  A confidence sequence (CS) is a sequence of confidence intervals that is
valid at arbitrary data-dependent stopping times. These are useful in
applications like A/B testing, multi-armed bandits, off-policy evaluation,
election auditing, etc. We present three approaches to constructing a
confidence sequence for the population mean, under the minimal assumption that
only an upper bound $\sigma^2$ on the variance is known. While previous works
rely on light-tail assumptions like boundedness or subGaussianity (under which
all moments of a distribution exist), the confidence sequences in our work are
able to handle data from a wide range of heavy-tailed distributions. The best
among our three methods -- the Catoni-style confidence sequence -- performs
remarkably well in practice, essentially matching the state-of-the-art methods
for $\sigma^2$-subGaussian data, and provably attains the $\sqrt{\log \log
t/t}$ lower bound due to the law of the iterated logarithm. Our findings have
important implications for sequential experimentation with unbounded
observations, since the $\sigma^2$-bounded-variance assumption is more
realistic and easier to verify than $\sigma^2$-subGaussianity (which implies
the former). We also extend our methods to data with infinite variance, but
having $p$-th central moment ($1<p<2$).
