A remark on comparison of the sum and the maximum of positive random variables

Abstract: We disprove a conjecture stated in a paper by Arnold and Villasenor concerning the sum and the maximum of independent and identically distributed half-normal random variables.

• The paper disproves the conjectured scaling rule equating the sum and maximum of i.i.d. positive random variables for n ≥ 3.
• It employs asymptotic tail analysis and Taylor expansions to rigorously reject distributional equivalence for half-normal and generalized gamma distributions.
• The findings impact modeling in reliability and risk management by clarifying the limits of applying simple scaling to extreme value analyses.

Disproof of Distributional Equivalence between the Sum and Maximum of Positive Random Variables

Introduction

This work addresses a conjecture by Arnold and Villaseñor concerning the distributional relationship between the sum and maximum of independent, identically distributed (i.i.d.) positive random variables, particularly those following the half-normal law. The original result established that, for $n=2$ i.i.d. half-normal variables $X_1, X_2$, the identity

$$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$

holds. Arnold and Villaseñor conjectured that for $n \ge 3$,

$$X_1 + \cdots + X_n \stackrel{d}{=} (n!)^{1/n} \max\{X_1, \dots, X_n\}$$

also holds under the same distributional assumptions. The present paper rigorously disproves this conjecture for $n \ge 3$, not only for the half-normal distribution but also for a subclass of generalized gamma distributions.

Statement of Main Result

Let $X_1, \dots, X_n$ be i.i.d. random variables with density $f(x) = c_1 \exp(-c_2 x^\beta)$ on $[0,\infty)$, where $c_1, c_2 > 0$ and $X_1, X_2$0 (generalized gamma class). Define $X_1, X_2$1 and $X_1, X_2$2. The main theorem asserts:

• For $X_1, X_2$3 and $X_1, X_2$4, there is no constant $X_1, X_2$5 such that $X_1, X_2$6.

Notably, for the half-normal law ($X_1, X_2$7), this precludes the conjecture for $X_1, X_2$8.

Technical Approach

The proof is executed via contradiction using asymptotic analysis of the lower and upper tails of the cumulative distribution functions of $X_1, X_2$9 and $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$0.

Behavior Near Zero

Using Taylor expansion and change of variables, it is shown that near zero:

• $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$1
• $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$2

Thus, any constant $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$3 satisfying $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$4 must be $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$5.

Right-Tail Asymptotics for $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$6

For large $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$7, explicit lower and upper bounds for $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$8 and $$X_1 + X_2 \stackrel{d}{=} \sqrt{2}\,\max\{X_1, X_2\}$$9 are derived. Convexity arguments allow bounding the sum's density via the maximum. Using the respective forms:

• $n \ge 3$0 falls off as $n \ge 3$1,
• $n \ge 3$2 is upper bounded by terms involving $n \ge 3$3.

By comparing the rates in the exponents, the ratio $n \ge 3$4 diverges, violating the conjectured equality in distribution.

Subexponential Case: $n \ge 3$5

For $n \ge 3$6, the underlying distribution becomes subexponential. Through reference to general results on subexponential distributions, notably that the tail of the sum mirrors that of the maximum asymptotically, it is shown that such a scaling can never produce distributional equality for $n \ge 3$7 with any constant $n \ge 3$8.

Further Remarks and Numerical Evidence

A detailed analysis for $n \ge 3$9, specifically for the half-normal distribution, is presented. Computing the second moments, it is shown that the required equality for distributional identity would demand an impossible algebraic relation—involving the transcendental number $$X_1 + \cdots + X_n \stackrel{d}{=} (n!)^{1/n} \max\{X_1, \dots, X_n\}$$0—thus providing an algebraic obstruction.

Additionally, the paper's framework applies to a strictly increasing family of $$X_1 + \cdots + X_n \stackrel{d}{=} (n!)^{1/n} \max\{X_1, \dots, X_n\}$$1's in generalized gamma distributions, further generalizing the disproof.

Implications

The findings invalidate a potential generalization proposed for half-normal and related distributions. This establishes strict limitations on the extent to which distributional identities involving the sum and maximum can be extended beyond the two-variable setting. Practically, this impacts the analysis and simulation of extremes in probabilistic models and their use in statistical inference where such distributional equivalences might be assumed.

Theoretically, the work exemplifies the power of asymptotic and analytic techniques in characterizing when certain algebraic identities in distribution can or cannot be extended to wider classes, providing clarity for future research in extremal theory and the theory of probabilistic representations.

Extensions to other domains, such as reliability theory and risk management (where maxima and sums play central roles), must account for the non-existence of simple scaling equivalences outside the two-dimensional half-normal case.

Conclusion

The conjecture asserting the existence of a distributional constant scaling between the sum and the maximum of $$X_1 + \cdots + X_n \stackrel{d}{=} (n!)^{1/n} \max\{X_1, \dots, X_n\}$$2 i.i.d. half-normal (and related) random variables is disproved. The failure of the conjecture is shown through asymptotic tail analysis and algebraic inconsistencies, with the results bearing implications for both theory and applications involving functionals of positive random variables and their extremal statistics.