A density and the average of i.i.d. random variables following that density

Question

Suppose $X_1,\ldots,X_n$ are i.i.d. continuous random variables with a common density $g$ and support $G=\{x:g(x)>0\}$. Let $$ Y=\frac{1}{n}\sum_{i=1}^nX_i. $$ It seems intuitive that $Y$ should have some density $f$ but how I prove that some such $f$ exists please? And does $f$ have the same support $G$ as $g$?

Proofs of course are welcomed but you don't have to give proofs if you can point me to references that contain the proofs. I'm particularly interested in books (easier to read than articles) that discuss the relationship between $g$ and $f$ (if the latter exists.) Thank you!

Answer 1

Density of a sum is convolution of densities $$f_{X_1+X_2}(z)=\int f_{X_1}(y)f_{X_2}(z-y)dy$$ To see why support of $Y$ doesn't have to be $G$, consider $X_i=Binomial(1,1/2)+Uniform(-\epsilon,\epsilon)$ for $\epsilon<1/3$. Then $Y$ has non-zero density on the neighborhood of $1/2$ for even $n$ while density of $X$ is zero on that neighborhood. In general, the union of supports of all $Y_n$ is convex hull of $G$.

Answer 2

We have $\{X_i\}_{i\in\Bbb N^+}$ as a sequence of iid random variables with probability density function $g$ over support $\Bbb G=\{x\mid x>0\}$.   That is: $\Bbb G=\Bbb R^+$

Let $Y_n \mathop{:=} \dfrac 1 n\sum\limits_{i=1}^n X_i$, and (assuming it exists) denote its density function as $f_n$.

The support of such a density function would also be $\Bbb R^+$, because any series of positive real numbers has a positive real value, and the quotient, of a positive real number and a positive integer, is a positive real number.

Now, clearly $f_1(y)=g(y)$.

By convolution we would find: $f_2(y) = \int\limits_0^{2y} g(x)\,g(2y-x)\operatorname d x$

And similarly we have the recursion: $f_n(y) = \int\limits_0^{ny} g(x)\, f_{n-1}(\frac{ny-x}{n-1})\operatorname d x$

So the existence of such probability density functions could be established by induction.