On the tails of Dickman-like perpetuities

Abstract: By using a probabilistic technique based on the exponential change of measure we find a precise tail asymptotic behavior of some perpetuities with distributions close to the Dickman distribution.

• The paper derives precise non-logarithmic tail asymptotics for Dickman-like perpetuities using exponential tilting and local limit approximation.
• It leverages saddle-point analysis, Fourier techniques, and a change-of-measure to obtain explicit polynomial prefactors in the asymptotic expansions.
• These findings bridge a longstanding gap in tail probability estimates, with practical implications for risk management and number theory.

Tail Asymptotics for Dickman-like Perpetuities: A Formal Summary

Introduction and Problem Setting

This work addresses the question of precise tail asymptotics for a class of perpetuities whose laws are close to the Dickman distribution. Specifically, the authors examine perpetuities of the form

$Z = Q_1 + \sum_{k\geq 1} M_1 \cdots M_k Q_{k+1},$

where $\{ (M_k,Q_k) \}$ are i.i.d. pairs, $M$ and $Q$ are independent, $M$ follows a Beta($\alpha$,1) distribution with $\alpha > 0$, and $Q$ is non-negative, with essential supremum $b \in (0,\infty)$. These "Dickman-like perpetuities" generalize the classic Dickman case ($\alpha=1, Q=1$), prominent in number theory and stochastic fixed-point equations.

Historically, logarithmic asymptotics for tails of such perpetuities were established (e.g., Goldie & Grübel 1996), showing Poissonian-type decay:

$\{ (M_k,Q_k) \}$0

However, explicit asymptotics—not just on the logarithmic scale—remained elusive under the general ("thin tail") regime.

Main Results: Precise Tail and Density Asymptotics

The central result is an explicit (i.e., precise, non-logarithmic) asymptotic expansion for the right tails and densities of Dickman-like perpetuities. The authors employ a probabilistic change-of-measure method, using exponential tilting and local limit approximation, to refine the earlier logarithmic description.

Let $\{ (M_k,Q_k) \}$1 and define

$\{ (M_k,Q_k) \}$2

For $\{ (M_k,Q_k) \}$3 (with $\{ (M_k,Q_k) \}$4 the critical point where $\{ (M_k,Q_k) \}$5 first becomes positive), the unique positive solution $\{ (M_k,Q_k) \}$6 to $\{ (M_k,Q_k) \}$7 governs the tail behavior.

The precise tail and density asymptotics for $\{ (M_k,Q_k) \}$8 as $\{ (M_k,Q_k) \}$9 are:

$M$0

and

$M$1

These results provide not only the exponential leading order but also the polynomial prefactors, filling a longstanding gap in the literature.

Analytical Approach and Technical Framework

The methodology leverages several advanced probabilistic and analytic features:

• Exponential Change of Measure: The authors tilt the original measure, focusing on the exponential tilted density, and analyze fluctuations under this new measure to obtain accurate local limit results.
• Selfdecomposability and Infinite Divisibility: The distribution of $M$2 is shown to be selfdecomposable, and its Lévy measure is derived explicitly, exploiting fine properties of infinitely divisible laws.
• Saddle-point and Fourier Techniques: A large deviations/saddle-point heuristic underlies the selection of $M$3, analogous to the characteristic function approach in classic works (de Bruijn, Tenenbaum) on Dickman-related variables.
• Regular and Gamma-like Tails: The results are compared against heavy-tailed and gamma-like (stretched-exponential) regimes, where slower decay or fundamentally different asymptotics arise.

A key technical achievement is characterizing the so-called "speed function" $M$4 explicitly, showing that, under the thin-tail regime, $M$5 up to subleading corrections. For $M$6 with an atom at $M$7 or regularly varying tails near $M$8, refined estimates for $M$9 are provided.

Examples and Explicit Asymptotics

The work includes several explicit examples:

• Discrete $Q$0 (e.g., $Q$1 with probability $Q$2): Asymptotic series for $Q$3 and the exponent are developed via Lagrange inversion, including terms in powers of $Q$4.
• $Q$5 with Gamma Distribution near $Q$6: The expansion incorporates the gamma parameter, affecting both the prefactor (exponent $Q$7) and the subleading asymptotic terms.
• Comparison to Classical Dickman Law: For $Q$8, the results recover and generalize the classical Dickman asymptotic, substantiated via analytic approximation.

Theoretical and Practical Implications

Theoretical Implications

These results rigorously establish the precise leading and next-to-leading order asymptotics of perpetuity tails under thin-tailed scenarios for a broad class of models. This bridges the gap between the previously understood logarithmic regime and exact, quantitative descriptions. The techniques affirm both the universality and the specificity (through prefactors and subexponential corrections) of Poisson-like tail decay in stochastic fixed-point equations of Dickman type. The analysis unifies probabilistic, analytic, and renewal-theoretic methods, with applications to number-theoretic structures, branching models, and random recursive trees.

Practical Implications

Knowledge of explicit tail probabilities and densities is essential for risk management in actuarial science, rare event simulation (where perpetuities model discounted sums of payments), and in algorithmic number theory (through Dickman-type distributions). The explicit asymptotics captured here enable precise tail risk quantification, rare event estimation, and refined numerical approximations for perpetuity-type models under light-tail constraints.

Open Directions and Future Research

• Multivariate Extensions: Generalizing the asymptotic analysis to vector-valued perpetuities or systems with interacting components.
• Refined Moderate Deviations: Quantifying transition regimes between regularly varying/gamma-like and thin tails.
• Algorithmic Implications: Leveraging explicit asymptotics for variance reduction in stochastic simulation of extreme events.

Conclusion

This paper provides exact tail and density asymptotics for Dickman-like perpetuities under thin-tail assumptions, leveraging probabilistic change-of-measure analysis and local limit approximation. The results constitute a comprehensive description of superexponential decay in these perpetuities, with broad implications for both the theory and applied modeling of perpetuities, stochastic recursions, and related objects.