🔬 mesoscale physics

A clarification of Fermi's formula for the maximum of partial sums

This paper corrects a calculation error in Fermi's 1926 note regarding the probability that partial sums remain below a threshold, clarifies the inconsistency in his final integration, and establishes that small individual variances alone are insufficient to guarantee the discrete limit without satisfying the Lindeberg condition.

Original authors: Alessandro Lodi

Published 2026-10-14

📖 4 min read☕ Coffee break read

CC0 1.0

Original authors: Alessandro Lodi

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of random processes, scientists often look at sequences of small, unpredictable steps to understand how a system evolves over time. Imagine a traveler taking a series of steps where each step is independent of the last, sometimes moving forward, sometimes backward, but with no overall tendency to drift in one direction. A central question in this field is to determine the likelihood that this traveler will stay within a certain boundary, never crossing a specific line no matter how many steps they take. This concept, known as the probability of a maximum, is crucial for understanding everything from the spread of heat to the fluctuations of financial markets. For nearly a century, a famous calculation by the physicist Enrico Fermi has been the standard reference for this problem, offering a formula to predict how often such a random walker stays below a threshold. However, a closer look at the mathematics reveals a subtle but significant error in how that formula was interpreted, one that flips the answer from the probability of staying safe to the probability of failing.

A recent analysis by independent researcher Alessandro Lodi revisits Fermi's 1926 note to clarify exactly what was calculated and where the logic went astray. The core of the issue lies in the difference between two related but opposite events: the chance that a random path stays entirely below a limit, and the chance that it crosses that limit at least once. Fermi's original text provided a mathematical expression that, upon inspection, actually describes the probability of crossing the line, not staying under it. The printed formula in the original note decreases as the boundary moves further away, which makes no sense for a probability of staying within bounds; if the wall is moved further out, the chance of staying inside should increase, not decrease. Lodi demonstrates that the correct probability is simply the complement of what was printed. By carefully tracing the steps of the original derivation, specifically how the final integration was performed, the author shows that the sign of the result was effectively inverted, turning the probability of survival into the probability of failure.

The paper does more than just correct a historical typo; it rigorously establishes the conditions under which the continuous mathematical model Fermi used actually applies to real, discrete steps. Fermi assumed that if the individual steps are small enough compared to the total distance traveled, the system behaves like a smooth, continuous flow. Lodi confirms that this is true, but only if a specific condition regarding the size of the steps is met. This condition ensures that no single step, no matter how rare, is large enough to disrupt the smooth pattern. To prove that this condition is necessary, the author constructs a specific counterexample involving a mix of tiny random movements and occasional, rare jumps of a fixed size. In this scenario, even though the individual steps are small on average, the rare jumps prevent the system from behaving like a smooth flow. The result is a probability that does not match the standard formula, proving that simply making the variances small is not enough to guarantee the limit holds.

The resolution offered in this work reconciles the physical intuition with the mathematical reality. It confirms that for a wide class of random processes where the steps are well-behaved, the probability of staying below a threshold is indeed given by a specific, smooth curve that approaches certainty as the threshold moves further away. However, it also serves as a warning that this elegant result relies on strict rules about the nature of the randomness. The paper concludes by showing how the discrete steps, when they follow the right rules, converge perfectly to the continuous model, but when those rules are broken by rare, large fluctuations, the standard approximation fails. This clarification ensures that future applications of Fermi's work, from physics to finance, use the correct probability for the event in question, distinguishing clearly between the chance of staying within bounds and the chance of breaking them.

Technical Summary: A Clarification of Fermi's Formula for the Maximum of Partial Sums

Problem Statement
This note addresses a discrepancy in Enrico Fermi's 1926 publication regarding the probability that all partial sums of independent random variables remain below a specific threshold $a$ . Fermi defined the event $Q_n(a) = P(S_1 \le a, \dots, S_n \le a)$ , where $S_k$ are partial sums of independent, zero-mean variables with finite variances. Fermi's printed formula (Formula 9 in his note) assigned the expression $\text{erfc}(a/\sqrt{2V_n})$ to this probability, where $V_n$ is the total variance.

