Assume $X$ is an absolutely continuous random variable with pdf $f:\mathbb{R}\to[0,\infty)$. Assume further there exists $M>0$ s.t. $|f(t)|\leq M \quad\forall t\in\mathbb{R}$.
Let $X_1,\dots,X_n$ be $n$ i.i.d copies of $X$, and let $x=(x_1,\dots,x_n)\in\mathbb{S}^{n-1}$ where $S^{n-1}\subseteq\mathbb{R}^n$ is the unit sphere, consider the random variable $$Y=\langle X,x\rangle=\sum_{j=1}^nx_jX_j$$
We know that $Y$ is also absolutely continuous with density $$f_y=g_1*\dots *g_n$$ where $$g_i(t)=\begin{cases}\frac{1}{|x_i|}f\left(\frac{t}{x_i}\right)& x_i\neq 0\\ \delta(t), &x_i=0 \end{cases}$$ I am pretty interested when $f_y$ is uniformly bounded, i.e. $\exists \tilde{M}$ s.t. $\Vert f_y\Vert\leq\tilde{M},\forall n\in\mathbb{N}$. This means $\tilde{M}$ should be independent of the choice of $n$.
A special case in which such an $\tilde{M}$ exists is that $X_i$ is a normal random variable. But I am curious if some other types of distribution also works. Thanks for any help.