Moment conditions in strong laws of large numbers for multiple sums and random measures

The validity of the strong law of large numbers for multiple sums $S_n$ of independent identically distributed random variables $Z_k$, $k\leq n$, with $r$-dimensional indices is equivalent to the integrability of $|Z|(\log^+|Z|)^{r-1}$, where $Z$ is the typical summand. We consider the strong law of large numbers for more general normalisations, without assuming that the summands $Z_k$ are identically distributed, and prove a multiple sum generalisation of the Brunk--Prohorov strong law of large numbers. In the case of identical finite moments of irder $2q$ with integer $q\geq1$, we show that the strong law of large numbers holds with the normalisation $\|n_1\cdots n_r\|^{1/2}(\log n_1\cdots\log n_r)^{1/(2q)+\varepsilon}$ for any $\varepsilon>0$. The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.Comment: 15 page

Strong Laws of Large Numbers for 𝔹

We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) for -valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors

Convergence rates in the SLLN for some classes of dependent random fields

AbstractLet {Xn,n∈Nr} be a random field i.e. a family of random variables indexed by Nr, r⩾2. We discuss complete convergence and convergence rates under assumption on dependence structure of random fields in the case of nonidentical distributions. Results are obtained for negatively associated random fields, ρ⁎-mixing random fields (having maximal coefficient of correlation strictly smaller then 1) and martingale random fields

Quadrature-Inspired Generalized Choquet Integral in an Application to Classification Problems

Correct classification remains a challenge for researchers and practitioners developing algorithms. Even a minor enhancement in classification quality, for instance, can significantly boost the effectiveness of detecting conditions or anomalies in safety data. One solution to this challenge involves aggregating classification results. This process can be executed effectively as long as the aggregation function is appropriately chosen. One of the most efficient aggregation operators is the Choquet integral. Furthermore, there exist numerous generalizations and extensions of the Choquet integral in the existing literature. In this study, we conduct a comprehensive analysis and evaluation of a novel approach for deriving an aggregate classification. The aggregation process applied to various classifiers is based on enhancements to the Choquet integral. These novel expressions draw inspiration from Newton-Cotes quadratures and other well-known formulae from numerical analysis. In contrast to previous approaches that exploit the generalization of the Choquet integral, our approach requires the utilization of two or three adjacent values associated with the membership of a specific element in different classes. This enables the use of more efficient enhancements in terms of accuracy measurement. Specifically, the t-norm following the integral symbol can be effectively replaced by mathematical expressions used in executing numerical integration formulae. This leads to more precise results and aligns with the concept of numerical integration. Furthermore, in a series of experiments, we thoroughly assess the performance of the proposed approach in terms of classification accuracy. We analyze the strengths and weaknesses of the new approach and establish the experimental settings that can be applied to similar tasks. In the series of experiments, we have demonstrated that the proposed Quadrature-Inspired Generalized Choquet Integral (QIGCI) can either outperform previous enhancements of the Choquet integral or at least achieve a similar level of accuracy measurement. However, we also highlight scenarios where previous approaches can still be a suitable choice. The number of QIGCI-based aggregation models that outperform others is convincing, indicating that this approach is worthy of consideration