A general tool to prove conditional strong laws of larger number is considered. It is shown that a conditional Kolmogorov type inequality implies a conditional Hajek–Renyi type inequality and this implies a strong law of large numbers. Both probability and moment inequalities are considered. Some applications are offered in the last section.

We introduce a Susceptible-Corrupted-Paying Well-Recovered (SCPwR) model specifically designed to analyze corruption dynamics within the public sector. This model is demonstrated to be well posed both epidemiologically and mathematically. Our results show that all model solutions remain positive and bounded given initial conditions within a meaningful set. We investigate the existence of unique corruption-free and endemic equilibrium points and calculate the basic reproduction number. The local and global stability of these equilibrium points is then analyzed. Our findings indicate that the system has a locally asymptotically stable corruption-free equilibrium when R_e < 1 and unstable when R_e > 1 . Additionally, the corruption endemic equilibrium point E^* is globally asymptotically stable only if dL/dt< 0 . Numerical implementation of the model suggests that corruption will persist in public sectors if civil servants are not adequately compensated.

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.

We introduce the notions of sub Gaussian random variables in sub-linear expectation spaces. To avoid the problem caused by the existence of two different expectations, i.e., the upper expectation and the lower expectation, we divide the definition of the sub-Gaussian property into an upper part and a lower part. It turns out that this approach fits well to the sub-linear setting; it provides a proper framework for extending Zajkowski's general result to sublinear expectation spaces. Within our framework, we establish a strong law of large numbers for sub-Gaussian sequences. We present an example showing the usefulness of our results.