Stochastic Linear Regression

We consider the following formulation of Lai, Robbins and Wei (1979), and Lai and Wei (1982). Consider the following regression problem,

for n=1,2,... where \epsilon_n are unobservable random errors and \beta_1,...,\beta_p are unknown parameters.

Typically for a regression problem, it is assumed that inputs x_{1},...,x_{n} are given and errors are IID random variables. However, we now want to consider a setting where we sequentially choose inputs x_i and then get observations y_i, and errors \epsilon_i are a martingale difference sequence with respect to the filtration \mathcal F_i generated by \{ x_j, y_{j-1} : j\leq i \}.

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