  We provide a general framework to study stochastic sequences related to
individual learning in economics, learning automata in computer sciences,
social learning in marketing, and other applications. More precisely, we study
the asymptotic properties of a class of stochastic sequences that take values
in $[0,1]$ and satisfy a property called "bounded expected relative hazard
rates." Sequences that satisfy this property and feature "small step-size" or
"shrinking step-size" converge to 1 with high probability or almost surely,
respectively. These convergence results yield conditions for the learning
models in B\"orgers, Morales, and Sarin (2004), Erev and Roth (1998), and
Schlag (1998) to choose expected payoff maximizing actions with probability one
in the long run.
