Is a distribution whose mean and variance are its only non-zero cumulants neccesarily Gaussian?

Question

I have a distribution. Its cumulants are all zero except for its mean and variance. Does that necessarily mean that my distribution is Gaussian?

Or in other words, are there non-Gaussian distributions whose only non-zero cumulants are mean and variance.

Answer 1

Since one can analytically reconstruct the distribution given all its cumulants, the answer is that a distribution with any mean, any non-zero variance, and all other cumulants zero is necessarily a Gaussian.

A stronger, and non-trivial, statement is that the cumulant generating function cannot be a finite polynomial of degree greater than 2; thus there is no distribution with any mean, any non-zero variance, and any non-zero skew but no non-zero higher cumulants.