Let $a \le b$ and $c\le d$ be real numbers and $f$ a real function. I am struggling with an integral of the form:
$$I = \int_a^b \mathrm{d}x_1 ...\mathrm{d}x_m \int_c^d \mathrm{d}y_1 ...\mathrm{d}y_n \prod_{i=1}^m \prod_{j=1}^n f(x_i y_j)$$
Can it be reduced to a single or double integration somehow?
I am interested in the large $m,n$ behavior of $I$. Specifically, suppose that the ratio $\alpha = m/n$ is fixed. Can we evaluate in general the limit:
$$\lim_{n\rightarrow\infty} \frac{1}{n} \ln I$$
So far, these are the only reductions I've been able to do:
$$\begin{aligned} I & = \int_a^b \mathrm d x_1 \dots \mathrm d x_m \left( \int_c^d \mathrm d y \prod_{i = 1}^m f (x_i y) \right)^n\\ & = \int_c^d \mathrm d y_1 \ldots \mathrm d y_m \left( \int_a^b \mathrm d x \prod_{j = 1}^n f (x y_j) \right)^m \end{aligned}$$