This chapter goes beyond bounding results on Bonferroni-type bivariate lower bounds. It extends the previous chapter in two directions. The first one is in the direction of multivariate lower bounds that focus on high dimension settings with any p sets of events. Literature in this regard includes the work of Chen and Seneta (1996, 1988, 2000), and Galambos and Lee (1992, 1994), among others. The second direction of extension improves bivariate lower bounds by using information beyond the Bonferroni summations. In particular, we concentrate on product-type lower bounds and lower bounds using information on Hamilton circuits in Sections 8.2 and 8.3.