Proof: The Lognormal Has Heavier Tails Than the Gamma

I put this proof together while studying T cell movement patterns, where I needed a clear comparison between the tail behaviours of the lognormal and gamma distributions. Although the result that the lognormal has heavier tails is often stated in textbooks, articles, and blog posts but the details are left as a heuristic or are implied from moment-based arguments. This specific proof, using a direct comparison of log-densities and asymptotic limits, does not appear to have been published in this form. I have written it out here both for clarity and as a reference for others looking for an explicit derivation.

Why Tail Behaviour Matters for T Cell Movement

1. Exploration Efficiency
   • A heavier-tailed distribution, like the lognormal, implies occasional very large steps.
   • This allows T cells to escape local areas and cover more space, which is useful in sparse or heterogeneous environments.
2. Search Strategy Implications
   • Gamma-distributed motion leads to more regular, exponentially bounded steps.
   • Lognormal motion leads to intermittent long-range jumps, similar to a Lévy walk, which is known to optimise search in environments where targets are rare and randomly located.
3. Biological Relevance
   • Experimental data from intravital imaging sometimes show that T cell velocities or displacements do not decay exponentially, suggesting lognormal or even power-law tails.
   • Choosing the wrong distribution can lead to incorrect predictions about how quickly T cells find their targets, how thoroughly they sample tissue, or how they respond to chemokine gradients.

Proof
Let \( f(x) \) be the gamma probability density function, and \( g(x) \) the lognormal probability density function. To compare their tail behaviours, we analyse the limit: \[ \log f(x) - \log g(x) \] as \( x \to \infty \).

The gamma density is asymptotically: \[ \log f(x) = (\alpha - 1)\log x - \frac{x}{\beta} + \text{const} \]

The lognormal density is: \[ \log g(x) = -\log x - \frac{(\log x - \mu)^2}{2\sigma^2} + \text{const} \]

Expanding the quadratic term: \[ \log g(x) = -\log x - \frac{(\log x)^2 - 2\mu \log x + \mu^2}{2\sigma^2} + \text{const} \]

Taking the difference: \[ \log f(x) - \log g(x) = (\alpha - 1 + 1)\log x - \frac{x}{\beta} + \frac{(\log x)^2 - 2\mu \log x + \mu^2}{2\sigma^2} + \text{const} \]

Simplifying: \[ \log f(x) - \log g(x) = -\frac{x}{\beta} + \frac{(\log x)^2}{2\sigma^2} + \text{lower-order terms} \]

As \( x \to \infty \), the linear term dominates: \[ -\frac{x}{\beta} + \frac{(\log x)^2}{2\sigma^2} \to -\infty \]

Therefore: \[ \log \left( \frac{f(x)}{g(x)} \right) \to -\infty \quad \Rightarrow \quad \frac{f(x)}{g(x)} \to 0 \]

Hence, the gamma density decays faster than the lognormal in the tail: \[ f(x) \in o(g(x)) \quad \text{as } x \to \infty \]

This confirms that the lognormal distribution has heavier tails than the gamma distribution.

Q.E.D.