Tail Index and Moment Existence in NIG and Variance Gamma Models

Summary

The document asks how to express the tail index of Normal Inverse Gaussian and Variance Gamma distributions in terms of their parameters. The response gives a moment-based interpretation: if a distribution has a finite tail index, moments above that index do not exist. Since all moments exist for both NIG and VG, the response concludes that their tail index is infinite, as it is for the Gaussian distribution.

This is a concise conceptual answer rather than a derivation. It does not provide parameter-specific tail formulas or discuss how the distributions’ tail decay compares in practice. The conclusion is tied to the stated definition of tail index through moment existence, so readers seeking a particular alternative tail parameterization will need additional detail.

Key ideas

• A finite tail index implies that moments above the index do not exist.
• NIG and Variance Gamma distributions have all moments under the response’s stated premise.
• The response therefore characterizes their tail index as infinite.
• The answer does not derive parameter-specific tail expressions.

Tags

# What is the tail index for NIG and/or VG?


# What is the tail index for NIG and/or VG?












...as a function of NIG (Normal Inverse Gaussian) or VG (Variance Gamma) parameters, obviously. I've read that the NIG $\alpha$ is related to the $\alpha$-stable tail parameter, which conversely maps to the tail index.

## Answer by Kiwiakos (score 1)

https://quant.stackexchange.com/a/20686

If the tail index is $\alpha$ then moments beyond that do not exist. The fact that all moments exist for NIG and VG indicates that the tail index is infinite (like the Gaussian).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.