  This paper unifies the design and the analysis of risk-averse Thompson
sampling algorithms for the multi-armed bandit problem for a class of risk
functionals $\rho$ that are continuous and dominant. We prove generalised
concentration bounds for these continuous and dominant risk functionals and
show that a wide class of popular risk functionals belong to this class. Using
our newly developed analytical toolkits, we analyse the algorithm $\rho$-MTS
(for multinomial distributions) and prove that they admit asymptotically
optimal regret bounds of risk-averse algorithms under CVaR, proportional
hazard, and other ubiquitous risk measures. More generally, we prove the
asymptotic optimality of $\rho$-MTS for Bernoulli distributions for a class of
risk measures known as empirical distribution performance measures (EDPMs);
this includes the well-known mean-variance. Numerical simulations show that the
regret bounds incurred by our algorithms are reasonably tight vis-\`a-vis
algorithm-independent lower bounds.
