Applications of the central limit theorem and other weak limit theorems are closely connected to the rate of convergence to the limit distribution. The rate of convergence was studied by many authors; see [6] and the references therein. The simplest result in this direction is the Berry–Esseen inequality. Let $\{\xi _{n},n\ge 1\}$ be a sequence of independent identically distributed random variables (iidrvs) with distribution function $F(x)$, $E\xi _{i}=0$, and $D\xi _{i}={\sigma }^{2}<\infty $. Let $\beta _{3}=\int _{\mathbb{R}}|x{|}^{3}dF(x)$ be the absolute 3rd moment, $\varPhi _{n}(x)=P\{\frac{\xi _{1}+\xi _{2}+\dots +\xi _{n}}{\sigma \sqrt{n}}<x\}$, and let $\varPhi (x)$, $x\in \mathbb{R},$ be the standard normal distribution function. Then the Berry–Esseen inequality states that This estimate gives the rate of convergence $O({n}^{-1/2})$ and the asymptotic expansions of the distribution function of the sum of iidrvs in terms of semiinvariants, presented in the book [6]. The same rate of convergence was obtained by Paulauskas [5] in terms of pseudomoments. Let $\sigma =1$. Then the “pseudomoment” function is defined as $H(x)=F(x)-\varPhi (x)$, the (absolute) third pseudomoment is defined as $\nu =\int _{\mathbb{R}}|x{|}^{3}|dH(x)|$, and we have However, this rate of convergence is slow, for instance, from the point of view of financial applications. The conditions that allow one to improve the rate of convergence were formulated by several authors. After the introduction of pseudomoments in [10], they are widely used in limit theorems. Zolotarev [11] obtained very general estimates in the central limit theorem using a different type of pseudomoments. Studnyev [9] obtained the following estimate of the rate of convergence in terms of pseudomoments. Let $\{\xi _{n},n\ge 1\}$ be centered iidrvs with unit variance and characteristic function $f(t)$, $\mu _{k}=\int _{R}{x}^{k}dH(x)$ be the kth-order pseudomoment, and $V(x)={V_{-\infty }^{x}}H(z)$ be the variation of the function H.

We can see that the condition $\mu _{k}=0$, $3\le k\le r$ supplies the rate of convergence $O({n}^{\frac{-r+1}{2}})$. The rate of convergence was also studied in [1, 3, 7]. In our work, we use a different type of pseudomoments and get the same rate of convergence avoiding the Cramer condition. Instead, we impose the boundedness of the truncated pseudomoments and integrability of the characteristic function.

Let, as before, $\{\xi _{n},n\ge 1\}$ be a sequence of iidrvs with $E\xi _{i}=0$, $D\xi _{i}={\sigma }^{2}\in (0,\infty )$, distribution function $F(x)$, and characteristic function $f(t)$, and let $\varPhi _{n}(x)$, $x\in \mathbb{R}$, be the distribution function of the random variable

We assume that, for some $m\ge 3$, there exist the pseudomoments where $H(x)=F(x\sigma )-\varPhi (x)$. The truncated pseudomoments are defined as (“truncation from above”) and (“truncation from below”).

Note assumption (i) implies the existence of the density $p_{n}(x)$ of the random variable $S_{n}$. Also, let $\phi (x)$ be the density of the standard normal law.

First we prove two auxiliary results. Denote $\omega (t)=|f(\frac{t}{\sigma })-{e}^{-\frac{{t}^{2}}{2}}|$.

Now, denote $T_{1}(n,m)=\sqrt{-2\ln (2e\nu _{n}(m))}$. Then, in turn, we have that $\nu _{n}(m)=\frac{1}{2e}\exp \{-\frac{1}{2}{T_{1}^{2}}(n,m)\}$. Note also that condition (ii) implies $T_{1}(n,m)\ge 1$.

Now we are in position to prove the main results.

We give an example of application of Theorem 1. It is similar to the example of [12], p. 375, where the discrete distribution was considered. Define the distribution function $F(x)$ as where $0<\epsilon <1$, and $0<\theta <1$ is the root of the equation This equation has a unique solution because ${\int _{0}^{\epsilon }}{x}^{2}d\varPhi (x)\le \frac{{\epsilon }^{2}}{3}(\varPhi (\epsilon )-\frac{1}{2})$. Indeed, on one hand, and therefore, So On the other hand, and we immediately get that

It is obvious that the density function is bounded. Moreover, $F(x)$ is symmetric. Therefore, $\mu _{1}=0$ and $\mu _{3}=0$. Furthermore, taking into account (15), consider This means that $\mu _{2}=0$. Consider further the pseudomoments where ϵ can be chosen so that $\nu _{4}\le \frac{1}{2}{e}^{-\frac{3}{2}}$ and ${\nu _{n}^{(1)}}(3)\le \frac{1}{2}{e}^{-\frac{3}{2}}$. Then Hence, condition (ii) of Theorem 1 holds. Therefore, the function $F(x)$ satisfies the conditions of Theorem 1 with $m=3$.