SupposethatEandFareseparableBanachspaces,XandYareindependentsymmetricEandF-valuedrandomvectorsrespectively.ThispaperisdevotedtothestudyofthecentrallimittheoremforX(○×)YintheinjoctiveandprojectivetensorproductspacesE(∨○×)FandE(∧○×)F.Specialattentionispaidtoι2(∧○×)ι2.Inaddition,twocounter-examplesaregiven.