Assume $X_i,Y_i,Z_i,i=1,2,3,...,m$ are normal random variables with the mean of zero and different variances. also $T_i=X_i+Y_iZ_i,i=1,2,3,...,m$ where $T_i$ are probably correlated. Then I would like to know
$$ E(max_{i=1,2,3,...,m}|X_i+Y_iZ_i|)\overset{?}{\leq} E(max_{i=1,2,3,...,m}|X_i|)+E(max_{i=1,2,3,...,m}|Y_iZ_i|) $$
Thanks.
we can prove it in this way: $$ E(max_{i=1,2,3,...,m}|X_i+Y_iZ_i|) =\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}max_{i=1,2,3,...,m}|X_i+Y_iZ_i|f(x,y,z)dxdydz\\ \leq \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}max_{i=1,2,3,...,m}|X_i|f(x)dx+ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}max_{i=1,2,3,...,m}|Y_iZ_i|f(y,z)dydz\\ = E(max_{i=1,2,3,...,m}|X_i|)+E(max_{i=1,2,3,...,m}|Y_iZ_i|)\\ $$