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

This paper refutes a conjecture by Arnold and Villasenor regarding the relationship between the sum and the maximum of independent and identically distributed half-normal random variables.

The Gist

Imagine you have a bag of identical, special dice. These aren't normal dice, though. They are "half-normal" dice, meaning they can only roll positive numbers, and they have a specific shape to their likelihood of rolling high or low numbers (like a bell curve cut in half).

Now, let's play two different games with a group of $n$ of these dice:

1. The Sum Game: You roll all $n$ dice and add up the total score.
2. The Max Game: You roll all $n$ dice, but you only care about the single highest number rolled.

The Big Question

A few years ago, two mathematicians (Arnold and Villasenor) noticed something magical. If you roll just two of these special dice, the total sum of the two dice behaves exactly like the highest single die, just scaled up by a specific factor ($\sqrt{2}$).

They thought, "This is so cool! It must work for any number of dice!" They made a bold guess (a conjecture):

"If you roll three or more of these dice, the total sum will always behave exactly like the highest single die, just scaled up by a specific mathematical factor."

The Plot Twist

This paper, written by Kazuki Okamura, is the "spoiler alert." Okamura says: "Stop right there. That guess is wrong."

He proves that while the magic works for two dice, it completely breaks down when you add a third (or more) dice to the mix. The sum and the maximum are no longer "twins" in disguise; they are actually very different animals.

How Did He Prove It? (The Detective Work)

Okamura didn't just guess; he used two different "magnifying glasses" to look at the dice behavior at the extremes.

1. The "Tiny Numbers" Test (Looking at the bottom)

Imagine looking at the very lowest possible scores.

• The Logic: If the Sum and the Max were truly the same (just scaled), then the chance of getting a tiny total sum should match the chance of getting a tiny maximum, perfectly.
• The Result: Okamura showed that for the math to work out, the scaling factor must be a specific number (the $n$-th root of $n$ factorial). This part actually worked out, confirming that if the magic were real, the scaling factor would have to be this specific number.

2. The "Huge Numbers" Test (Looking at the top)

This is where the magic trick falls apart. Okamura looked at the odds of rolling incredibly high numbers.

• The Scenario: Imagine rolling 3 dice. What are the odds that the sum is huge? What are the odds that the highest single die is huge?
• The Reveal:
  • The Max: For the highest single die to be huge, one die just needs to roll a massive number. The others can be anything.
  • The Sum: For the total sum to be huge, you usually need all the dice to roll high numbers, or at least a few of them to be very high.
  • The Analogy: Think of it like a relay race.
    • The Max is like asking, "Did the fastest runner finish in under 10 seconds?" If one person is a superstar, the answer is yes.
    • The Sum is like asking, "Did the total time of the whole team finish in under 10 seconds?" Even if one person is a superstar, if the other three are slow, the total time is bad.
  • Okamura proved that for these specific dice, the "Sum" becomes incredibly rare much faster than the "Max" as the numbers get huge. They don't match up.

The "Smoking Gun" (The Pi Proof)

For the specific case of 3 dice, Okamura used a clever trick involving the number $\pi$ (3.14159...).

• He calculated the "average squared score" for the Sum game.
• He calculated the "average squared score" for the Max game.
• If the original guess were true, these two calculations would have to be equal.
• But when he set them equal, the math required $\pi$ to be a specific type of number that it simply isn't (it would have to be related to square roots of 3 and cube roots of 6 in a way that contradicts the fact that $\pi$ is a "transcendental" number).
• Conclusion: The equation is impossible. The Sum and the Max are not the same.

The Takeaway

The paper is a correction of a mathematical "urban legend."

• For 2 dice: The Sum and the Max are perfect twins (scaled).
• For 3 or more dice: They are strangers. The Sum behaves differently than the Max.

It's a reminder that in math, just because a pattern works for a small example (like 2), it doesn't mean it will hold true for the whole group. Sometimes, adding just one more element changes the rules of the game entirely.

Technical Summary

1. Problem Statement

The paper addresses a conjecture proposed by Arnold and Villasenor regarding the distributional relationship between the sum ($S_n$) and the maximum ($M_n$) of independent and identically distributed (i.i.d.) half-normal random variables.

• Context: For $n=2$, it is known that if $X_1, X_2$ are i.i.d. half-normal variables, then $X_1 + X_2 \stackrel{d}{=} \sqrt{2} \max\{X_1, X_2\}$.
• The Conjecture: Arnold and Villasenor conjectured that this relationship generalizes for $n \geq 3$. Specifically, they hypothesized that for i.i.d. half-normal variables $X_1, \dots, X_n$:
  $$S_n = X_1 + \dots + X_n \stackrel{d}{=} (n!)^{1/n} \max\{X_1, \dots, X_n\} = (n!)^{1/n} M_n$$
• Objective: The author aims to disprove this conjecture for $n \geq 3$ and extend the analysis to a broader subclass of generalized gamma distributions.

2. Methodology

Okamura employs a proof by contradiction, analyzing the asymptotic behavior of the tail probabilities and the small-value behavior of the cumulative distribution functions (CDFs) for the sum and the maximum.

