  We introduce a stochastic version of Taylor's 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).
