Let $\{(X_i,Y_i), 1\leq i\leq n\}$ be $n$ i.i.d. random variable distributed as $(X,Y)$. and
$$P_{n} =\sum_{j=1}^{n}\mu_{nj}(x,y)$$
where $$\mu_{nj}(x,y)=\frac{\sqrt{nh_H\phi_{x}(h_{K})}\mathbb{E}(\beta_{1}^{2}K_{1})}{h_{H}(n-1)\mathbb{E}(\beta^{2}_{1}K_{i}K_{2}-\beta_{1}\beta_{2}K_{1}K_{2})}\left(K_{j}(H_{j}-{h_{H}f(y,x))-\mathbb{E}(K_{j}(H_{j}-h_H}f(y,x)))\right)$$ $K$ and $H$ are kernel functions,$\int H^2(t)\mathrm{d}t<+\infty$ and $h_K:= h_{n,K}$ (resp $h_H:= h_{n,H}$) is a sequence that decrease to zero as $n$ goes to infinity. $H_j=H\left(\frac{y-Y_j}{_Hh}\right)$, $K_{j}=K(h_{K}^{-1} d(x, X_j))$, $d$ is semi-metric and $\beta_{i}=\beta(X_i, x)$, $\beta$ is measurable function, $\phi_{x}(h_{K})=P(d(x, X)\leq h_{K})$. I want to proof, by using Lindeberg- Fuller theorem, that $$P_n \mbox{ converge in ditribution to } N(0,V_{HK}),$$ where $$V_{HK}(x,y)=\displaystyle\frac{M_2f(y,x)}{M_1^2}\left(\int H^2(t)\mathrm{d}t\right),$$
and $$M_j=K^{j}(1)-\int_{0}^{1}(K^{j})^{'}(s)\chi_{\theta,x}(s)\mathrm{d}s \,\,\mbox{for}\,\,\ j=1,2.$$ I have $$\mu_{nj}(x,y)$$ is a sequence of random variables indépendantes and identically distributed of mean $\mathbb{E}(\mu_{n1}(x,y))=0$ and $$\lim_{n\to+\infty}var(\mu_{n1}(x,y))=\lim_{n\to+\infty}E(\mu^2_{n1}(x,y))=V_{HK}$$ My question is how to prove that $$\sum_{j=1}^{n}\mathbb{E}\left(\mu^{2}_{nj}1_{(|\mu_{nj}|>\eta)}\right)=n\mathbb{E}(\mu^{2}_{n1}1_{(|\mu_{n1}|>\eta)})=\mathbb{E}\left((\sqrt{n}\mu_{n1})^{2}1_{(|\sqrt{n}\mu_{n1}|>\sqrt{n}\eta)}\right)$$ converge to zero. Thank you for helping me.