Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 36 / 2 / a / Solution

Choose a collection of statistical paths through $P_f$ that are differentiable in quadratic mean. A statistical tangent set at $P_f$ is the set of their score functions. In particular each member belongs to $L_0^2(P_f)$, by the mean-zero score identity under quadratic-mean differentiability.

A statistical tangent set records which first-order directions the chosen statistical paths can realize. Its closed linear span in L2 space is the statistical tangent space. The tangent set consists of attainable scores; the tangent space also includes their linear combinations and $L^2(P_f)$ limits.