Statistical Convergence of Random Variables

Statistical convergence of random variables provides a rigorous framework to describe how sequences of random quantities approach limiting behaviour in probability, in distribution and in more refined senses. Classical convergence modes include almost sure convergence, where outcomes align pointwise in the long run; convergence in probability, which quantifies the shrinking probability of large deviations; and convergence in distribution, capturing the stabilisation of cumulative distribution functions. Beyond these, complete convergence and complete moment convergence impose summability conditions on tail probabilities and moments, ensuring stronger control over convergence rates. Recent years have witnessed advances in extending these concepts to dependent structures, such as negatively associated or mixing sequences, and to non-additive probability frameworks. Such developments underpin applications as diverse as risk assessment under model uncertainty, non-parametric regression with dependent errors, and the analysis of stochastic processes in finance and machine learning.

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Research from all publishers

Emerging work in sublinear expectation theory has redefined central limit theorems under model uncertainty by constructing G-normal distributions. In this setting, a nonlinear central limit theorem generalises Kolmogorov’s classical result, yielding robust limit laws when probability measures themselves are uncertain. Complementary research has derived precise asymptotics for complete convergence under sublinear expectations, identifying exact rates in terms of boundary and weight functions and establishing integral conditions that guarantee almost sure convergence. A further advance has developed a general form of such precise asymptotics, unifying convergence rate, weighted functions and limit values within the sublinear framework. Parallel contributions have extended invariance principles and central limit theorems to linear processes under this new notion of independence, demonstrating that long-range dependent structures also admit analogous statistical convergence results when expectation is non-additive.

Statistical Convergence of Random Variables publication trend

The graph below shows the total number of articles in statistical convergence of random variables across all publications each year (not limited to Nature Index journals).

Technical terms

Almost sure convergence: A sequence converges almost surely if the probability that it eventually stays arbitrarily close to its limit is one.

Convergence in probability: A sequence converges in probability if the chance that its deviation from the limit exceeds any positive threshold tends to zero.

Convergence in distribution: A sequence converges in distribution if its cumulative distribution functions converge pointwise to that of the limit variable at continuity points.

Complete convergence: A sequence satisfies complete convergence if the sum of tail probabilities with scaled deviations is finite, ensuring rapid decay of large deviations.

Sublinear expectation: A functional generalising classical expectation that is monotonic, positively homogeneous and subadditive, allowing for model uncertainty.

G-normal distribution: The limit law emerging in the sublinear central limit theorem, characterised by a nonlinear variance operator reflecting distributional uncertainty.

References

1. Complete convergence and complete moment convergence for negatively associated sequences of random variables. Journal of Inequalities and Applications (2016).
2. On complete convergence and complete moment convergence for weighted sums of ρ∗-mixing random variables. Journal of Inequalities and Applications (2018).
3. Bernstein‐Type Inequality for Widely Dependent Sequence and Its Application to Nonparametric Regression Models. Abstract and Applied Analysis (2013).
4. Law of large numbers and central limit theorem under nonlinear expectations. Probability, Uncertainty and Quantitative Risk (2019).
5. Precise Asymptotics for Complete Integral Convergence under Sublinear Expectations. Mathematical Problems in Engineering (2020).
6. A general form for precise asymptotics for complete convergence under sublinear expectation. AIMS Mathematics (2022).
7. Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation. Applied Mathematics-A Journal of Chinese Universities (2021).