Suppose I have the random variables $X_1, .., X_n$$X_1, \dots, X_n$ with the marginal distributions are not normal (in fact, unknown marginal distribution).
Will there be any difference between the assumption $X_1, ..., X_n$$X_1, \dots, X_n$ are independent and $X_1, ... X_n$$X_1, \dots, X_n$ are modeled with $t$Student-Student$t$ copula with the correlation matrix is all zeros (with 1 in$1$ on the diagonal) ?