A dominated differentiable-in-quadratic-mean submodel

Let $\mathcal{M}$ be a model of probability measures on a measurable space $(\mathcal{X},\mathcal{A})$, let $ϵ>0$, and let $t\mapsto P_t\in \mathcal{M}$ be a path for $0\le t<ϵ$ with $P_0=P$. It is dominated by a common measure when one supplied sigma-finite positive measure $\mu$ satisfies $P_t\ll \mu$ for every $t$ (Dominated statistical experiments). Write $p_t=d{P}_t\mathrm{/}d\mu$ and $p=p_0$ for simultaneously supplied nonnegative density versions; the existence and normalization of each density are the Radon–Nikodym properties in Likelihood densities and A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density. The Axiom of Choice is the library convention for selecting such a family of versions along a path (The Axiom of Choice). The DQM property below is unchanged by replacing any $p_t$ by another version, since each $L^2(\mu )$ norm is unchanged by a $\mu$-null modification.

The path is differentiable in quadratic mean at $0$, with score $s\in L^2(P)$, when

${\parallel \sqrt{p_t}-\sqrt{p}-\frac{t}{2}s\sqrt{p}\parallel }_{L^2(\mu )}=o(t)(t\downarrow 0)\mathrm{.}$

Here $o(t)$ means that the displayed norm divided by $t$ tends to $0$ as $t\downarrow 0$; $L^2(P)$ and $L^2(\mu )$ are quotient spaces modulo almost-everywhere equality (The space $L^p(\mu )$ as the quotient by null functions). The definition is directional. A two-sided regular submodel qualifies by restricting it to $t\ge 0$. No mean-zero condition is assumed in this definition.