I am a bit confused about the following statements:
The probability distribution of the sum of two or more "independent" random variables is the convolution of their individual distributions.
The probability mass function or probability density function of a sum of random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
Does this mean that for the second statement, statistical independence is not required anymore?
Independence is very essential in the discrete case also. The second statement is wrong.
Counterexample: let $X$ take values $0$ and $1$ with probability $\frac 1 2$ each and $Y=-X$. Then $P(X+Y=1)=0$ but the convolution assigns possitive probability for this.