I am interested in the PDF $p_\sigma(\mathbf{x})$ for the sum of two random variables, with PDFs $p(\mathbf{x})$ and $N(\mathbf{x}; 0, \sigma)$, respectively.
$N$ is the normal distribution PDF. $p$ is another PDF for which we do not have an analytic form, but can evaluate for specific points $\mathbf{x}$ with moderate cost.
Actually I'm only interested evaluating $p_\sigma$ at the single point $\mathbf{x}$ - I do not need the full PDF.
$\mathbf{x}$ is high-dimensional, say 10 dimensions.
The PDF for a sum of random variables is the convolution of the individual PDFs:
$$p_\sigma(\mathbf{x}) = \int p(\mathbf{y}) \; N(\mathbf{x}; \mathbf{y}, \sigma) \;d\mathbf{y}$$
I am aware of the techniques in computer vision of separating the multidimensional gaussian for more efficient convolution, but this is still intractable because I'm not dealing with the low dimensions of 2D images.
Are there established methods for a local approximation of the convolution for the specific point $\mathbf{x}$?