Study of super-Gaussian or super-Exponential distributions?

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the terms may be weird, but they just mean distributions with heavier tail with Gaussian/exponential respectively, in a way that is exactly the opposite of sub-Gaussian and sub-exponential.

There's apparently a lot of study on sub-Gaussian and sub-exponential distributions, but I wonder if there's any on super-Gaussian and super-exponential distributions?

Thanks.