This is called the Darmois-Skitovich theorem. Of course, one needs to add the
condition that  $a_jb_j\neq 0$.

The reference is
MR0346969 Kagan, A. M.; Linnik, Yu. V.; Rao, C. R., Characterization problems in mathematical statistics. Translated from the Russian by B. Ramachandran. John Wiley & Sons, New York-London-Sydney, 1973.