  Computing market equilibria is a problem of both theoretical and applied
interest. Much research focuses on the static case, but in many markets items
arrive sequentially and stochastically. We focus on the case of online Fisher
markets: individuals have linear, additive utility and items drawn from a
distribution arrive one at a time in an online setting. We define the notion of
an equilibrium in such a market and provide a dynamics which converges to these
equilibria asymptotically. An important use-case of market equilibria is the
problem of fair division. With this in mind, we show that our dynamics can also
be used as an online item-allocation rule such that the time-averaged
allocations and utilities converge to those of a corresponding static Fisher
market. This implies that other good properties of market equilibrium-based
fair division such as no envy, Pareto optimality, and the proportional share
guarantee are also attained in the online setting. An attractive part of the
proposed dynamics is that the market designer does not need to know the
underlying distribution from which items are drawn. We show that these
convergences happen at a rate of $O(\tfrac{\log t}{t})$ or $O(\tfrac{(\log
t)^2}{t})$ in theory and quickly in real datasets.
