A note on limiting behaviour of constrained sums of two variables

TL;DR

This paper investigates the asymptotic behavior of a normalized sum conditioned on a fixed total, revealing how the tail properties of the underlying distribution influence the limiting distribution and challenging common assumptions about tail decay.

Contribution

It provides new insights into the limiting behavior of constrained sums of i.i.d. variables, highlighting the role of log-convexity or log-concavity of the density and presenting surprising results about tail behavior.

Findings

Distributional limits depend on the log-convexity or log-concavity of the density.

Tail decay rate does not strictly determine the limit distribution.

Light-tailed distributions can exhibit heavy-tailed-like behavior.

Abstract

This note studies the asymptotic properties of the variable $$Z_d:=\frac{X_1}{d}|\{X_1+X_2=d\},$$ as $d\to \infty$. Here $X_1$ and $X_2$ are non-negative i.i.d. variables with a common twice differentiable density function $f$. General results concerning the distributional limits of $Z_d$ are discussed with various examples. Eventual log-convexity or log-concavity of $f$ turns out to be the key ingredient that determines how the variable $Z_d$ behaves. As a consequence, two surprising discoveries are presented: Firstly, it is noted that the distributional limit is not strictly determined by the decay rate of the tail function. Secondly, it is shown that there exists a light-tailed distribution exhibiting behaviour that is commonly associated with heavy-tailed distributions i.e. the principle of a single big jump.