On sequential maxima of exponential sample means, with an application to ruin probability

Dimitris Cheliotis; Nickos Papadatos

arxiv: 1906.09377 · v1 · pith:4XOQFYH2new · submitted 2019-06-22 · 🧮 math.PR

On sequential maxima of exponential sample means, with an application to ruin probability

Dimitris Cheliotis , Nickos Papadatos This is my paper

Pith reviewed 2026-05-25 18:37 UTC · model grok-4.3

classification 🧮 math.PR

keywords exponential random variablessample meansmaximal averageruin probabilityclosed form distributionrisk theorymemoryless propertysequential maxima

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The pith

For i.i.d. exponential random variables the inverse distribution of the maximal average admits a simple closed form.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives the distribution of the largest value among the successive sample means formed from a sequence of independent exponential random variables. It shows that the inverse of this distribution function possesses a simple closed-form expression. The result is then applied to obtain an explicit formula for ruin probability in a classical risk model. A sympathetic reader would care because exponential distributions model many arrival and lifetime processes, and an explicit inverse turns a previously intractable calculation into a direct one.

Core claim

The inverse distribution of the maximal average in a sequence of i.i.d. exponential random variables admits a simple closed form. This is obtained by exploiting the memoryless property to track the distribution of the sequential maxima of the partial averages, and the same expression yields the ruin probability in the associated risk process.

What carries the argument

The sequential maximum of the sample means of partial sums, whose distribution is derived recursively using the memoryless property of the exponential distribution.

If this is right

Ruin probability in the risk model is given by a direct closed-form expression rather than an integral or limit.
The probability that any partial average exceeds a fixed level x follows immediately from the inverse distribution.
Exact computation replaces simulation or numerical approximation for quantities governed by the maximal average.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The closed form may permit an explicit distribution for the index at which the maximum average is first attained.
The same recursive technique could be tested on the joint law of the maximal average and the total sum at the stopping time.
In large-sample regimes the expression might yield precise rates for how quickly the maximal average converges to its almost-sure limit.

Load-bearing premise

The random variables are independent and identically distributed as exponential so that the memoryless property can be used to derive the distribution.

What would settle it

Generate many independent sequences of exponential random variables, compute the empirical distribution of the maximal average for each sequence, invert the resulting cdf, and check whether the values match the claimed closed-form expression at multiple points.

read the original abstract

We obtain the distribution of the maximal average in a sequence of independent identically distributed exponential random variables. Surprisingly enough, it turns out that the inverse distribution admits a simple closed form. An application to ruin probability in a risk-theoretic model is also given.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives an explicit finite-sum formula for the tail of the maximal sample mean of i.i.d. exponentials via Poisson embedding and memorylessness.

read the letter

The main result is a closed-form expression for P(1/M_n > t) where M_n is the max of the first n sample means from i.i.d. Exp(1) variables; it reduces to a finite sum of elementary terms. That is the concrete new item. The derivation embeds the partial sums in a Poisson process and tracks first-passage times, which is a standard move for exponentials but applied cleanly here to get an explicit inverse distribution. The ruin-probability application follows directly from the same expression without extra assumptions. The math looks internally consistent on the terms given; no circularity or hidden fitting appears. The i.i.d. exponential assumption is used exactly where the memoryless property is invoked, so the scope is narrow but the claim inside that scope holds up. A minor soft spot is that the paper does not compare the new formula numerically against simulation for moderate n, which would make the result easier to check quickly. Overall the work is a short, self-contained distributional calculation rather than a broad methodological advance. It is the sort of note that belongs in a probability journal if the derivation is fully rigorous. I would send it to referees rather than desk-reject; the explicit form is verifiable and the application is immediate. A reader working on sequential maxima or classical risk models would get direct use from the formula.

Referee Report

0 major / 2 minor

Summary. The paper derives the distribution of the maximal sample mean M_n = max_{k≤n} (S_k/k) where S_k are partial sums of i.i.d. Exp(1) random variables. It shows that P(1/M_n > t) admits an explicit closed-form expression as a finite sum of elementary terms, obtained by embedding the sequence in a Poisson process and applying the memoryless property to track normalized first-passage times. The same expression is used to obtain an exact formula for ruin probability in a classical risk model.

Significance. The explicit finite-sum representation for the inverse distribution is a clear advance, as it replaces simulation or recursive approximation with direct computation for any fixed n and t. The derivation relies only on standard properties of exponentials and Poisson processes with no free parameters or ad-hoc assumptions, and the ruin-probability application follows immediately from the same expression. This combination of closed form and direct applicability strengthens the contribution.

minor comments (2)

[Introduction] The introduction states the result for general n but the explicit sum is first displayed only after the Poisson embedding; adding a forward reference to the final expression in §2 would improve readability.
[Application to ruin probability] In the ruin-probability section the initial capital is denoted u without restating its relation to the threshold t used earlier; a single sentence linking the two would avoid any momentary ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. We are pleased that the closed-form result and its application were viewed favorably.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The manuscript derives the distribution of the maximal average M_n = max_{k≤n} (S_k/k) for i.i.d. Exp(1) variables by embedding the partial sums in a Poisson process and applying the memoryless property to obtain first-passage probabilities. The resulting finite-sum expression for P(1/M_n > t) follows directly from these probabilistic assumptions and the i.i.d. exponential structure; no parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the ruin-probability application is an immediate substitution of the same expression. The derivation therefore contains no step that reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard assumption that the variables are i.i.d. exponential; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The sequence consists of i.i.d. exponential random variables.
Explicitly stated in the abstract as the setting for the maxima.

pith-pipeline@v0.9.0 · 5552 in / 1018 out tokens · 21993 ms · 2026-05-25T18:37:44.887380+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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[1] [1]

Abel series distributions with applications to ﬂuctua- tions of sample functions of stochastic processes

Charalambides, Ch.A. Abel series distributions with applications to ﬂuctua- tions of sample functions of stochastic processes. Communications in Statis- tics – Theory & Methods , 19(1), (1990), 317–335

work page 1990

[2] [2]

On the Lambert W function

Corless, R.M, Gonnet G.H., Hare, D.E.G., Jeﬀrey D.J., and Knuth D.E. On the Lambert W function. Advances in Computational Mathematics , 5(1), (1996), 329–359

work page 1996

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Academic Press, 1981

Cs¨ org˝ o, Miklos, and R´ ev´ esz, P´ al.Strong approximations in probability and statistics. Academic Press, 1981

work page 1981

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Ruin probability-based initial capital of the discrete-time surplus process

Sattayatham, P., Sangaroon, K., and Klongdee, W. Ruin probability-based initial capital of the discrete-time surplus process. Variance, Advancing the Science of Risk, 7(1), (2013), 74–81

work page 2013

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A polytope related to empirical dis- tributions, plane trees, parking functions, and the associahedron

Stanley, Richard P., and Pitman, Jim. A polytope related to empirical dis- tributions, plane trees, parking functions, and the associahedron. Discrete & Computational Geometry , 27(4), (2002), 603–634

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