Probabilistic Sums and Limit Theorems in Statistical Distributions

Probabilistic sums arise when aggregating random variables whose number of terms may itself be random, a scenario common in fields as diverse as insurance claims modelling, queueing theory and genomic sequence analysis. Classical limit theorems such as the central limit theorem and law of large numbers describe the convergence of normalized sums of independent identically distributed variables to stable laws under growing sample sizes. Extensions to random sums—where the count of summands follows distributions such as geometric, negative binomial or Poisson—have revealed a rich tapestry of limit behaviours, including compound Poisson and scale-mixture limits. Modern advances explore refinements of normal and non-normal approximations, bounds on convergence rates, and multivariate generalisations involving mixtures of stable or elliptical laws. These developments underpin practical applications in extreme-value analysis, long-term forecasting, anomaly detection and likelihood-based classification, demonstrating the global significance of probabilistic sums and their asymptotic properties.

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Probabilistic Sums and Limit Theorems in Statistical Distributions publication trend

The graph below shows the total number of articles in probabilistic sums and limit theorems in statistical distributions across all publications each year (not limited to Nature Index journals).

Technical terms

Random sum: A sum in which the number of terms is a random variable, modelling situations with stochastic sample sizes.

Limit theorem: A mathematical result that describes the convergence in distribution of a sequence of random variables under suitable normalisation.

Equilibrium transform: An operator that maps a distribution to its renewal or size-biased form, used to assess approximation accuracy.

Total variation distance: A metric quantifying the maximal difference between probabilities assigned by two distributions.

Stein’s method: A technique for deriving quantitative bounds on the distance between probability distributions and their approximations.

References

1. Convergence in Total Variation of Random Sums. Mathematics (2021).
2. A Generalized Equilibrium Transform with Application to Error Bounds in the Rényi Theorem with No Support Constraints. Mathematics (2020).
3. Multivariate Scale-Mixed Stable Distributions and Related Limit Theorems. Mathematics (2020).
4. A Rényi-Type Limit Theorem on Random Sums and the Accuracy of Likelihood-Based Classification of Random Sequences with Application to Genomics. Mathematics (2023).