[Paper Review] The almost sure limits of the minimal position and the additive martingale in a branching random walk

Yueyun Hu|HAL (Le Centre pour la Communication Scientifique Directe)|Nov 22, 2012

Stochastic processes and statistical mechanics28 references3 citations

TL;DR

This paper establishes integral tests to characterize the almost sure lower limits of the minimal position and the upper limits of the additive martingale in a branching random walk under the boundary case. Using techniques from Aïdékon and Shi, it proves that the liminf of the minimal position minus 1/2 log n is -1 times log log n almost surely, and derives a precise integral criterion for the decay rate of the additive martingale.

ABSTRACT

Consider a real-valued branching random walk in the boundary case. Using the techniques developed by Aïdékon and Shi [5], we give two integral tests which describe respectively the lower limits for the minimal position and the upper limits for the associated additive martingale.

Motivation & Objective

• To characterize the almost sure lower limits of the minimal position in a branching random walk in the boundary case.
• To derive an integral test for the decay rate of the additive martingale in the boundary case.
• To resolve the open question of whether the liminf of the minimal position minus 1/2 log n can be bounded below by a deterministic sequence.
• To establish the exact rate of convergence for the additive martingale via a Seneta-Heyde norming.
• To extend results on fluctuation behavior of the minimal position and additive martingale using martingale techniques and size-biased measures.

Proposed method

• Applies size-biased martingale measures and spine decomposition techniques to analyze the branching random walk under the boundary case.
• Uses the derivative martingale $ D_n $ and its limit $ D_ au $ to control the behavior of the additive martingale $ W_n $.
• Employs a change of measure via $ bQ^{( heta)} $ to study the asymptotic behavior of $ rac{ ilde{W}_n^{( heta)}}{D_n^{( heta)}} $, where $ ilde{W}_n^{( heta)} $ is a size-biased version of $ W_n $.
• Applies Doob's $ L^2 $-inequality and variance bounds to prove almost sure convergence of $ rac{ ilde{W}_n^{( heta)}}{D_n^{( heta)}} $ to a constant $ heta $ under $ bQ^{( heta)} $.
• Derives a precise integral test involving $ rac{1}{t ext{exp}(f(t))} $ to determine whether $ bP^*(bM_n - rac{1}{2} ext{log} hinspace n < -f(n) ext{ i.o.}) $ is 0 or 1.
• Uses Borel-Cantelli arguments and exponential moment estimates to control the probability of rare events in the lower tail of the minimal position.

Experimental results

Research questions

• RQ1 What is the exact almost sure lower limit of $ bM_n - rac{1}{2} ext{log} hinspace n $ in the boundary case of branching random walks?
• RQ2 Can a deterministic sequence $ f(n) $ be found such that $ bP^*(bM_n - rac{1}{2} ext{log} hinspace n < -f(n) ext{ i.o.}) $ is either 0 or 1?
• RQ3 What is the precise rate of convergence of the additive martingale $ W_n $, and can it be normalized using the derivative martingale $ D_n $?
• RQ4 How does the behavior of the additive martingale relate to the minimal position through the spine decomposition and size-biased measures?
• RQ5 What integral condition determines whether the minimal position can drop below $ -f(n) $ infinitely often?

Key findings

• The liminf of $ bM_n - rac{1}{2} ext{log} hinspace n $ is almost surely $ - ext{log} hinspace ext{log} hinspace n $, i.e., $ bP^* ext{-a.s.} \ ext{liminf}_{n oty} rac{1}{ ext{log} hinspace ext{log} hinspace n} ig( bM_n - frac{1}{2} ext{log} hinspace n ig) = -1 $.
• An integral test determines whether $ bM_n - rac{1}{2} ext{log} hinspace n $ drops below $ -f(n) $ infinitely often: $ bP^*( ext{event i.o.}) = 1 $ iff $ extstyleigintsss^ty rac{dt}{t ext{exp}(f(t))} = ty $.
• The additive martingale $ W_n $ satisfies $ bP^* ext{-a.s.} \ ext{liminf}_{n oty} rac{ ext{log} hinspace W_n}{ ext{log} hinspace n} = -rac{1}{2} $, and $ bP^* ext{-a.s.} \ ext{liminf}_{n oty} rac{1}{ ext{log} hinspace ext{log} hinspace n} ext{log} hinspace W_n = -rac{1}{2} $.
• Under the size-biased measure $ bQ^{( heta)} $, $ rac{ ilde{W}_n^{( heta)}}{D_n^{( heta)}} o heta $ almost surely, where $ heta $ is a positive constant defined via the derivative martingale.
• The variance of $ rac{ ilde{W}_n^{( heta)}}{D_n^{( heta)}} $ decays polynomially as $ n^{- heta_2} $, enabling the use of Borel-Cantelli lemmas to prove almost sure convergence.
• The results confirm that the minimal position fluctuates at the $ ext{log} hinspace ext{log} hinspace n $ scale, and the additive martingale decays at a rate governed by $ ext{log} hinspace ext{log} hinspace n $, with precise probabilistic thresholds provided by the integral test.