  This paper considers a class of real-time decision making problems to
minimize the expected value of a function that depends on a random variable
$\xi$ under an unknown distribution $\mathbb{P}$. In this process, samples of
$\xi$ are collected sequentially in real time, and the decisions are made,
using the real-time data, to guarantee out-of-sample performance. We approach
this problem in a distributionally robust optimization framework and propose a
novel Online Data Assimilation Algorithm for this purpose. This algorithm
guarantees the out-of-sample performance in high probability, and gradually
improves the quality of the data-driven decisions by incorporating the
streaming data. We show that the Online Data Assimilation Algorithm guarantees
convergence under the streaming data, and a criteria for termination of the
algorithm after certain number of data has been collected.
