Consistency of an asymptotically linear estimator

Question

An estimator $\hat{\boldsymbol{\gamma}}\triangleq \hat{\boldsymbol{\gamma}}(\mathbf{x}_1,\ldots,\mathbf{x}_M)$ of the $q$-dimensional vector $\boldsymbol{\gamma}_0$ is called asymptotically linear (AL) iff there exists a $q$-dimensional random function $\boldsymbol{\varphi}_0(\mathbf{x}) \triangleq \boldsymbol{\varphi}(\mathbf{x},\boldsymbol{\gamma}_0)$ such that:

1. $E_{p_0}\{\boldsymbol{\varphi}_0(\mathbf{x})\}=\mathbf{0}$,
2. $E_{p_0}\{\boldsymbol{\varphi}_0^T(\mathbf{x})\boldsymbol{\varphi}_0(\mathbf{x})\} < +\infty$,
3. The $q \times q$ matrix $\mathbf{C}_{p_0}(\boldsymbol{\varphi}_0)\triangleq E_{p_0}\{\boldsymbol{\varphi}_0(\mathbf{x})\boldsymbol{\varphi}_0^T(\mathbf{x})\}$ is non-singular,
4. For every AL estimator $\hat{\boldsymbol{\gamma}}$, we have: \begin{equation} \sqrt{M}(\hat{\boldsymbol{\gamma}} - \boldsymbol{\gamma}_0)=\frac{1}{\sqrt{M}}\sum_{m=1}^{M}\boldsymbol{\varphi}_0(\mathbf{x}_m) + o_P(1), \end{equation} where $o_{P_X}(1)$ is a term that converges in probability to zero as $M$ goes to infinity.

How can I obtain a formal proof of the consistency of $\hat{\boldsymbol{\gamma}}$?

Thanks!

Answer 1

From 4 $$\hat{\gamma}-\gamma = \frac{1}{M}\sum_{i=1}^M \phi_0(X_m) + o_P(1),$$ since $(1/\sqrt{M})o_P(1) = o_P(1/\sqrt{M})$ which converges in probability to 0. By the WLLN the average on the RHS of the displayed equation converges to its expectation which is 0. Thus the RHS of the displayed equation is $o_P(1) + o_P(1)$ which is $o_P(1)$, and hence $\hat{\gamma} \stackrel{P}{\rightarrow}\gamma$.