[Paper Review] Sharp moment estimates for polynomial martingales

E. Ostrovsky, L. Sirota|arXiv (Cornell University)|Oct 3, 2014

Mathematical functions and polynomials10 references15 citations

TL;DR

This paper establishes sharp non-asymptotic moment estimates for polynomial martingales using unconditional and relative moments of martingale differences. It derives optimal bounds of the form $ U(p;d,n) \leq \gamma(d) \cdot \frac{p^d}{(\ln p)^d} \cdot V_d(p) $ in the martingale case and $ \kappa(d) \cdot \frac{p^d}{(\ln p)^d} \cdot W_d(p) $ in the independent case, proving their exactness via lower bounds and applications to Grand Lebesgue spaces and tail probability estimates.

ABSTRACT

In this paper non-asymptotic moment estimates are derived for tail of distribution for discrete time polynomial martingale by means of martingale differences as a rule in the terms of unconditional and unconditional relative moments and tails of distributions of summands. We show also the exactness of obtained estimations.

Motivation & Objective

• To derive non-asymptotic, sharp moment estimates for discrete-time polynomial martingales with centered martingale differences.
• To express these estimates in terms of unconditional and relative $ L^p $ moments of the summands, avoiding dependence on higher-order moments.
• To establish the exactness of the bounds via lower estimates and counterexamples.
• To generalize prior results by improving the $ p^d / \ln^d p $ rate over previous $ p^d $ and $ p^d / \ln p $ rates.
• To provide tail probability estimates under sub-Gaussian-type tail assumptions using Grand Lebesgue space theory.

Proposed method

• The authors use recursive construction of constants $ \gamma(d) $ based on Osekowski’s constant $ K_{\text{Os}} \approx 15.7858 $, defined via $ \gamma(d+1) = \gamma(d) \cdot K_{\text{Os}} \cdot (1 + 1/d)^d $.
• They define polynomial martingales $ Q(d,n,b) = \sum_{\vec{i} \in I(d,n)} b(\vec{i}) \prod_{s=1}^d \xi(i_s,s) $ as homogeneous polynomials of degree $ d $ in martingale differences.
• Key norms are introduced: $ \mu_m(p) = \sup_i |\xi(i,m)|_p $, $ V_d(p) = \prod_{m=1}^d \mu_m(dp) $, and $ W_d(p) = \prod_{m=1}^d \mu_m(p) $ for independent and martingale cases respectively.
• The main inequality is derived using martingale inequalities, triangle inequalities, and Rosenthal-type estimates, with sharpness confirmed via examples and lower bounds.
• Grand Lebesgue space techniques are applied to connect tail decay conditions to moment growth, particularly under sub-Weibull tail assumptions.
• The relative moment norms $ \tilde{\mu}_m(p) $ are introduced for normalized variables, leading to normalized moment estimates $ \tilde{U}(p;d,n) \leq \gamma(d) \cdot \frac{p^d}{(\ln p)^d} \cdot \tilde{V}_d(p) $.

Experimental results

Research questions

• RQ1 Can sharp non-asymptotic $ L^p $ moment estimates be derived for polynomial martingales in terms of unconditional moments of the summands?
• RQ2 Is the $ p^d / (\ln p)^d $ rate of growth in the moment bound optimal, and can it be proven exact via lower bounds?
• RQ3 How do tail decay properties of martingale differences (e.g., sub-Weibull) affect the tail behavior of the polynomial martingale?
• RQ4 Can the moment estimates be refined using normalized (relative) moments to improve practical applicability?
• RQ5 What is the role of Osekowski’s constant and recursive construction in achieving sharpness?

Key findings

• The paper establishes the sharp upper bound $ U(p;d,n) \leq \gamma(d) \cdot \frac{p^d}{(\ln p)^d} \cdot V_d(p) $ for polynomial martingales, where $ \gamma(d) $ is recursively defined via Osekowski’s constant.
• For independent summands, the bound $ U(p;d,n) \leq \kappa(d) \cdot \frac{p^d}{(\ln p)^d} \cdot W_d(p) $ is proven sharp, with $ \kappa(d) $ defined analogously to $ \gamma(d) $.
• Lower bounds confirm the optimality: $ K_I(p;d) \geq \frac{C(d) \cdot p^d}{\ln^d p} $, showing the $ p^d / \ln^d p $ rate cannot be improved.
• Under sub-Weibull tail assumptions $ \mathbb{P}(|\xi(i,m)| \geq x) \leq \exp(-C_1 x^q (\ln x)^{-qr}) $, the tail of $ Q(d,n,b) $ decays as $ \exp(-C_2 x^{q/(dq+1)} (\ln x)^{-(q(r-d))/(dq+1)}) $.
• The normalized moment estimate $ \tilde{U}(p;d,n) \leq \gamma(d) \cdot \frac{p^d}{(\ln p)^d} \cdot \tilde{V}_d(p) $ is derived, improving practical usability by using relative moments.
• The results are exact in the sense that the constants $ \gamma(d) $, $ \kappa(d) $, and the $ p^d / \ln^d p $ rate are shown to be optimal via counterexamples and asymptotic analysis.