According　to　in-depth　research,　a　wide　range　of　problems　in　applied　science　involve　estimating　the　probability　of　compound　stochastic　sums　of　heavy-tailed　risks　over　a　large　threshold.　Many　researchers　have　explored　this　issue　from　different　aspects　in　recent　times.　There　are　two　main　difficulties　here:　one　is　how　to　deal　with　the　heavy　tail　of　risk,　and　another　is　how　to　handle　the　dependence　of　the　aggregated　processes.　Aimed　at　these　two　main　problems,　we　investigate　the　asymptotic　properties　of　the　tail　of　compound　stochastic　sums　of　heavy-tailed　risks　in　a　general　dependence　framework,　and　some　approximate　bounds　and　key　characteristics　related　to　value-at-risk　are　also　derived.　Several　practical　examples　are　given　to　demonstrate　the　effectiveness　of　the　approximation　results.　Furthermore,　the　main　results　in　this　paper　can　be　applied　to　studies　of　stochastic　models　in　finance　and　econometrics　and　studies　of　dependent　netput　processes　of　the　M/G/1　queuing　systems,　etc.