[Paper Review] On the Law of Large Numbers for Discrete Fourier Transform

Na Zhang|arXiv (Cornell University)|Feb 23, 2016

Stochastic processes and financial applicationsEconomics, Econometrics and Finance9 references21 citations

TL;DR

This paper establishes the rate of convergence in the strong law of large numbers for the discrete Fourier transform (DFT) of identically distributed random variables with finite $p$th moment, where $1 < p < 2$. It extends the result to weakly dependent sequences satisfying $P(|X_n| /geq x) \leq P(|X_1| \geq x)$, proving almost sure convergence of $S_n(t)/n^{1/p} \to 0$ for almost all $t \in [-\pi, \pi)$, with precise moment conditions and tail probability bounds via truncation and martingale techniques.

ABSTRACT

We establish the rate of convergence in the strong law of large numbers of discrete Fourier Transform of the identically distributed random variables with finite moment of order p, where 1

Motivation & Objective

• To extend the strong law of large numbers to the discrete Fourier transform of dependent random variables.
• To establish almost sure convergence rates for $S_n(t)/n^{1/p}$ under finite $p$th moment conditions.
• To generalize results beyond i.i.d. sequences to weakly dependent sequences satisfying stochastic dominance.
• To provide moment-based tail bounds for the maximum of partial DFT sums.
• To unify results from Baum-Katz and Stoica via Fubini and Carleson-type convergence arguments.

Proposed method

• Truncation of random variables via $Y_k = X_k I\{|X_k| \leq k\}$ to control large deviations.
• Application of Carleson's Theorem on almost sure convergence of $\sum_{k=1}^\infty \frac{e^{ikt}Y_k}{k}$ for a.e. $t$.
• Use of Kronecker's Lemma to derive $\frac{1}{n}S_n^*(t) \to 0$ a.s. for a.e. $t$.
• Decomposition of $S_k(t)$ into truncated ($S_k'$) and remainder ($S_k''$) components for tail probability analysis.
• Application of Fubini's Theorem to interchange integration and summation over $t \in [-\pi, \pi)$.
• Use of moment bounds and inequalities (Markov, Cauchy-Schwarz) to control $\mathbb{E}[|S_k'|^2]$ and $\mathbb{E}[|S_k''|]$.

Experimental results

Research questions

• RQ1 Does the discrete Fourier transform of identically distributed random variables satisfy a strong law of large numbers?
• RQ2 What is the rate of convergence of $S_n(t)/n$ to zero for almost all $t \in [-\pi, \pi)$ under finite $p$th moment conditions?
• RQ3 Can the result be extended to sequences where $P(|X_n| \geq x) \leq P(|X_1| \geq x)$ but $X_n$ are not identically distributed?
• RQ4 What moment conditions ensure the summability of tail probabilities $\sum_n n^{p/r - 2} P(\max_{k \leq n} |S_k(t)| > \epsilon n^{1/r})$?
• RQ5 How does the DFT behavior compare to classical sums in terms of almost sure convergence and moment rates?

Key findings

• For $1 < p < 2$, if $\mathbb{E}|X_1|^p < \infty$, then $\sum_{n=1}^\infty n^{p/r - 2} P(\max_{1 \leq k \leq n} |S_k(t)| > \epsilon n^{1/r}) < \infty$ for almost all $t \in [-\pi, \pi)$ and all $1 \leq r \leq p$.
• Under the same moment condition, $S_n(t)/n^{1/p} \to 0$ almost surely for almost all $t \in [-\pi, \pi)$.
• The strong law $S_n(t)/n \to 0$ holds a.s. for a.e. $t$ if $\mathbb{E}|X_1| < \infty$, even without identical distribution, provided $P(|X_n| \geq x) \leq P(|X_1| \geq x)$.
• The result extends to weakly dependent sequences satisfying stochastic dominance, with the same convergence rates and tail bounds.
• The proof relies on truncation, Carleson's Theorem, Fubini's Theorem, and moment estimates via $\mathbb{E}[|X_k|^2 I\{|X_k| \leq n^{1/r}\}]$ and $\mathbb{E}[|X_k| I\{|X_k| > n^{1/r}\}]$.
• The summability condition implies almost sure convergence via Borel-Cantelli and a maximal inequality argument, with the final result derived using Fubini and integrability of the series in $t$.