The setup of the question is maximum likelihood estimation from stochastic processes. Let $\{(\Omega_{t}, \mathcal{A}_{t}, P_{\theta}^{t})\ \colon t\in T \}$ be a family of probability spaces, where we assume that $T=\mathbb{R}_{+} = \{x\in \mathbb{R} \ \colon x \geq 0\}$ and $\theta \in \Theta$, and open set of $\mathbb{R}^{k}$. We assume that for any $t$, $P_{\theta}^{t}$ is absolutely continuous with respect to a $\sigma$-finite measure $\lambda_{t}$ and call $p_{t}(\theta)$ the density of $P_{\theta}^{t}$ with respect to $\lambda_{t}$. Further, define ($a.e. \lambda_{t}$) the log likelihood function $l_{t}(\theta) = \operatorname{log}p_{t}(\theta)$ and let $l_{t}'(\theta)$ be the vector of first-order derivatives of $l_{t}(\theta)$. Let $M_{k}$ be the space of all $k\times k$ matrices equipped with the norm $\vert A \vert = \left(\operatorname{tr}\left(A^{T}A\right)\right)^{1/2}$ for $A\in M_{k}$. For the following, assume the existence of a family $\{A_{t}(\theta) \ \colon t \in T \}$ of non-random $k \times k$ matrices which are assumed to be continuous in $\theta$. In addition, we will need the definition of a uniformly stochastically bounded (u.s.b.) $\mathcal{A}_{t}$-measurable function.
A family $\{g_{t}(\theta) \ \colon t \in T \}$ of $\mathcal{A}_{t}$-measurable functions is u.s.b. if for any $\epsilon > 0$ and for any compact set $K \subset \Theta$, there exists real $\bar{c} > 0$ and $t_{0} \in T$ such that $P_{\theta}^{t}(\vert g_{t}(\theta)\vert>\bar{c})<\epsilon$ for all $t>t_{0}$ and $\theta \in K$.
Let now $\theta \in \Theta$ and consider a sequence $(\theta_{t})$ in $\Theta$ such that $\theta_{t} \to \theta$ as $t \to \infty$. Let $\left(P_{\theta_{t}}^{t}\right)$ be the respective sequence of probability measures. Define for some real constant $c > 0$ the set $$S_{t}^{c} = \left\{\phi \in \mathbb{R}^{k} \ \colon \vert \{A_{t}(\theta_{t})\}^{T}\left(\phi-\theta_{t}\right)\vert = c\right\}.$$ It can now be shown that if $f_{t}(\phi) := \left(\phi-\theta_{t}\right)^{T}l_{t}'(\phi)<0$ for any $\phi \in S_{t}^{c}$, then there exists a local maximum $\theta_{t}^{*}$ of $l_{t}(\theta)$ satisfying $\vert Y_{t}(\theta_{t})\vert \leq c$, where $Y_{t}(\theta_{t}) = \left\{A_{t}(\theta_{t}\right\}^{T}\left(\theta_{t}^{*} - \theta_{t}\right)$. This result shall be called Result (1) and will be used later.
Finally, define $$\pi_{t,c} := P_{\theta_{t}}^{t}\left(\sup_{\phi \in S_{t}^{c}}f_{t}(\phi) \geq 0 \right).$$ Now the following statement is subject to my question: Suppose that $$\limsup_{t\to\infty}\pi_{t,c} \to 0 \quad \text{as $c \to \infty$}.$$ Then there exists a local maximum $\theta_{t}^{*}$ of $l_{t}(\theta)$ with the family $\{Y_{t}(\theta)\}$ u.s.b..
I do not quite see how to proof the above statement.
My thoughts: From the assumption of the statement we know that for any given $\epsilon >0$ there exists $t_{0}$ such that for any $t>t_{0}$ $\pi_{t,c} < \epsilon$ for $c$ large enough. Then in this environment, $$ P_{\theta_{t}}^{t}\left(\exists \ \phi \in S_{t}^{c} \text{ s.t. } f_{t}(\phi) \geq 0 \right) \leq P_{\theta_{t}}^{t}\left(\sup_{\phi \in S_{t}^{c}}f_{t}(\phi) \geq 0 \right) < \epsilon,$$ which is equivalent to $$P_{\theta_{t}}^{t}\left(\forall \ \phi \in S_{t}^{c} \: f_{t}(\phi) < 0\right) > 1-\epsilon.$$ But according to Result (1) $$\left\{\forall \ \phi \in S_{t}^{c} \: f_{t}(\phi) < 0\right\} \subset \left\{\text{$\exists$ local maximum $\theta_{t}^{*}$ of $l_{t}(\theta)$ s.t. $\vert Y_{t}(\theta_{t}) \vert \leq c$}\right\}.$$ So we would have $$P_{\theta_{t}}^{t}\left(\text{$\exists$ local maximum $\theta_{t}^{*}$ of $l_{t}(\theta)$ s.t. $\vert Y_{t}(\theta_{t}) \vert \leq c$}\right) > 1-\epsilon.$$
For a reference of the above result, consider Lemma 4 in T.J. Sweetings uniform asymptotic normality of the maximum likelihood estimator.