Conditional and Quantitative Strong Laws of Large Numbers

Abstract

This dissertation investigates advanced generalizations of the Strong Law of Large Numbers (SLLN) within conditional, multi-indexed, and nonlinear probabilistic frameworks. The first part establishes a general conditional SLLN by proving that conditional Kolmogorov-type maximal inequalities imply conditional Hájek–Rényi inequalities, which in turn yield almost sure convergence of normalized partial sums. The second part develops quantitative SLLNs for double-indexed random variables, deriving explicit probability bounds and convergence rates for pairwise independent and quasi-uncorrelated arrays. The third part extends the theory to conditional sub-additive expectations and capacities, where strong laws are formulated in terms of quasi-sure convergence in non-additive probability spaces. Further results are obtained for φ-sub-Gaussian random variables under sublinear expectations, showing that exponential tail control is sufficient for strong convergence without classical moment assumptions.