[Paper Review] Asymptotics of the occupancy scheme in a random environment and its applications to tries

Silvia Businger|Zurich Open Repository and Archive (University of Zurich)|Sep 11, 2016

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper studies the asymptotic behavior of the occupancy scheme in a random environment, modeling $ m $ copies of a Markov chain via a nested box system. It derives almost sure limit laws for the time $ H_{m,j} $ when all $ m $ trajectories differ and $ G_{m,j} $ when all possible trajectories are observed at least $ j $ times, extending classical results on tries to random transition probabilities using martingale and large deviation techniques in a branching process framework.

ABSTRACT

Consider $ m $ copies of an irreducible, aperiodic Markov chain $ Y $ taking values in a finite state space. The asymptotics as $ m $ tends to infinity, of the first time from which on the trajectories of the $ m $ copies differ, have been studied by Szpankowski (1991) in the setting of tries. We use a different approach and model the $ m $ trajectories by a variant of the occupancy scheme, where we consider a nested sequence of boxes. This approach will enable us to extend the result to the case when the transition probabilities are random. We moreover use the same techniques to study the asymptotics as $ m $ tends to infinity of the time up to which we have observed all the possible trajectories of $ Y $ in random and nonrandom scenery.

Motivation & Objective

• To extend classical results on the height of $ K $-ary tries to the case where transition probabilities are random, modeling the system as an occupancy scheme in a random environment.
• To analyze the first time $ H_{m,j} $ when $ m $ trajectories of an irreducible, aperiodic Markov chain are all distinct, in a random environment.
• To study the saturation time $ G_{m,j} $, defined as the minimal time by which all possible trajectories of the Markov chain have been observed at least $ j $ times.
• To establish almost sure asymptotic limits for $ H_{m,j} $ and $ G_{m,j} $ as $ m \to \infty $, under general conditions on the random environment.
• To generalize Szpankowski's results on $ j $-tries to the setting of random transition matrices using a nested box model and extreme value analysis of box sizes.

Proposed method

• Models $ m $ i.i.d. copies of a Markov chain $ Y $ using a nested sequence of boxes indexed by a $ K $-ary tree, where each box corresponds to a path in the state space.
• Represents the size of a box at generation $ n $ as the product of transition probabilities along the path, forming a multiplicative cascade in a random environment.
• Uses a martingale argument and Borel-Cantelli-type lemmas to control the size of the largest and smallest boxes at each generation.
• Applies extreme value theory and exponential tail bounds on i.i.d. exponential random variables to estimate the number of balls (trajectories) required to fill or saturate boxes.
• Introduces auxiliary random variables $ \textbf{T}_n $ representing the number of trajectories reaching a given box at generation $ n $, and controls their growth via coupling with exponential variables.
• Establishes almost sure convergence of $ \frac{1}{\ln m} H_{m,j} $ and $ \frac{1}{\ln m} G_{m,j} $ by analyzing the interplay between box sizes and the number of balls thrown, using large deviation estimates.

Experimental results

Research questions

• RQ1 What is the almost sure asymptotic behavior of $ H_{m,j} $, the first time all $ m $ trajectories of a Markov chain are distinct, when transition probabilities are random?
• RQ2 How does the saturation time $ G_{m,j} $, the time by which all possible trajectories have been observed at least $ j $ times, behave asymptotically in a random environment?
• RQ3 Can the classical results on $ j $-tries with deterministic transition probabilities be extended to the case of random transition matrices using a nested box model?
• RQ4 What role do the maximum modulus eigenvalues $ \rho(j) $ of the matrix $ (p_{ik}^j) $ play in the asymptotic scaling of $ H_{m,j} $ when transition probabilities are random?
• RQ5 How do the sizes of the smallest and largest boxes at each generation influence the threshold times $ H_{m,j} $ and $ G_{m,j} $ in the random environment?

Key findings

• The paper establishes that $ \liminf_{m \to \infty} \frac{1}{\ln m} H_{m,j} \geq \zeta^* $ almost surely, where $ \zeta^* $ is a constant derived from the random environment's spectral properties.
• Under conditions (8), (9), and (10), the paper proves $ \lim_{m \to \infty} \frac{1}{\ln m} G_{m,j} = \zeta_* $ almost surely, with $ \zeta_* $ related to the minimal box size growth rate.
• The asymptotic behavior of $ H_{m,j} $ is governed by the largest box size at each generation, with the threshold time scaling logarithmically in $ m $, similar to the deterministic case but with a modified constant.
• The saturation time $ G_{m,j} $ is characterized by the smallest box size at each generation, and its asymptotic growth is determined by the inverse of the minimal box size's exponential decay rate.
• The results extend Szpankowski’s classical $ j $-trie results to the random environment case by showing that the logarithmic scaling of $ H_{m,j} $ and $ G_{m,j} $ persists, albeit with constants dependent on the random transition structure.
• The analysis relies on coupling the occupancy process with exponential random variables and applying Borel-Cantelli lemmas to control rare events in the size of extreme boxes.