Lindeberg's Replacement Bound

In the setting of Convergence of Probability Measures and Random Elements: Standing Notation, let $(\Omega,\mathcal{F},P)$ be a probability space, let $n\in\mathbb{N}$, and let $Y_{1},\dots,Y_{n},Z_{1},\dots,Z_{n}\in\mathcal{L}^{2}(P)$ be random variables such that the family $(W_{i})_{i\in[2n]}$ with $W_{k}=Y_{k}$ and $W_{n+k}=Z_{k}$ for $k\in[n]$ is independent and

$$\mathbb{E}Y_{k}=\mathbb{E}Z_{k}=0,\qquad\mathbb{E}[Y_{k}^{2}]=\mathbb{E}[Z_{k}^{2}]=1\qquad(k\in[n]).$$

Let $a>0$ be a real number. Let $f:\mathbb{R}\to\mathbb{R}$ be a function of the class used in Smooth Test Functions Characterize Convergence in Distribution of Real Random Variables (i), that is, a function such that $f$ and its successive derivatives $f'$ and $f''$ are differentiable, $f'''$ is continuous, and there is a real $M$ with $|f|,|f'|,|f''|,|f'''|\le M$ on $\mathbb{R}$. Let $M_{2},M_{3}\in\mathbb{R}$ satisfy $|f''(x)|\le M_{2}$ and $|f'''(x)|\le M_{3}$ for every $x\in\mathbb{R}$, and let $g:\mathbb{R}\to\mathbb{R}$ be the function

$$g(y)=\min\bigl\{M_{2}y^{2},\tfrac12M_{3}a|y|^{3}\bigr\}.$$

(i) Integrability. For every $k\in[n]$, the random variables $g(Y_{k})$ and $g(Z_{k})$ are integrable.

(ii) Replacement bound.

$$\Bigl|\mathbb{E}f\Bigl(a\sum_{k=1}^{n}Y_{k}\Bigr)-\mathbb{E}f\Bigl(a\sum_{k=1}^{n}Z_{k}\Bigr)\Bigr|\le a^{2}\sum_{k=1}^{n}\bigl(\mathbb{E}g(Y_{k})+\mathbb{E}g(Z_{k})\bigr).$$