Strong Laws of Large Numbers for General Random Variables Under Conditional Sub-Additive Expectation and Capacity

We study strong laws of large numbers in a non-linear framework based on conditional sub-additive expectations and conditional sub-additive capacities. Using an axiomatic approach to conditional sub-additive expectation, we establish a conditional Hájek–Rényi-type maximal inequality assuming a general conditional Kolmogorov-type maximal inequality but without imposing any weak dependence structure on the underlying sequence. As a consequence, we derive a general conditional strong law of large numbers. Finally, we introduce a notion of conditional negative dependence under sub-additive expectations and obtain the corresponding conditional Kolmogorov-type maximal inequality, leading to a conditional strong law of large numbers for conditionally negatively dependent random variables.