If $B_1, ..., B_k$ are $k$ random variables that are dependent. And $R_1, ..., R_k$ are independent variables, such that the total variational distance satisfies $d(B_i, R_i)\leq \epsilon_i$ for all $i$. Can we say that
$$d((B_1, ..., B_k), (R_1, ..., R_k))\leq \epsilon_1 + ... + \epsilon_k$$
In particular, if $B_1, ..., B_k$ are independent, this this follows. However, my random variables are dependent.