In probability theory we have mean, variance, skewness, kurtosis and etc. Can we use the same terms for a convolution (kernel) that sums to 1?
For example, can we say a $1$-D convolution kernel $[3/2,-2,3/2]$ has mean $0$ and $\sigma= 3$?
I'm wondering what basic results generalize from probability distributions to convolutions, or if there are any references that people would recommend (background: math and phy undergrad and grad.). Thanks!
If you work with non-negative kernels, then after the normalization, you can treat it as a distribution of a delta-signal after one convolution step. So it's natural to talk about statistics of such distribution. Important: it's not the values statistics, it's position statistics assuming the values are probabilities. So if you have a kernel $[0.25, 0.5, 0.25]$ and assume the positions are $[-1,0,1]$, then the mean and sigma are $$m=-1\times0.25 +0\times 0.5+1\times0.25 = 0\\ \sigma=(-1)^2\times0.25+0^2\times0.5+1^2\times0.25=0.5$$
However, if you want to treat the numbers in the kernel as a set and calculate statistics over those, it makes little sense.