On the Sum of Symmetric Random Variables

Let U and V be two symmetric (about zero) random variables with U + V symmetric about C; here C is a constant.It is easy to see that if U and V are mutually independent, or if both U and V satisfy the weak law of large numbers, then C = 0. So, intuitively, we would suspect that C = 0 in general.However, we show that there exist two random variables U and V symmetric about 0 with U + V symmetric about C . 0.The example given is closely related to one given by Alejandro D. De Acosta in another context.