# Kronecker Products and Powers notation

We are all familiar with the notation for powers, which represent repeated multiplication: $$x^n = \underbrace{x \times x \times \cdots \times x}_{n \text{ times}}$$ Is there something similar to represent a repeated Kronecker product? $$\text{?} = \underbrace{\mathbf{x} \otimes \mathbf{x} \otimes \cdots \otimes \mathbf{x}}_{n \text{ times}}$$

The application for this is in formulating a concise representation for the higher-order moments of multivariate probability distributions. Just as univariate moments can be expressed as the expected value, $\mathbb E_{x}[x^n]$, I believe multivariate moments can be conveniently represented in matrix form as $\mathbb E_{\mathbf{x}}[\mathbf{x} \otimes \mathbf{x} \otimes \cdots \otimes \mathbf{x}]$, or in centralized form, $\mathbb E_{\mathbf{x}}[(\mathbf{x} - \boldsymbol\mu) \otimes (\mathbf{x} - \boldsymbol\mu) \otimes \cdots \otimes (\mathbf{x} - \boldsymbol\mu)]$. Please correct me if this formulation is incorrect.

• I use $\mathbf{x}^{\otimes n}$. – darij grinberg Feb 28 '15 at 17:50
• The body of a Question should be as self-contained as possible, not requiring the title for clarity. Please add a few words, perhaps mentioning the field in which this notation will find its application. – hardmath Feb 28 '15 at 18:30

$x^{\otimes\mkern1.5mu n}=\underbrace{x\otimes x\otimes \dots\otimes x}_{n\enspace\text{times}}$