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Consider two random variables $Z$ and $W$. Given the variances of $Z$ and $W$, how can we compute the variance of their convolution $Z \circledast W $?

As an example, please consider the case of noise ($Z$) in an image being modeled by a Gaussian distribution of zero mean and a given variance $\sigma_Z^2$. Suppose we perform a convolution of $Z$ with a Gaussian kernel denoted by $W$. How can we find the variance of $Z \circledast W $? I am not able to find a good reference for this. Thanks for any help.

user409495
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    Can you explicitly show us the mathematical definition of $Z \circledast W $? To the best of my knowledge, in probability, the operands of a convolution operator are two distributions, i.e., the convolution of $F$ and $G$ is defined as $(F * G)(y) = \int G(y - x)dF(x)$, which corresponds to the distribution of the sum of two independent r.v.s $X$ (whose distribution is $F$) and $Y$ (whose distribution is $G$). How do you define the convolution of two r.v.s directly at the r.v.-level? – Zhanxiong Mar 18 '24 at 13:18
  • Are you convolving the variables or their densities? The two things are very different! The convolution of the densities is the density of the sum of the variables: see https://stats.stackexchange.com/questions/331973 Your example suggests the latter, in which case the variances add provided the "noise" is independent of the image. See https://stats.stackexchange.com/questions/38721 – whuber Mar 18 '24 at 15:03
  • @Zhanxiong Thanks for your comment. My apologies for the confusion. I was referring to the convolution of random variables. I think $W$ may be treated as a constant kernel without any statistics associated with it. I am wondering how the convolution of $W$ with $Z$ would affect the variance of $Z$. – user409495 Mar 19 '24 at 18:38
  • @whuber Thanks for the reply, and sorry for the confusion. I was referring to convolution of the variables. I think in this particular case, $W$ may be treated as a constant kernel. I am not sure how would the convolution with $W$ affect the variance of $Z$. – user409495 Mar 19 '24 at 18:42
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    @user409495 Can you define $Z \circledast W$ in mathematics symbols, as opposed to plain English? Or can you provide any reference to such definition? – Zhanxiong Mar 19 '24 at 18:45

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Like the previous answer posted, there's talk of convolution of the distributions of the 2 random variables. If these are independent, the convolution gives the distribution of the sum of 2 random variables. The characteristic function is the product of the 2 given ones. You can get the variance from this product; it is the coefficient of lambda^2. More precisely it is the second moment of the sum Z=X+Y. Or you can compute it directly as E[(X+Y)^2]. I hope this will help.

Ismail
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  • Re "get the variance from this product:" that's unnecessarily complicated. It is basic (and easily proven) that the variances simply add. – whuber Mar 19 '24 at 13:25