  The traditional requirement for a randomized streaming algorithm is just {\em
one-shot}, i.e., algorithm should be correct (within the stated $\eps$-error
bound) at the end of the stream. In this paper, we study the {\em tracking}
problem, where the output should be correct at all times. The standard approach
for solving the tracking problem is to run $O(\log m)$ independent instances of
the one-shot algorithm and apply the union bound to all $m$ time instances. In
this paper, we study if this standard approach can be improved, for the
classical frequency moment problem. We show that for the $F_p$ problem for any
$1 < p \le 2$, we actually only need $O(\log \log m + \log n)$ copies to
achieve the tracking guarantee in the cash register model, where $n$ is the
universe size. Meanwhile, we present a lower bound of $\Omega(\log m \log\log
m)$ bits for all linear sketches achieving this guarantee. This shows that our
upper bound is tight when $n=(\log m)^{O(1)}$. We also present an
$\Omega(\log^2 m)$ lower bound in the turnstile model, showing that the
standard approach by using the union bound is essentially optimal.
