Concentration inequalities for random tensors

We show how to extend several basic concentration inequalities for simple random tensors $X=x_1\otimes \cdots \otimes x_d$ where all $x_k$ are independent random vectors in ${\mathbb{R}}^n$ with independent coefficients. The new results have optimal dependence on the dimension $n$ and the degree $d$. As an application, we show that random tensors are well conditioned: $(1-o(1))n^d$ independent copies of the simple random tensor $X\in {\mathbb{R}}^{n^d}$ are far from being linearly dependent with high probability. We prove this fact for any degree $d=o(\sqrt{n\mathrm{/}\log n})$ and conjecture that it is true for any $d=O(n)$.