definition complete convergence the sequence of random variables is said to converge completely to if for all [|xn xen cpl we denote the complete convergence as xn - complete convergence example (complete and almost sure convergencewe show that complete convergence implies almost sure convergence we can bound the criterion for almost sure convergence as followsp[sup |xk xep[ > {|xk xe} > [|xk xeby union bound in ( > | = [|xk xe{ cpl - = [|xk xecfrom xn -- cn- = [|xk xe- henceby definition xn - the next theorem shows how the different types of convergence are related to each > other for examplein the diagram belowthe notation means that -norm convergence implies lq -norm convergence under the assumption that theorem modes of convergence the most general relationships among the various modes of convergence for numerical random variables are shown on the following hierarchical diagramcpl xn - xn - lp > xn - xn - lq xn - xn -