[Paper Review] Optimized estimates of the regularity of the conditional distribution of the sample mean

Victor Chulaevsky|arXiv (Cornell University)|Aug 19, 2014

Stochastic processes and financial applications2 references3 citations

TL;DR

This paper provides optimized estimates for the regularity of the conditional distribution of the sample mean given fluctuations in i.i.d. random variables, replacing probabilistic bounds with distributional ones to achieve the optimal regularity exponent. The key result establishes a sharp bound of the form $\mathbb{P}(\xi_N \in I_s(\eta)) \leq C N s$ for smooth, compactly supported densities with bounded logarithmic derivatives, improving prior results in the context of eigenvalue concentration in random operators.

ABSTRACT

We give an improved estimate for the regularity of the conditional distribution of the empiric mean of a finite sample of IID random variables, conditional on the sample "fluctuations", extending the well-known property of Gaussian IID samples. Specifically, we replace the bounds in probability, established in our earlier works, by those in distribution, and this results in the optimal regularity exponent in the final estimate.

Motivation & Objective

• To improve regularity estimates for the conditional distribution of the sample mean given fluctuations in i.i.d. random variables.
• To replace previous probabilistic bounds with stronger distributional bounds to achieve optimal regularity exponents.
• To establish sharp Wegner-type bounds for eigenvalue concentration in multi-particle Anderson-type Hamiltonians.
• To analyze the conditional density of the sample mean when the underlying distribution has smooth, compactly supported density with bounded logarithmic derivative.

Proposed method

• Introduces a partition of the sample space into cubes based on a discretization of the support of the density, with mesh size $M_N = N^2$.
• Uses the bounded logarithmic derivative condition to control the relative variation of the joint density within each cube.
• Establishes equivalence between the original measure and a uniform measure on each cube up to a factor $e^{\pm 2\alpha_N}$ with $\alpha_N = \ell N^{-1}$.
• Applies the bound $\mathbb{P}_{\mathbf{k}}(\xi \in I_s(\eta)) \leq C N s$ on each cube and combines via union bound.
• Uses integration by parts and Stieltjes integral estimates to control tail behavior and refine the final bound.
• Derives the final estimate $\mathbb{P}(\xi_N \in I_s(\eta)) \leq C N s$ for $s \in (0, \ell N^{-2})$ under smoothness assumptions.

Experimental results

Research questions

• RQ1 What is the optimal regularity exponent for the conditional distribution of the sample mean given fluctuations in i.i.d. random variables?
• RQ2 How can probabilistic bounds on the conditional density be improved to achieve sharp estimates in the context of random operator theory?
• RQ3 To what extent does the bounded logarithmic derivative of the density ensure regularity of the conditional distribution of the sample mean?
• RQ4 Can sharper Wegner-type bounds be derived for eigenvalue concentration in multi-particle systems using refined regularity estimates?
• RQ5 How does the choice of discretization and measure equivalence within cubes affect the final bound on the conditional probability?

Key findings

• The paper achieves the optimal regularity exponent in the bound for the conditional density of the sample mean, improving upon earlier probabilistic estimates.
• For smooth, compactly supported densities with bounded logarithmic derivative, the conditional probability satisfies $\mathbb{P}(\xi_N \in I_s(\eta)) \leq C N s$ for $s \in (0, \ell N^{-2})$, with $C$ depending on the distribution and support length.
• The bound is derived via a cube partitioning scheme with $M_N = N^2$ and measure equivalence within each cube up to a factor $e^{\pm 2\alpha_N}$, $\alpha_N = \ell N^{-1}$.
• The method replaces earlier bounds in probability with bounds in distribution, leading to a tighter and optimal estimate.
• The result is applied to derive improved Wegner-type bounds for eigenvalue concentration in random DSO Hamiltonians.
• The final bound $\mathbb{P}(\xi_N \in I_s(\eta)) \leq C N s$ is shown to be sharp under the given assumptions, with $C$ finite and dependent on the density and support length.