[Paper Review] A note on some sub-Gaussian random variables

Romeo Meštrović|arXiv (Cornell University)|Mar 7, 2018

Sparse and Compressive Sensing Techniques13 references3 citations

TL;DR

This paper investigates the sub-Gaussian properties of complex-valued discrete random variables $X_l(m,N)$, defined as sums of $m$-th roots of unity selected uniformly without replacement from $\{1, \dots, N\}$. It establishes tight upper and lower bounds for the expected values of $|X_l(m,N)|$, $|U_l(m,N)|$, and $|V_l(m,N)|$ using the sub-Gaussian norm $\|\cdot\|_{\psi_2}$, with explicit expressions derived via trigonometric summation identities and case analysis based on $m$ and $N$ modulo 4.

ABSTRACT

In [8] the author of this paper continued the research on the complex-valued discrete random variables $X_l(m,N)$ ($0\le l\le N-1$, $1\le M\le N)$ recently introduced and studied in [24]. Here we extend our results by considering $X_l(m,N)$ as sub-Gaussian random variables. Our investigation is motivated by the known fact thatthe so-called Restricted Isometry Property (RIP) introduced in [4] holds with high probability for any matrix generated by a sub-Gaussian random variable. Notice that sensing matrices with the RIP play a crucial role in Theory of compressive sensing. Our main results concern the proofs of the lower and upper bound estimates of the expected values of the random variables $|X_l(m,N)|$, $|U_l(m,N)|$ and $|V_l(m,N)|$, where $U_l(m,N)$ and $U_l(m,N)$ are the real and the imaginary part of $X_l(m,N)$, respectively. These estimates are also given in terms of related sub-Gaussian norm $\Vert \cdot\Vert_{ψ_2}$ considered in [28]. Moreover, we prove a refinement of the mentioned upper bound estimates for the real and the imaginary part of $X_l(m,N)$.

Motivation & Objective

• To extend prior work on complex-valued discrete random variables $X_l(m,N)$ by analyzing them as sub-Gaussian random variables.
• To establish sharp upper and lower bounds for $\mathbb{E}[|X_l(m,N)|]$, $\mathbb{E}[|U_l(m,N)|]$, and $\mathbb{E}[|V_l(m,N)|]$ in terms of the sub-Gaussian norm $\|\cdot\|_{\psi_2}$.
• To refine existing upper bound estimates for the real and imaginary parts of $X_l(m,N)$ using trigonometric summation techniques.
• To provide explicit closed-form expressions for the $L^\infty$-norms of the imaginary part $V_l(m,N)$ across all parity and modular cases of $m$ and $N$.

Proposed method

• The random variable $X_l(m,N)$ is defined as the sum of $m$ distinct $N$-th roots of unity selected uniformly at random without replacement.
• The real and imaginary parts $U_l(m,N)$ and $V_l(m,N)$ are analyzed separately using trigonometric identities and summation formulas.
• The analysis proceeds by case differentiation based on the parity of $m$ and $N \mod 4$, leading to distinct trigonometric expressions for the maximum absolute values of $V_l(m,N)$.
• The sub-Gaussian norm $\|X_l(m,N)\|_{\psi_2}$ is used to bound the tail behavior and derive expected value estimates.
• The identity $\sum_{k=a}^{b} \sin(2k\pi/N) = \frac{\sin(m\pi/N) \sin(\text{adjustment term})}{\sin(\pi/N)}$ is applied to compute extremal values of $V_l(m,N)$.
• The results are unified across cases by showing that $\|V_l(m,N)\|_{\infty}$ consistently equals $\frac{\sin(m\pi/N) \sin((2\lfloor N/4\rfloor + 1)\pi/N)}{\sin(\pi/N)}$ for even $m$, and $\frac{\sin(m\pi/N) \sin(2\lfloor(N+1)/4\rfloor\pi/N)}{\sin(\pi/N)}$ for odd $m$.

Experimental results

Research questions

• RQ1 What are the tightest possible upper and lower bounds for $\mathbb{E}[|X_l(m,N)|]$ in terms of the sub-Gaussian norm $\|\cdot\|_{\psi_2}$?
• RQ2 How do the expected magnitudes of the real and imaginary parts $U_l(m,N)$ and $V_l(m,N)$ scale with $m$ and $N$?
• RQ3 Can the $L^\infty$-norm of $V_l(m,N)$ be expressed in a unified closed form across all modular cases of $m$ and $N$?
• RQ4 What refinements can be made to existing upper bound estimates for $\mathbb{E}[|U_l(m,N)|]$ and $\mathbb{E}[|V_l(m,N)|]$?
• RQ5 How does the structure of the multiset $\Phi(l,N)$ influence the sub-Gaussian behavior of $X_l(m,N)$?

Key findings

• The $L^\infty$-norm of the imaginary part $V_l(m,N)$ is bounded by $\|V_l(m,N)\|_{\infty} = \frac{\sin(m\pi/N) \sin((2\lfloor N/4\rfloor + 1)\pi/N)}{\sin(\pi/N)}$ when $m$ is even and $N \equiv 1$ or $3 \pmod{4}$.
• For even $m$ and $N \equiv 0$ or $2 \pmod{4}$, the same expression holds: $\|V_l(m,N)\|_{\infty} = \frac{\sin(m\pi/N) \sin((2\lfloor N/4\rfloor + 1)\pi/N)}{\sin(\pi/N)}$, derived via case analysis of trigonometric sums.
• When $m$ is odd, $\|V_l(m,N)\|_{\infty} = \frac{\sin(m\pi/N) \sin(2\lfloor(N+1)/4\rfloor\pi/N)}{\sin(\pi/N)}$, with $M_2 = -M_1$ in all cases, simplifying the maximum to $M_1$.
• The expected value of $|X_l(m,N)|^2$ is exactly $\frac{m(N-m)}{N-1}$, confirming the variance structure under uniform sampling.
• The real and imaginary parts satisfy $\mathbb{E}[(U_l(m,N))^2] = \mathbb{E}[(V_l(m,N))^2] = \frac{m(N-m)}{2(N-1)}$ when $N \neq 2l$, confirming symmetry.
• The paper establishes that $X_l(m,N)$ is sub-Gaussian with $\|X_l(m,N)\|_{\psi_2} \lesssim \sqrt{\frac{m(N-m)}{N-1}}$, linking the result to the Restricted Isometry Property (RIP) in compressive sensing.