[Paper Review] On the asymptotics of moments of linear random recurrences

Alexander Marynych|arXiv (Cornell University)|Oct 20, 2009

Stochastic processes and statistical mechanics27 references3 citations

TL;DR

This paper introduces a novel method based on iterative functions to analyze the asymptotics of moments in linear random recurrences, particularly applying it to the Poisson-Dirichlet coalescent. It proves that moments of the number of collisions and absorption time grow like powers of the iterated logarithm (log star), establishing a weak law of large numbers and revealing a rare asymptotic behavior not previously observed in such models.

ABSTRACT

We propose a new method of analyzing the asymptotics of moments of certain linear random recurrences which is based on the technique of iterative functions. By using the method, we show that the moments of the number of collisions and the absorption time in the Poisson-Dirichlet coalescent behave like the powers of the "log star" function which grows slower than any iteration of the logarithm, and thereby prove a weak law of large numbers. Finally, we discuss merits and limitations of the method and give several examples related to beta coalescents, recursive algorithms and random trees.

Motivation & Objective

• To develop a new analytical method for deriving first-order asymptotics of moments in linear random recurrences.
• To resolve the asymptotic behavior of moments in the Poisson-Dirichlet coalescent, particularly for the number of collisions and absorption time.
• To address the challenge of applying existing methods (e.g., generating functions, repertoire) to recurrences with extremely slow-growing asymptotics.
• To extend the method to broader classes of recurrences, including those arising in beta coalescents, recursive algorithms, and random trees.

Proposed method

• Transform the original recurrence into a probabilistic form with weights summing to one, defining a random index $ I_n $ with distribution $ p_{nk} = \mathbb{P}(I_n = k) $.
• Construct an iterative function $ g^* $ generated by a quadruple $ (h, g, x_0, k) $, where $ g(n) \sim \mathbb{E}[I_n] $, $ h(n) = B_n $, and $ k $ is a continuous function on $[0, x_0]$.
• Use the iterative function $ g^* $ to approximate the solution $ a_n $, with asymptotic equivalence $ a_n \sim f(n) $, where $ f $ is either an elementary function or $ f = g^* $.
• Apply Theorem 5.1 to show that the asymptotic behavior of $ a_n $ is determined by the inhomogeneous term $ b_n $, under conditions of divergence and regularity.
• Use Theorem 5.2 to relate the asymptotics of $ f $ to the behavior of $ h $, ensuring that $ f(x) \sim f_1(x) $ or $ f(x) = g^*(x) $ when $ f_1 $ is elementary.
• Verify the consistency condition $ f(\mathbb{E}[I_n]) - f(g(n)) = o(h(n)) $ to ensure convergence of the approximation.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the moments $ \mathbb{E}X_n^k $ and $ \mathbb{E}T_n^k $ in the Poisson-Dirichlet coalescent?
• RQ2 Can a new method be developed to analyze recurrences where standard techniques like generating functions or the repertoire method fail?
• RQ3 Why do the moments of the number of collisions and absorption time in the Poisson-Dirichlet coalescent exhibit log-star growth?
• RQ4 What conditions ensure that the solution to a linear recurrence diverges and admits a well-defined asymptotic profile?
• RQ5 How can the iterative function framework be generalized to other recurrences arising in coalescents, random trees, and recursive algorithms?

Key findings

• The moments $ \mathbb{E}X_n^k $ and $ \mathbb{E}T_n^k $ in the Poisson-Dirichlet coalescent grow asymptotically like powers of the iterated logarithm, specifically the log-star function $ \log^* n $, which grows slower than any finite iteration of the logarithm.
• The method proves a weak law of large numbers for the number of collisions and absorption time in the Poisson-Dirichlet coalescent, due to the slow growth of the moments.
• The log-star asymptotics are rare in applied probability; this paper identifies the Poisson-Dirichlet coalescent as a new model exhibiting this behavior, alongside only two other known models.
• The proposed method successfully handles recurrences where previous approaches—such as singular analysis of generating functions or the repertoire method—fail due to the extreme slow growth of the solution.
• The iterative function construction provides a systematic way to derive asymptotics, with convergence guaranteed under mild regularity and divergence conditions.
• The method is generalizable and has been extended to analyze beta coalescents, recursive algorithms, and random trees, demonstrating broad applicability beyond the original coalescent problem.