A Note on Taylor's Expansion and Mean Value Theorem With Respect to a Random Variable

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 (Ω,F , P ). Fix a ∈ I , there exists a random variable ξ such that ξ(ω) ∈ I for every ω ∈ Ω and f(X(ω)) = f(a) + f (ξ(ω))(X(ω)− 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 the maximum likelihood estimator. In particular, when mentioning “there exists θ between θ̂ (a maximum likelihood estimator) and θ0 (the true value)”, a stochastic version of Mean Value Theorem guarantees θ is a random variable (or a random vector).