Collaborative recommendation is an information-filtering technique that
attempts to present information items (movies, music, books, news, images, Web
pages, etc.) that are likely of interest to the Internet user. Traditionally,
collaborative systems deal with situations with two types of variables, users
and items. In its most common form, the problem is framed as trying to estimate
ratings for items that have not yet been consumed by a user. Despite
wide-ranging literature, little is known about the statistical properties of
recommendation systems. In fact, no clear probabilistic model even exists
allowing us to precisely describe the mathematical forces driving collaborative
filtering. To provide an initial contribution to this, we propose to set out a
general sequential stochastic model for collaborative recommendation and
analyze its asymptotic performance as the number of users grows. We offer an
in-depth analysis of the so-called cosine-type nearest neighbor collaborative
method, which is one of the most widely used algorithms in collaborative
filtering. We establish consistency of the procedure under mild assumptions on
the model. Rates of convergence and examples are also provided.