In　this　paper,　we　consider　the　discrete-time　constrained　average　stochastic　games　with　independent　state　processes.　The　state　space　of　each　player　is　denumerable　and　one-stage　cost　functions　can　be　unbounded.　In　these　game　models,　each　player　chooses　an　action　each　time　which　influences　the　transition　probability　of　a　Markov　chain　controlled　only　by　this　player.　Moreover,　each　player　needs　to　pay　some　costs　which　depend　on　the　actions　of　all　the　players.　First,　we　give　an　existence　condition　of　stationary　constrained　Nash　equilibria　based　on　the　technique　of　average　occupation　measures　and　the　best　response　linear　program.　Then,　combining　the　best　response　linear　program　and　duality　program,　we　present　a　non-convex　mathematic　program　and　prove　that　each　stationary　Nash　equilibrium　is　a　global　minimizer　of　this　mathematic　program.　Finally,　a　controlled　wireless　network　is　presented　to　illustrate　our　main　results.　©　2019　by　the　authors.