A. General Framework

The paper considers random variables $X_i$ with a density function of the form:
$$f(x) = c_1 \exp(-c_2 x^\beta), \quad x \geq 0$$
where $c_1, c_2 > 0$ and $\beta > 0$. The standard half-normal distribution corresponds to $\beta = 2$.

B. Step 1: Small-Value Asymptotics ($x \to 0^+$)

The author first establishes a necessary condition for the equality in distribution $S_n = C M_n$.

• Lemma 2.1: By analyzing the behavior of $P(S_n \leq x)$ and $P(M_n \leq x)$ as $x \to 0$, the author proves that if such a constant $C$ exists, it must be $C = (n!)^{1/n}$.
• Technique: This involves Taylor expansion of the density near 0 and applying the Dominated Convergence Theorem to integrals over the simplex.

C. Step 2: Large-Value Asymptotics ($x \to +\infty$)

The proof proceeds by showing that the necessary constant derived in Step 1 fails to satisfy the distributional equality when examining the tails ($x \to \infty$). The analysis is split into two cases based on the parameter $\beta$:

Case 1: $\beta \geq 1$

• Lower Bound for Maximum: Using the Mean Value Theorem, a lower bound for $P(M_n > x)$ is derived, showing it decays roughly as $\exp(-x^\beta)$.
• Upper Bound for Sum: Using convexity arguments (since $\beta \geq 1$), an upper bound for $P(S_n > x)$ is derived.
• Contradiction: Under the condition $\beta < \frac{n \log n}{n \log n - \log(n!)}$, the ratio of the tail probabilities $\frac{P(M_n > x/(n!)^{1/n})}{P(S_n > x)}$ tends to infinity as $x \to \infty$. This contradicts the assumption that $S_n$ and $(n!)^{1/n}M_n$ have the same distribution.

Case 2: $0 < \beta < 1$

• Subexponential Property: The author demonstrates that for $0 < \beta < 1$, the distribution is subexponential. This is verified by checking the concavity of $-\log f(x)$ and satisfying conditions from existing literature (Foss et al.).
• Tail Equivalence: For subexponential distributions, it is a known property that $P(M_n > x) \sim P(S_n > x)$ as $x \to \infty$.
• Contradiction: If $S_n = (n!)^{1/n} M_n$, then the tail of $X_1$ must satisfy a specific scaling limit. However, by applying L'Hôpital's rule to the specific density form, the author shows that the limit of the ratio of tails is 0, not 1, leading to a contradiction.

D. Alternative Proof for $n=3$ (Half-Normal Case)

For the specific case of the half-normal distribution ($\beta=2$) with $n=3$, the author provides an algebraic disproof:

1. Calculate the second moment of the sum: $E[S_3^2] = 3 + \frac{12}{\pi}$.
2. Calculate the second moment of the maximum: $E[M_3^2] = 1 + \frac{2\sqrt{3}}{\pi}$.
3. If the conjecture held, $E[S_3^2]$ would equal $(3!)^{2/3} E[M_3^2]$.
4. This equality implies that $\pi$ is an algebraic number (specifically in the field $\mathbb{Q}(\sqrt{3}, 6^{1/3})$), which contradicts the fact that $\pi$ is transcendental.

3. Key Contributions

1. Disproof of Conjecture: The paper definitively proves that the distributional equality $S_n \stackrel{d}{=} (n!)^{1/n} M_n$ does not hold for i.i.d. half-normal variables when $n \geq 3$.
2. Generalization: The result is extended to a subclass of generalized gamma distributions defined by the density $f(x) \propto e^{-x^\beta}$.
3. Threshold Condition: The paper identifies a precise threshold for $\beta$ and $n$ (Equation 1.3) under which the equality fails. For the half-normal case ($\beta=2$), the condition holds for all $n \geq 3$.
4. Moment-Based Counterexample: The algebraic proof for $n=3$ using moments and the transcendence of $\pi$ offers a distinct, elegant method of disproof separate from the asymptotic analysis.

4. Results

• Theorem 1.1: For $X_i$ with density $c_1 \exp(-c_2 x^\beta)$, if $n \geq 2$ and $\beta < \frac{n \log n}{n \log n - \log(n!)}$, there is no constant $C$ such that $S_n = C M_n$ in distribution.
• Specific Application: For the standard half-normal distribution ($\beta=2$), the inequality holds for all $n \geq 3$. Thus, the conjecture by Arnold and Villasenor is false for $n \geq 3$.
• Numerical Observation: While the distributions are not identical, the author notes that the ratio of the second moments for $n=3$ is numerically close ($3.24$ vs $3.30$), explaining why the conjecture might have been plausible initially.

5. Significance

• Clarification of Distributional Properties: The paper corrects a misunderstanding in the literature regarding the relationship between sums and maxima of positive random variables. It highlights that while such relationships exist for $n=2$ in specific cases, they do not generalize linearly or via simple factorial scaling for $n \geq 3$.
• Methodological Insight: The work demonstrates the power of combining asymptotic analysis of tails (for $\beta < 1$ and $\beta \geq 1$) with small-value behavior analysis to disprove distributional equalities.
• Implications for Extreme Value Theory: The results reinforce the distinction between the behavior of sums and maxima in subexponential and heavy-tailed (or light-tailed with specific decay) regimes, cautioning against assuming simple scaling laws without rigorous verification.