[Paper Review] Bide - Side Exponential and Moment Inequalities for Tails of Distributions of Polynomial Martingales

E. Ostrovsky|ArXiv.org|Jun 25, 2004

Stochastic processes and financial applications12 references15 citations

TL;DR

This paper establishes non-asymptotic exponential and moment inequalities for the tail distributions of polynomial martingales, deriving uniform bounds over coefficient sequences in the unit sphere. It shows that the tail decay of the polynomial martingale $ Q_d $ is governed by a generalized moment index $ M(d, \vec{q}) = \left(\frac{d}{2} + \sum_{m=1}^d \frac{1}{q(m)}\right)^{-1} $, with sharp bounds in terms of the marginal tail behavior of the martingale differences $ \xi(i,m) $, extending to applications in weak compactness and U-statistics.

ABSTRACT

In this paper non-asymptotic exponential estimates are derived for the tail distribution of polynomial martingale differences in terms unconditional tails distributions of summands. Applications are considered in the theory of polynomials on independent random variables, to the theory of U-statistics, multiply martingale series and in the theory of weak compactness measures on the Banach spaces. Partially supported by the Israel Ministry of Absorbtion Mathematics Subject Classification (2000): 47A45, 47B10, 60F10, 60G42.

Motivation & Objective

• To derive non-asymptotic, uniform-in-coefficient exponential bounds for the tail distribution of polynomial martingales of order $ d $.
• To express the tail decay of $ Q_d $ in terms of the marginal tail and moment behavior of the underlying martingale differences $ \xi(i,m) $.
• To establish sharp bounds that are optimal in the sense of matching lower and upper bounds up to constants.
• To extend the results to applications in weak compactness of random fields and $ U $-statistics via metric entropy conditions.

Proposed method

• Define $ Q_d = \sum_{I \in I(d,n)} b(I) \xi(I) $, where $ \xi(I) = \prod_{m=1}^d \xi(i_m, m) $, with $ \xi(i,m) $ centered martingale differences.
• Introduce the index $ M(d, \vec{q}) = \left(\frac{d}{2} + \sum_{m=1}^d \frac{1}{q(m)}\right)^{-1} $, which governs the tail decay rate of $ Q_d $.
• Use the space $ G(q) $ and generalized $ G(q,r) $ spaces to characterize the tail and moment behavior of $ \xi(i,m) $.
• Establish uniform bounds on $ S(\vec{q}, x) = \sup_{b \in B} \sup_{\{\xi(i,m)\}} T(Q_d, x) $, where $ T(\cdot, x) $ is the symmetric tail probability.
• Apply metric entropy and majorizing measure techniques to derive conditions for almost sure uniform convergence of random fields $ \tau(t) $ and $ \zeta(t) $.
• Use the identity for differences of products to bound $ \zeta(t_1) - \zeta(t_2) $ in terms of differences in $ \xi(i,m,t) $, enabling application of the main inequality.

Experimental results

Research questions

• RQ1 What is the sharp non-asymptotic tail decay rate of a homogeneous polynomial martingale $ Q_d $ of degree $ d $, in terms of the marginal tail behavior of its components?
• RQ2 Can the tail behavior of $ Q_d $ be bounded uniformly over all coefficient sequences $ b \in B(d,n) $, given only marginal tail and moment conditions on $ \xi(i,m) $?
• RQ3 How do the results extend to random fields $ \tau(t) $ and $ \zeta(t) $ with parameter-dependent coefficients or random fields?
• RQ4 Under what metric entropy conditions on the parameter space $ V $ does the random field $ \zeta(t) $ converge uniformly and have sub-Gaussian-type tail decay?

Key findings

• For $ \sup_i T(\xi(i,m), x) \leq \exp\left(-(x/K(m))^{q(m)}\right) $, the tail of $ Q_d $ satisfies $ \exp\left(-\left[x/(C_1 K)\right]^M\right) \leq S(\vec{q}, x) \leq \exp\left(-\left[x/(C_2 K)\right]^M\right) $ with $ M = M(d, \vec{q}) $.
• The bound is sharp: inequality (1.6) cannot be improved, as the lower bound matches the upper bound up to constants.
• The result extends to random fields $ \tau(t) $, where $ T(\sup_t |\tau(t)|, x) \leq \exp\left(-C_{10} x^{M(d,q)}\right) $ under a metric entropy condition on $ (V, r_1) $.
• For parameter-dependent $ \xi(i,m,t) $, the same tail decay holds under a similar entropy condition on $ (V, r_2) $, with $ r_2(t_1,t_2) = \sum_m \sup_i ||\xi(i,m,t_1) - \xi(i,m,t_2)||_{q(m)} $.
• The family of distributions of $ \zeta_\alpha(t) $ is weakly compact under appropriate boundedness and entropy conditions, via majorizing measure techniques.