Convolution of multivariate distributions

Question

I have seen many questions on convolution of probability distributions (i.e., uniform, Gaussian, etc.). However, so far I have only seen 1-d case, even in wikipedia. I wonder what is the formulation of convolution for multivariate distributions. The case I am interested in particular is multivariate uniform distribution (i.e., uniform distribution in high-dimension with a well-defined region as its support). I am curious if its behavior is similar to its 1-d counterpart (e.g., looks like either a high-dimensional analogue of a trapezoid or a triangle, with the maximum probability density being equal to the min of the probability amplitude of the two multivariate uniform distributions used for convolution, etc.). My current gut feeling is to do the convolution dimension-by-dimenstion,and the order shouldn't matter.

Answer 1

Consider first the two-dimensional case of planar distributions. In general by definition, $h=g*f = \int_x \int_y g(t_1-x, t_2 -y) f(x,y) \ dx dy$. How to unpack this expression? Convolution consists of taking a moving average of one pattern $f(x,y)$, using the "mirror-flipped" version of the second pattern: $g(-x,-y)$.Here $(t_1, t_2)$ is the amount of translation applied as you move one pattern against the other.

To be more concrete consider the case of probability distributions that are uniformly distributed on two regions of different shapes and areas. The convolution has this geometric-probabilistic interpretation: the convolution is the "fractional overlap" of one region with another as you translate one across the other.

Consider the case in which the regions are rectangles whose edges are parallel to the coordinate axes. Each has an associated probability distribution, which is a characteristic function, normalized by dividing by an appropriate constant (area) The convolution of a rather small rectangle S centered at $c_S$ with a big one B centered at $c_B$ will produce a distribution that is centered at $c_S+ c_B$, and will look like a smudged version of $B$ re-positioned at that new center. Let $\tilde B$ denote this translated version of $B$. The smudges version has different behavior in various regions: constant deep inside the interior of $\tilde B$, beveled like a linear ramp near the vertical and horizontal edges of $\tilde B$, and a product of two linear factor in different variables near the corners, producing a quadratic behavior there. Far from $\tilde B$ the convolution is zero. As the size of $S$ shrinks, the transition zones where these changes occur get narrower. The overall shape of the convolution resembles a mesa that has a flat central top, sides that decline linearly away from the upper plateau, and rounded corners that have a quadratic decay. (The graph at the corners are ruled surfaces that look like the flank of a saddle surface.) A three-fold convolution of planar patterns smooths this out even more.

Consider next two convex polygonal regions in the plane. The complement of each polygon decomposes as a disjoint collection of exterior wedges. Complementation changes $ \chi_A$ to $\chi_{A^c}= 1- \chi_A$. The convolution of a constant with any probability distribution is the constant 1. So focus now on the complements. The convolution of such wedges will generally be piecewise quadratic expressions. Thus the convolution of two convex polygons will decompose as a sum of many terms that are piecewise of degree two or less. Obviously the patterns are a pain to write down explicitly!