Autoconvolution-type Integral

Question

I have an integral of the type \begin{align} \int_{\mathbb{R}^d} f(y)f(y-x_1)f(y-x_2)\cdots f(y-x_n) \text{d}y \end{align} where $f:\mathbb{R}^d\to \mathbb{R}$ is a function such that $\sup_{x\in\mathbb{R}^d}\frac{|f(x)|}{1+|x|^{d+\varepsilon}}\leq C$ for some $C>0$. What I am interested in is the asymptotic behaviour of the integral above. In the case $n=1$ it follows that the autoconvolution of $f$ with itself has the same asymptotic behaviour as $f$, but starting at $n=2$ I can only find a bound of the type \begin{align} \int_{\mathbb{R}^d} f(y)f(y-x_1)f(y-x_2)\text{d}y \leq \begin{cases}B\, f\ast f(x_1-x_2) \\ B\, f\ast f(x_1) \\ B\, f\ast f(x_2) \end{cases} \end{align} which does not really help me as the bound grows like $B^n$ for $n\to\infty$. I am thankful for any kind of hints of where I might find integrals of this type or how to handle this.