A Note on Taylors Expansion and Mean Value Theorem With Respect to a Random Variable

We introduce a stochastic version of Taylors expansion and Mean Value Theorem, originally proved by Aliprantis and Border (1999), and extend them to a multivariate case. For a univariate case, the theorem asserts that suppose a real-valued function $f$ has a continuous derivative $f$ on a closed interval $I$ and $X$ is a random variable on a probability space $(Omega, mathcal{F}, P)$. Fix $a in I$, there exists a textit{random variable} $xi$ such that $xi(omega) in I$ for every $omega in Omega$ and $f(X(omega)) = f(a) + f(xi(omega))(X(omega) - a)$. The proof is not trivial. By applying these results in statistics, one may simplify some details in the proofs of the Delta method or the asymptotic properties for a maximum likelihood estimator. In particular, when mentioning there exists $theta ^ *$ between $hat{theta}$ (a maximum likelihood estimator) and $theta_0$ (the true value), a stochastic version of Mean Value Theorem guarantees $theta ^ *$ is a random variable (or a random vector).