Hunter's positivity theorem and random vector norms

Abstract

A theorem of Hunter ensures that the complete homogeneous symmetric polynomials of even degree are positive definite functions. A probabilistic interpretation of Hunter's theorem suggests a broad generalization: the construction of so-called random vector norms on square complex matrices. This paper surveys these ideas, starting from the fundamental notions and developing the theory to its present state. We study numerous examples and present a host of open problems.

Figures (6)

• Figure 1: A bipartite graph (left) and a $V_1$-perfect matching (right).
• Figure 2: Unit circles for $\|\cdot\|_{\boldsymbol{X},d}$, in which $X_1$ and $X_2$ are Bernoulli with varying parameter $q$ and with $d=2$ (Left) and $d=10$ (Right).
• Figure 3: (Left) Unit circles for $\|\cdot\|_{\boldsymbol{X},d}$ with $d=1, 2, 4, 20$, in which $X_1$ and $X_2$ are standard normal random variables. (Right) Unit circles for $\| \cdot \|_{\boldsymbol{X},d}$ with $d=2, 4,8, 20$, in which $X_1$ and $X_2$ are Bernoulli with $q=0.5$.
• Figure 4: (Left) Unit circles for $\|\cdot\|_{\boldsymbol{X},d}$ with $d=1, 2, 4.5, 10, 18$, in which $X_1$ and $X_2$ are normal random variables with $\mu=\sigma=1$. (Right) Unit circles for $\|\cdot\|_{\boldsymbol{X},10}$, in which $X_1$ and $X_2$ are normal random variables with means $\mu=-2, -1, 0, 1, 6$ and variance $\sigma^2=1$.
• Figure 5: (Left) Unit circles for $\| \cdot \|_{\boldsymbol{X},2}$, in which $X_1$ and $X_2$ are independent Pareto random variables with $\alpha=2.1, 3, 4, 10$ and $x_m=1$. (Right) Unit circles for $\| \cdot \|_{\boldsymbol{X},d}$, in which $X_1$ and $X_2$ are independent Pareto random variables with $\alpha=5$ and $d=1, 2, 4$.

Theorems & Definitions (76)

• remark 2.1
• theorem 3.1: Hall
• proof
• theorem 3.2: Birkhoff
• proof
• theorem 3.3: Hardy, Littlewood, Pólya
• proof
• theorem 3.4: Schur--Ostrowski
• proof
• theorem 4.1: Hunter