A liquid equity book gets no support from a theorem whose qualifying losses have E|X| = ∞. Liu and Mao study bounded-variation Lévy loss processes and identify exactly which jump measures make every allocation across n ⩾ 2 iid copies at least as risky as one copy, across every loss level and horizon. Risk is ordered by first-order stochastic dominance: at each loss threshold x, the spread book has at least the same probability of losing more than x. Every increasing utility function gives the same ranking. The mathematics is exact. The examples belong to catastrophe, cyber and operational losses, while the paper supplies no evidence that liquid equity returns meet its conditions.
The result
The setting is infinitely divisible distributions. For every n, an infinitely divisible law can be represented as the sum of n iid pieces, and each such law is the time-1 distribution of a Lévy process. The paper includes α-stable laws, Pareto and Fréchet laws with shape at most one, and their convolutions in this class.
A sequence of results for individual distributions provides the starting point. Chen, Embrechts and Wang (2025a) proved that a non-trivial convex combination of iid infinite-mean Pareto risks is strictly riskier under first-order dominance. Chen et al. (2026) obtained the phenomenon for iid compound Poisson processes, described by Liu and Mao as the only earlier work on processes. The new paper reaches general Lévy processes, dependent and heterogeneous multivariate components, and path functionals. It also asks for dominance at every horizon t > 0. We read it as a theory paper: there are no data, no sample and no mechanism for making money. Its use lies in risk sharing and allocating infinite-mean exposures.
Lévy processes are loss processes with stationary independent increments. A Gaussian component, a drift and a Lévy measure ν determine each one. The main results remove the Gaussian component, leaving ν and the drift to determine the comparison. The measure ν records how often jumps of each size arrive. A convex combination of iid copies remains a Lévy process, with a Lévy measure formed from rescaled copies of ν.
Two facts drive the proofs. Suppose one Lévy measure assigns more mass to large positive jumps than another. Its process is stochastically larger when its drift is no smaller and it assigns no more mass to large negative jumps. The authors use the bounded-variation version from Samorodnitsky and Taqqu. Conversely, P(X_t > x)/t converges to ν((x,∞)) as t shrinks. Requiring dominance at every horizon therefore imposes dominance on the jump measures themselves.
Theorem 1 follows. Set h(x) = ν((1/x,∞)), which is the arrival rate of jumps exceeding 1/x, and assume bounded-variation paths. A combination of n ⩾ 2 iid copies is never less risky than one copy at every t > 0 exactly when h is subadditive and negative jumps are absent. Reverse diversification order is the stronger result. Majorization orders weight vectors by the evenness of their allocations; under reverse diversification order, greater evenness can never lower risk. This property holds exactly when h is concave. Theorem 1 also derives the absence of negative jumps. With bounded variation, any positive mass on the negative half-line defeats the result.
The theorem characterizes the failure of the usual risk reduction from spreading capital.
Can finite-mean losses pass?
No. Proposition 2 rules out ordinary return models because either condition entails E|X| = ∞. Every substantive example in the paper lies beyond that boundary. Positive one-sided α-stable laws satisfy both properties strictly for α in (0,1). Compound Poisson losses with Pareto severities satisfy non-diversification for shape α in (0,1]. Fréchet and inverse-Gamma severities satisfy both properties with α in (0,1].
The authors present the infinite-mean implication as part of the intended regime and call it consistent with the literature. Their introduction names financial returns from some technological innovations, earthquakes, nuclear accidents, cyber losses and operational risk as applications involving infinite means. They cite Fama (1965) on stable laws in portfolio analysis as an early precedent. That defence works for loss classes where published fits support infinite means. The paper gives no evidence extending it to liquid equity returns.
Symmetry makes the restriction sharper. Theorem 2 leaves only the Cauchy process among symmetric Lévy processes, with ν(dx) = c|x|^{-2}dx. A combination of that process has the same distribution as one copy. Thus dominance at every horizon produces zero diversification effect throughout the qualifying symmetric class. The abstract likewise identifies the symmetric 1-stable process as the sole symmetric Lévy process with the non-diversification property.
The multivariate analysis narrows the admissible cases again. For α-stable vectors with α in (0,1) whose mass lies on the positive orthant, Theorem 4 requires a permutation-invariant spectral measure to obtain the reverse order. Put plainly, the components must be exchangeable. Proposition 7 considers independent stable components and requires one common index, α_1 =... = α_n. Two-sided jumps place equity returns beyond Theorem 1 and Theorem 4. The paper's two-sided cases either imply zero diversification effect under Theorem 3 or yield only Schur-type conditions under Proposition 14.
