Suppose that $X_1,\dots, X_n$ are iid $P$ on $\mathcal{X}$. The empirical measure $\mathbb{P}_n$ is defined by $$\mathbb{P}_n:=\frac{1}{n}\sum_{i=1}^n\delta_{X_i}$$
For a real-valued function $f$ on $\mathcal{X}$ we write $$\mathbb{P}_n(f)=\int f d\mathbb{P}_n=\frac{1}{n}\sum_{i=1}^n f(X_i).$$
If $\mathcal{F}$ is a collection of real-valued functions defined on $\mathcal{X}$, then $\{\mathbb{P}_n(f): f\in \mathcal{F}\}$ is the empirical measure indexed by $\mathcal{F}$. Let's assume that $$ Pf:=\int fdP. $$ Let $\mathbb{G}_n := \sqrt{n}(\mathbb{P}_n − P)$ be empirical process and the collection of random variables $\{\mathbb{G}_n(f): f\in \mathcal{F}\}$ as $f$ varies over $\mathcal{F}$ is called empirical process indexed by $\mathcal{F}$.
Question: Define $\phi_\theta(x,y):=yI[x\ge \theta]$. Assume that $\hat{\theta}_n\to \theta_0$ convergence in probability as $n\to \infty$. Do we need to assume that $y$ is bounded so that $$ \mathbb{G}_n\phi_{\hat{\theta}_n}(x,y)=\mathbb{G}_n\phi_{\theta_0}(x,y)+o_p(1)? $$