  In insertion-only streaming, one sees a sequence of indices $a_1, a_2,
\ldots, a_m\in [n]$. The stream defines a sequence of $m$ frequency vectors
$x^{(1)},\ldots,x^{(m)}\in\mathbb{R}^n$ with $(x^{(t)})_i = |\{j : j\in[t], a_j
= i\}|$. That is, $x^{(t)}$ is the frequency vector after seeing the first $t$
items in the stream. Much work in the streaming literature focuses on
estimating some function $f(x^{(m)})$. Many applications though require
obtaining estimates at time $t$ of $f(x^{(t)})$, for every $t\in[m]$. Naively
this guarantee is obtained by devising an algorithm with failure probability
$\ll 1/m$, then performing a union bound over all stream updates to guarantee
that all $m$ estimates are simultaneously accurate with good probability. When
$f(x)$ is some $\ell_p$ norm of $x$, recent works have shown that this union
bound is wasteful and better space complexity is possible for the continuous
monitoring problem, with the strongest known results being for $p=2$ [HTY14,
BCIW16, BCINWW17]. In this work, we improve the state of the art for all
$0<p<2$, which we obtain via a novel analysis of Indyk's $p$-stable sketch
[Indyk06].
