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Suppose that we have a normal distribution with known centre and variance.

I would like to express it in terms of a sum of infinitely many gaussians whose centres are evenly, and infinitely densely, separated from minus to plus infinity.

How can I find the amplitudes and variances of those infinitely many gaussians?

edit: a remark: the reason that I did not ask the possibility of such a decomposition is that I tried it in python and sum of infinitely many gaussians perfectly fit a gaussian. But you are very welcome to enlighten me on the possibility of such a decomposition; whether it be a proof or a counterexample that states "not necessarily".

artmyb
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  • What does it mean that the centers (means) are "evenly and infinitely densely separated from minus to plus infinity"? Do you want a countable collection of components whose means are dense over the whole real line? – Jukka Kohonen Oct 04 '21 at 11:28
  • There must be infinitely many components. The infinity comes from both being "from minus to plus infinity" and being infinitely many in a finite portion of real line. i.e. the decomposition should be sum of gaussians whose means are each real number on the line. i.e. it is an integral. – artmyb Oct 04 '21 at 11:31
  • So, in particular, there must be an infinite number of components whose mean is in $[5,6]$. The expectation of their sum must be $+\infty$. Also there must be an infinite number of components whose mean is in $[-6,-5]$, and the expectation of their sum is $-\infty$. (And same for all finite intervals.) I'm not sure how you can meaningfully make the sum of all these a well-defined finite quantity. Are all components also required to be independent? – Jukka Kohonen Oct 04 '21 at 14:56
  • artmyb, are you sure you don't want a weighted sum of gaussians? An infinite sum of gaussians that have large positive and negative means seems unwieldy. – Jukka Kohonen Oct 04 '21 at 17:16

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