The author identifies a fundamental inconsistency: the printed formula describes a decreasing function of $a$ that tends to zero as $a \to \infty$ . However, the event $Q_n(a)$ is nondecreasing in $a$ and must tend to 1 as $a \to \infty$ . The printed expression actually corresponds to the probability of the complementary event (the crossing probability), not the survival probability.

Methodology
The paper employs a combination of probabilistic analysis, diffusion approximation, and limit theorems to resolve the inconsistency and clarify the conditions for the discrete limit.

Diffusion Equation and Reflection Principle: The author reconstructs Fermi's derivation using standard Brownian motion ( $B_t$ ) and the reflection principle. By defining the density of paths that have not yet reached the threshold $a$ as $q_a(t, x) = g_t(x) - g_t(2a-x)$ , where $g_t$ is the free Gaussian density, the paper derives the correct survival probability $Q(a, t) = 2\Phi(a/\sqrt{t}) - 1$ .
Integration Analysis: The paper traces the error in Fermi's original calculation. While Fermi's formula (19) correctly sets up the density of surviving paths by subtracting the first-passage density, the subsequent integration on page 318 incorrectly identifies the integral of the surviving density with the integral of the absorbed mass (the first-passage probability). The author demonstrates that integrating the correct density recovers the survival probability, whereas the printed calculation yields the crossing probability.
Discrete Limit Analysis: The paper investigates the convergence of discrete partial sums to the Brownian limit. It utilizes the functional Lindeberg theorem, parameterizing the sums by accumulated variance to establish convergence to Brownian motion.
Counterexample Construction: To test the sufficiency of "small individual variances" as a condition for the limit, the author constructs a counterexample involving independent increments with smooth densities. These increments consist of a mixture of rare unit jumps and Gaussian noise.

Key Contributions and Results

Correction of Fermi's Formula: The paper explicitly corrects Fermi's Formula (9) and (19). It establishes that the correct probability for the maximum of partial sums remaining below $a$ is $2\Phi(a/\sqrt{V_n}) - 1$ (or $\text{erf}(a/\sqrt{2V_n})$ ), while the printed formula $\text{erfc}(a/\sqrt{2V_n})$ corresponds to the probability that the maximum exceeds $a$ .
Explicit Limit Theorem (Proposition 1): The paper states a precise limit for independent increments under the Lindeberg condition. It proves that if the Lindeberg condition holds and the normalized threshold $a_n/\sqrt{V_n}$ converges to $b$ , then:
$\lim_{n \to \infty} P(M_n \le a_n) = 2\Phi(b) - 1$
This result controls the absolute error of the approximation.
Necessity of the Lindeberg Condition: The paper demonstrates that the condition of "small individual variances" relative to the sum is insufficient to justify the diffusion limit. The constructed counterexample shows that even with smooth densities and vanishing individual variances relative to the total, the presence of rare, large jumps (unit size) that survive normalization can cause the discrete probability to deviate significantly from the Brownian limit. In the example, the discrete probability remains bounded away from the Gaussian prediction because the rare jumps prevent the Lindeberg condition from being satisfied.
Discrete vs. Continuous Absorption: The author clarifies that while the continuous model imposes a zero boundary condition ( $q_a(t, a) = 0$ ), the discrete recursion for path densities does not strictly enforce a zero value at the threshold in the same manner, though the probabilities converge under the Lindeberg condition.

Significance and Scope
The paper claims modest significance, focusing on reconciling the stated event in Fermi's note with the printed calculation and making the sufficient hypotheses for the discrete limit explicit. The probabilistic results themselves (such as the reflection principle and the Lindeberg theorem) are described as classical. The primary contribution is the correction of a historical calculation error and the rigorous delineation of the conditions (specifically the Lindeberg condition) required for the diffusion approximation to hold for the maximum of partial sums. The author notes that the limit of 1 mentioned at the end of Fermi's note correctly belongs to the crossing probability, not the survival probability.

The paper does not propose new applications or future experimental directions but serves as a technical clarification of a specific historical derivation in mathematical physics.

Technical Summary: A Clarification of Fermi's Formula for the Maximum of Partial Sums

More like this