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Assume that we have two independent random variables $X_1$ and $X_2$ with distribution functions $F_1$ and $F_2$ respectively. Then Theorem 6.1.1. on Kai Lai Chung's "A course in probability theory" states that:

The sum $X_1+X_2$ has distribution function $F_1*F_2$.

On this set of notes the author states that actually the distribution of the sum is NOT the convolution of the distribution functions, but rather the convolution of the $F_1$ and the density $f_2$.

For what know I about the definition of convolution

$\int F_1(x-x_1) dF_2(x_2)$ looks more like the convolution $F_1*f_2$, I think that this is more a problem of notation but still I am curious.

EDIT: Another example on which the convolution is taken between the distribution functions is "Probability Theory and examples" by Rick Durret.

Chaos
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    Yes, the density is a convolution of densities, not of CDFs. Indeed the CDF would be $$P[X_1+X_2\leq x] = \int_{y=-\infty}^{\infty} P[X_1 + y\leq x|X_2=y]f_{X_2}(y)dy = \int_{-\infty}^{\infty} \underbrace{P[X_1\leq x-y]}{F{X_1}(x-y)}f_{X_2}(y)dy$$ where independence is used to claim $P[X_1+y\leq x|X_2=y]=P[X_1+y\leq x]$. Indeed this is a convolution of $F_1$ and $f_2$, not of $F_1$ and $F_2$. – Michael Nov 29 '19 at 16:36
  • @Michael I don't really understand why I've seen this $F_1*F_2$ in so many good books. – Chaos Nov 29 '19 at 16:41
  • I don't understand why the authors would want to write that, either. I don't think I have ever seen that (or ever noticed that). If I did, I would assume it to be a typo. – Michael Nov 29 '19 at 16:41

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This question hinges on the definition of "convolution" and of the $*$ symbol. A glance at the wikipedia article gives one formulation for the convolution of two probability measures: it is the probability measure of the sum of two independent random variables, equivalent to $(\mu*\nu)(A)=P(X+Y\in A)$ if $X\sim\mu$ and $Y\sim\nu$. From this one has $$F_{X+Y}(t)=P(X+Y\le t)=E[(P(X+Y\le t|Y)]=E[ P(X\le t-Y|Y)] = E[ F_X(t-Y)|Y] = \int_{\mathbb{R}}F_X(t-y) F_Y(dy) = \int_{\mathbb{R}}F_X(t-y) dF_Y(y),$$ which one might as well take to be the definition of the convolution of distributions functions of probability measures, regardless of whether the random variables $X$ and $Y$ are continuous or discrete, with densities or without.

The notes the OP cites are not reliable on this issue: they adopt the dumbing-down restriction to probability laws with density functions. I do not know what Ross says, but I think Chung (the most trustworthy of the lot) does not make mistakes like this.

kimchi lover
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