Serving as an essential tool in the development of statistical methodologies, probability inequalities play a critical role in statistics and have a plethora of applications (see, for example, Gumbel (1958), Galambos (1978, 1995), Hoppe (1993)). The elegance of probability inequalities stems from their generality (no special model assumptions), applicability (adaptable to various fields), and reliability (conservative in controlling error rate). Investigation of probability inequalities has always been a primary research area in statistics (Wong and Shen (1995), Block, Costigan, and Sampson (1997)). The discovery of a novel probability inequality correspondingly results in refinements of the associated statistical methodologies, advancing the field of statistical inference. Today, the Bonferroni procedure directly affects many applied fields such as analyzing extreme values in actuarial science (Galambos (1984, 1987, 1994)), establishing bounds for convergence rates of random sequences in limiting theory (Seneta and Weber (1982), Hoppe (2006, 2009)), comparing drug effects for two or more treatments in medical research (Seneta (1993), Chen, Walsh, Comerota, et al (2011), Chen and Comerota (2012)), evaluating business markets in financial analysis (Chen (2010), Gupta, Chen and Troskie (2003)), and assessing system reliabilities in engineering (Castillo (1988), Guerriero, Pozdnyakov, Glaz, et al. (2010), Block, Costigan, and Sampson (1992)), to list just a few.