  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 $d = o(\sqrt{n/\log n})$ and conjecture that it is true for any
$d = O(n)$.
