For a matrix $A$ and sequence of its singular values $s(A)=(s_1(a), \ldots, s_n(A))$ the Shatten norm is defined as
$$ \|A\|_{S_p}=\|s(A)\|_p, \quad 1\leq p<\infty. $$ Denote by $\bar r=(r_1, \ldots, r_N)$ random variables such that $Proba(r_i)=1=Proba(r_j)=-1=1/2$ and with condition that $r_1+\ldots r_N=B$.
Find lower bound of $$ E\|\sum_{i=1}^Nr_iA_i\|^p_{S_p}\geq ? $$
Note: I would expect that $E\|\sum_{i=1}^Nr_iA_i\|^p_{S_p}\geq C(p, B) min \{ \|(\sum_{i=1}^NA^*_iA_i)^{1/2}\|^p_{S_p}, \|\sum_{i=1}^NA_iA^*_i)^{1/2}\|^p_{S_p}\}$, but not sure how to work with condition on random variables.