Bounded Returns, Sub-Gaussian Tails, and a Logical Misconception

Summary

The document asks whether financial returns can be bounded given that every bounded random variable is sub-Gaussian. It states the sub-Gaussian condition through a bound on the moment-generating function of a centered variable and points to a textbook example for the bounded-implies-sub-Gaussian result.

The question then attempts to infer that returns are unbounded from the premise that financial returns are not Gaussian. That inference does not follow: being sub-Gaussian is not the same as being Gaussian, and the stated implication only says bounded variables are sub-Gaussian. The document offers no empirical return analysis or resolution, so it is best read as a probability question highlighting distinctions between boundedness, tail behavior, and distributional shape.

Key ideas

• A bounded random variable satisfies a sub-Gaussian moment-generating-function bound.
• Sub-Gaussian distributions are a broader class than Gaussian distributions.
• The fact that returns are non-Gaussian does not by itself show that they are unbounded.
• The document poses a probability question but provides no financial data or conclusive answer.

Tags

# A bounded random variable is sub-Gaussian, thus financial returns are not bounded?


# A bounded random variable is sub-Gaussian, thus financial returns are not bounded?












Bounded random variables are sub-Gaussian, yet I, intuitively, assume financial returns are bounded random variables; however, they are not sub-Gaussian. Am I wrong to assume financial returns are bounded random variables? How could that be so?

For a proof of bounded RV $\implies$ sub-Gaussian, see Example 2.4 on page 24 in Wainwright 2019 textbook, which is freely available online. Ex 2.3 in Chapter 2 link here.

For a reminder on sub-Gaussian definition: a RV $X$ with $\mu = E_X [X]$ is sub-Gaussian if $\exists \sigma > 0 \ \text{s.t.} \ E[e^{\lambda (X-\mu)}] \leq e^{\sigma^2 \lambda^2 / 2}, \ \forall \lambda \in \mathbb{R}.$

Edit: To make sure I am not making a logical error here. The statement $P \implies Q$ is equivalent to $\lnot P \implies \lnot Q.$ Then we have that financial returns are not Gaussian implying that they are not bounded...I assume this is something that has been discussed by the literature?

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