\in\mathbb{R}$, $b>0$ is the asymmetric logistic family of Korolev (2024), which reduces to the logistic distribution $F(x)=1/(1+e^{-x})$ when $a=0$ and $b=1$.

The problem comprises three linked deliverables that jointly constitute this methodology.

(1) Estimation and inference for the mixing law. Given observations $Y_1,\dots,Y_m$ assumed to be drawn — exactly, or as finite-$n$ pre-limit values of random-size statistics $T_{N_n}$ — from the normal scale mixture $\mathsf{E}\,\Phi(xU^{-1/2})$ with unknown mixing law $U$ (in the logistic case $U\stackrel{d}{=}4D^2$ with distribution function $K^\ast$), construct statistically justified procedures: nonparametric estimators of the mixing law $U$ (equivalently of $K^\ast$), or estimators of the parameters $(a,b)$ of the asymmetric logistic family $F_{a,b}$, together with consistency statements, convergence rates, and asymptotic distributions. The analysis must state explicitly which observation model is assumed — exact draws from the limit mixture, or pre-limit data where each observation is a statistic computed from a sample whose size is random — and, in the latter case, control the interplay between the transfer-theorem approximation error and the statistical estimation error.

(2) Validity domain of the limit laws. Characterize what replaces the logistic limit when the hypotheses of the source theorems fail: summands with heavy tails violating the Lindeberg condition, dependent summands, or index sequences that do not tend to infinity in probability. Identify the limit classes that arise and the boundary conditions under which the Kolmogorov-type mixing ($4D^2$, or $K^\ast$ in the double-array scheme) still emerges.

(3) Model selection and diagnostics. Provide goodness-of-fit tests and discrimination procedures for data with random sample sizes that separate the logistic law, the normal law (degenerate mixing), and neighboring normal scale / variance–mean mixtures arising as random-sum limits (for example Student-type or generalized-hyperbolic-type laws), with quantified asymptotic approximation quality.

References

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