  Conditional Value-at-Risk (CVaR) is a widely used risk metric in applications
such as finance. We derive concentration bounds for CVaR estimates, considering
separately the cases of light-tailed and heavy-tailed distributions. In the
light-tailed case, we use a classical CVaR estimator based on the empirical
distribution constructed from the samples. For heavy-tailed random variables,
we assume a mild `bounded moment' condition, and derive a concentration bound
for a truncation-based estimator. Notably, our concentration bounds enjoy an
exponential decay in the sample size, for heavy-tailed as well as light-tailed
distributions. To demonstrate the applicability of our concentration results,
we consider a CVaR optimization problem in a multi-armed bandit setting.
Specifically, we address the best CVaR-arm identification problem under a fixed
budget. We modify the well-known successive rejects algorithm to incorporate a
CVaR-based criterion. Using the CVaR concentration result, we derive an
upper-bound on the probability of incorrect identification by the proposed
algorithm.
