On Linearity of $s$-Predictors

Abstract

Let $(\Omega, \mathscr{a}, P)$ be a probability space and $\mathscr{B} \subset \mathscr{a}$ be a $\sigma$-field. Let $s$ with $1 < s < \infty$ be fixed. If $f \in L_s(\Omega, \mathscr{a}, P)$ and $\mathscr{B} \subset \mathscr{a}$ is a $\sigma$-field, the unique element $g_f \in L_s(\Omega, \mathscr{a}, P)$ such that $\|f - g_f\|_s = \inf\{\|f - h\|_s: h \in L_s(\Omega, \mathscr{B}, P)\}$ is called the $s$-predictor of $f$ relative to the $s$-norm and the $\sigma$-field $\mathscr{B}$. Such $g_f$ exists and is uniquely determined. The mapping $P_s^\mathscr{B}: f\rightarrow g_f$ is called a prediction operator. The prediction operator is not necessarily a linear operator. The problem is to characterize the $\sigma$-fields $\mathscr{B}$ in terms of $P\mid\mathscr{a}$ for which $P_s^\mathscr{B}$ is a linear operator. We show that, for a fixed $\sigma$-field $\mathscr{B}$, the prediction operators $^\mathscr{B}_s$ are linear for all $s$ or for no $s \neq 2$. We give a necessary and sufficient condition for the linearity of $s$-predictors in terms of conditional expectations only. If moreover regular conditional probabilities given $\mathscr{B}$ exist, the $s$-predictors are linear if and only if the regular conditional probabilities of $P\mid\mathscr{a}$ given $\mathscr{B}$ consist only of measures concentrated on at most two points. Furthermore we obtain a simple criterion that $s$-prediction coincides with the usual conditional expectation (i.e., with 2-prediction): the conditional expectations of indicator functions may only assume the values $0, \frac{1}{2}$ and 1.