  We study the classical problem of moment estimation of an underlying vector
whose $n$ coordinates are implicitly defined through a series of updates in a
data stream. We show that if the updates to the vector arrive in the
random-order insertion-only model, then there exist space efficient algorithms
with improved dependencies on the approximation parameter $\varepsilon$. In
particular, for any real $p > 2$, we first obtain an algorithm for $F_p$ moment
estimation using $\tilde{\mathcal{O}}\left(\frac{1}{\varepsilon^{4/p}}\cdot
n^{1-2/p}\right)$ bits of memory. Our techniques also give algorithms for $F_p$
moment estimation with $p>2$ on arbitrary order insertion-only and turnstile
streams, using $\tilde{\mathcal{O}}\left(\frac{1}{\varepsilon^{4/p}}\cdot
n^{1-2/p}\right)$ bits of space and two passes, which is the first optimal
multi-pass $F_p$ estimation algorithm up to $\log n$ factors. Finally, we give
an improved lower bound of $\Omega\left(\frac{1}{\varepsilon^2}\cdot
n^{1-2/p}\right)$ for one-pass insertion-only streams. Our results separate the
complexity of this problem both between random and non-random orders, as well
as one-pass and multi-pass streams.
