x[i] =new Integer(x[i]+y[i]).byteValue(); will work

When 2 bytes are added, the addition returns an int. Just cast the
addition back to a byte.

x[i] = (byte)(x[i]+y[i]);

A "byte" variable in Java can store numbers of 8 bits,
if you add two bytes you can obtain a number of 9 bits.
As you can see, a "byte" can not store that number.

I didn't see the program you are doing, but in general,
a simple solution is to use a "short" (16 bits) to store
the addition.

But, there are some special cases (on compression
algorithms as an example) where you can substract 256
if the addition is greater than 256, and take a
similar decision on the decompression process, but
like I said, that will depend on what your algorithm
does.