One jump-rate function does the work
For bounded-variation processes, the operational gain is compression. Deciding whether every split fails to help at every horizon requires two checks: negative jumps must be absent, and h must have the required shape. Anyone already using a jump model can apply that screen. The paper states its boundary directly. When bounded variation is dropped, the conditions remain necessary and may cease to be sufficient.
A tail index cannot answer the question by itself. As the paper observes, changing a Lévy measure on bounded sets leaves its tail asymptotics intact and can still destroy dominance. The authors write that "tail behavior alone does not determine the non-diversification phenomenon." A Hill estimate below one supplies only an initial screen.
Time horizon can reverse the answer. Proposition 13 constructs a compound Poisson process whose total jump mass is log 4 and for which ν({1}) = 3/4. At t = 1/10, dominance fails for every weight λ in (0,1). Given any ε in (0,1/2), dominance holds for all sufficiently large t when λ belongs to [ε, 1−ε]. The proof of failure is simple: P(X_s = 0) = 4^{-s}, while 4^{1/10} < 23/20. A result at one horizon need not survive at another, and this example makes the direction flip.
The paper provides no procedure for recovering ν from data. We did not find an estimator for h, a test for its subadditivity or concavity, or an analysis of turnover and costs. Its ranking is ordinal as well. A penalty is established, without a VaR or ES gap that would measure its size.
Where the extensions matter
Ruin, storage and volatility factors give the theory its practical reach. Corollary 2 covers n ⩾ 2 iid loss processes with bounded-variation paths, no negative jumps and the required subadditivity. Every convex combination then has weakly higher finite-horizon ruin probability over [0,T]. The statement holds for all u > 0 and c > 0. Corollary 3 transfers the ordering to storage content. Corollary 4 carries both phenomena into Lévy-driven Ornstein-Uhlenbeck variance and integrated variance. In the authors' words, "the mean-reverting mechanism of the OU process does not necessarily restore diversification benefits."
Several cited risk classes have been modeled or estimated with infinite means: financial returns from some technological innovations, earthquakes, nuclear accidents, cyber losses and operational risk. Corollary 2 gives a reinsurer sizing quota shares across catastrophe lines a precise warning. Replacing one full exposure with a 1/n share of each of n iid catastrophe exposures can increase finite-horizon ruin probability when their loss processes satisfy the condition. Corollary 4 supplies the corresponding ordering for a variance model assembled from iid OU factors driven by this kind of Lévy noise.
The pathwise statements cover less ground than the headline theorem. Propositions 10 and 12 establish non-diversification for running maxima and integrated convex functionals, without the reverse order. Proposition 12 additionally requires g to be increasing and convex. Proposition 11 assumes a nonnegative integrand.
Our long-only construction
The paper proves population statements about infinitely divisible processes, chiefly in the infinite-mean regime. A finite record of US stock returns cannot establish those properties. The stocks are neither identically distributed nor demonstrably independent Lévy processes. The theory supplies no estimation, selection, rebalancing or sizing rules. Our exercise is a descriptive construction inspired by the idea and has no bearing on the theorems. Because the paper reports no performance, there are no author results to compare with ours.
We used daily bars from 2020-01-01 to 2024-07-01. At the start of each year, the universe was the top 100 non-ADR US stocks by dollar volume. Every month, we fitted positive-loss tails using Hill and GPD methods at 5%, 7.5% and 10% tail fractions, requiring a joint common α in (0,1). The resulting cohorts contained 20 names. Equal weighting assigned 5% to each name; the concentrated vectors were capped at 10%. Orders filled at the next close through market-on-close. We charged commissions of $0.004 a share, subject to a $1 minimum and a cap of 1% of trade value. Every reported figure is net of those commissions. Market impact is not modeled.
Across the 54 months from January 2020 through July 2024, the construction returned 46.73% in total. Annualised volatility reached 23.26%, with a Sharpe ratio of 0.45 and a Sortino ratio of 0.59. The maximum drawdown was -40.39%, leaving a Calmar ratio of 0.22. This is weak performance for a fully invested long-only portfolio of liquid large caps. A 40% drawdown purchased less than half a unit of Sharpe.
Our 10% single-name ceiling excludes the 100% endpoints compared in Theorem 1. The implemented trade is consequently a mild tilt, far weaker than the contrast between one copy and a combination. Screening the 100 most heavily traded names for α in (0,1) also compels the method to locate infinite-mean tails, exactly where Proposition 2 places the theory. The fitted tails carry most of the argument in our run, so we would inspect that choice first. This result reflects one automated pass and should not be read as a verdict on the authors' work.
A larger backtest is unlikely to alter our conclusion. An estimator that recovered h from loss data and tested its subadditivity directly could do so, especially on a catastrophe or cyber book where the infinite mean required by Proposition 2 is plausible.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.