Asymptotic Theory in Probability and Statistics

Asymptotic theory examines the limiting behaviour of probabilistic and statistical quantities as the sample size or problem dimension grows without bound. Key pillars include the law of large numbers, which guarantees convergence of sample averages to true expectations, and the central limit theorem, which establishes that suitably normalised sums converge in distribution to the Gaussian law. Beyond these classical results, modern asymptotic theory addresses rates of convergence and refinements via Berry–Esseen bounds and Edgeworth expansions, allowing precise control of approximation errors. Developments in moderate and large deviation principles quantify the exponential decay of tail probabilities, while self-normalised and Studentised statistics provide pivotal inference without external scale parameters. High-dimensional and non-identically distributed settings have spurred new techniques to control asymptotic normality of estimators such as maximum likelihood estimators and U-statistics under complex dependence structures. The interplay between theoretical advances and computational practice has expanded applications in machine learning, signal processing, finance and the biological sciences, underscoring the global significance of asymptotic methods for robust inference and risk assessment.

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Asymptotic Theory in Probability and Statistics publication trend

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Technical terms

Asymptotic normality: Convergence in distribution of a sequence of estimators or test statistics to a normal law as sample size increases.

Berry–Esseen bound: A quantitative bound on the error in approximating a distribution by the normal law, typically of order O(n⁻¹/²).

Edgeworth expansion: A series expansion that refines the central limit theorem by adding correction terms to the normal approximation.

Self-normalised sum: A sum of random variables divided by a data-dependent scale, often their own sample standard deviation.

Studentisation: Adjustment of a statistic by an estimate of its standard deviation to achieve a pivotal limit distribution.

Delta method: A technique for deriving the asymptotic distribution of smooth functions of estimators.

Moderate deviation: A regime between the central limit theorem and large deviations, describing probabilities of deviations shrinking slower than n¹/² but faster than constant order.

References

1. Self-normalized limit theorems: A survey. Probability Surveys (2013).
2. A weak Cramér condition and application to Edgeworth expansions. Electronic Journal of Probability (2017).
3. Another Look at Stein’s Method for Studentized Nonlinear Statistics with an Application to U-Statistics. Journal of Theoretical Probability (2024).
4. Assessing the multivariate normal approximation of the maximum likelihood estimator from high-dimensional, heterogeneous data. Electronic Journal of Statistics (2018).