In　determining　structural　responses　under　random　excitation　using　the　Monte　Carlo　method,　simulation　of　non-Gaussian　processes　is　always　a　challenge.　Various　methods　have　been　developed　to　simulate　non-Gaussian　processes　but　literature　suggests　that　the　complete　expression　of　the　Gaussian　correlation　function　matrix　(CFM)　and　the　corresponding　applicable　range　of　the　target　non-Gaussian　CFM　have　not　been　attempted　in　the　transformation　based　on　the　Hermite　polynomial　model　(HPM)　to　date.　The　intention　of　this　paper　is　to　derive　a　complete　transformation　model　of　CFM　from　non-Gaussian　to　Gaussian　processes　with　the　applicable　range　of　the　target　non-Gaussian　CFM　based　on　the　unified　Hermite　polynomial　model　(UHPM).　An　efficient　procedure　for　simulating　stationary　non-Gaussian　processes　is　also　presented　for　easy　application.　It　is　found　in　the　paper　that　the　complete　transformation　model　of　Gaussian　CFM　is　necessary　because　HPMs　are　not　always　monotonic　and　that　the　proposed　method　can　simulate　stationary　non-Gaussian　processes　efficiently　and　accurately.　The　proposed　method　can　equip　researchers　and　engineers　with　a　more　efficient　and　accurate　tool　to　handle　non-Gaussian　processes　frequently　encountered　in　engineering　practices.