On expansion of the characteristic function of symmetric distributions

The article presents a new asymptotic expansion of the characteristic function of a symmetric distribution with a clear assessment of the accuracy of the residual part of the asymptotic expansion. The asymptotic expansion of the characteristic function can be used to build a new asymptotic expansion in the Central limit theorem with an explicit estimate for the remainder. In the article the main part of the decomposition of the characteristic function contains the moments of the Chebyshev–Hermite.
When constructing asymptotic expansions in the Central limit theorem, the expansions of the characteristic function are often used. For example, a Taylor expansion of the characteristic function
$$f(t)=\sum^{m}_{k=0}\frac{\alpha_k}{k!}(it)^{k}+\rho_{m}(t), \,(1)$$
where $\alpha_k$ — the $k$-th moment of the probability distribution $P$ with characteristic function $f(t)$ and $\left|\rho_{m}(t)\leq\frac{\beta_{m+1}}{(m+1)!}|t|^{m+1}\right|$, where $\beta_{m+1}$ —– the absolute moments of order $(m+1)$.
It would be quite natural to assume that $\displaystyle{f(t)=\sum^{m+1}_{k=0}\frac{\alpha_k}{k!}(it)^{k}+\rho_{m+1}(t)}$ is better (1). However, if the moment $\alpha_{m+2}$ does not exist, then the question arises of estimating the remainder.
Asymptotic expansions using the last known moment in the main part of the decomposition were proposed by Prawitz [9] and investigated by Shevtsova [10]. A modification of these expansions for the characteristic function of symmetric distributions (we further assume that $\alpha_{2j+1}$ for $j=0,1,\ldots$ and $(m + 1)$ — even integer value) is proposed by Senatov [2]
$$f(t)=\sum^{m}_{j=0}\frac{\alpha_j}{j!}(it)^{j}+\lambda\frac{\alpha_{m+1}}{(m+1)!}(it)^{m+1}+\gamma(t)\overline{\lambda}\frac{\alpha_{m+1}}{(m+1)!}(it)^{m+1}, \,(2)$$
where $\lambda\in[0,1], \overline{\lambda}=\max{\lambda,1-\lambda}$, a function $\gamma$ such that $|\gamma|\leq1$.
This paper presents new expansions of the characteristic functions of symmetric distributions similar to those constructed in [8] (see also [3; 7]), but with a different estimate of the remainder. To do this, we will use the formula (2) and normalized moments of the Chebyshev – Hermite
$$\theta_{l}=\sum^{\lfloor l/2\rfloor}a_{l-2j}b_{2j}.$$
where $\displaystyle{a_{k}=\frac{\alpha_k}{k!}, k=0,1,...,m+1}$ and $\displaystyle{b_{2j}=\frac{(-1)^j}{2^jj!}}$,$j=0,1,... .$. In this context, also occur incomplete moments of the Chebyshev – Hermite of order $l\geq(m+2)$
$$\theta^{(m+1)}_l=\sum^{\lfloor l/2\rfloor}_{j=1;2 j\geq l-m-1}a_{l-2j}b_{2j},\, l>m+1.$$
The asymptotic expansion of the characteristic function (4) from the following statement can be used to build a new asymptotic expansions in the Central limit theorem with an explicit estimate for the remainder. For example, it is enough to repeat the course of evidence from [1; 4–6] using a new expansion (4).
Theorem. Suppose that the characteristic function of symmetric distributions $f(t)$ has a moment of even order $(m+2)\geq2$. Then
$$\displaystyle{f(t)=e^{-t^2/2}\left(\sum^{m+1}_{k=0}\theta_k(it)^k-(1-\lambda)a_{m+1}(it)^{m+1}\right)+R_m,\,(4)}$$
and the remainder term satisfies the inequality
$$|R_m|\leq\overline{\lambda}\cdot a_{m+1}\cdot|t|^{m+1}+\|\theta_m\|\cdot|b_2|\cdot|t|^{m+2}+\lambda\cdot a_{m+1}\cdot|b_2|\cdot|t|^{m+3}+\|\theta_{m-1}\|\cdot b_4\cdot|t|^{m+3},$$
where
$$\|\theta_l\|=\Sigma^{\lfloor l/2\rfloor}_{j=0}|a_{l-2j}|\cdot|b_{2j}|,\,0\leq l\leq n.$$
Remark 1. The expansion (4) can be written in the form
$$f(t)=e^{-t^2/2}\left(\sum^m_{k=0}\theta_k(it)^k+\theta^{(m+1,\lambda)}_{m+1}(it)^{m+1}\right)+R_m.\,(7)$$
where $\displaystyle{\theta^{(m+1,\lambda)}_{m+1}=\lambda a_{m+1}+\theta^{(m-1)}_{m+1}}$ (for more details, see [2]).
Remark 2. For $\displaystyle{\lambda=\frac{1}{2}}$ expansion (4) takes the form
$$f(t)=e^{-t^2/2}\left(\sum^{m+1}_{k=0}\theta_k(it)^k-\frac{a_{m+1}}{2}(it)^{m+1}\right)+R_m,\,(8)$$
where
$$|R_m|\leq \frac{a_{m+1}}{2}\cdot |t|^{m+1}+\|\theta_m\|\cdot|b_2|\cdot|t|^{m+2}+\left(\frac{a_{m+1}}{2}\cdot b_2+\|\theta_{m-1}\|\cdot|b_4|\right)\cdot|t|^{m+3}.$$
The minimum value of $\overline{\lambda}=\max{\lambda,1-\lambda}$ on the closed interval $[0,1]$ is $\displaystyle{\frac{1}{2}}$ at a point $\displaystyle{\lambda=\frac{1}{2}}.$ Therefore, this assessment $\displaystyle{\lambda=\frac{1}{2}}.$ is minimal.
Remark 3. For $\lambda=1$ expansion (4) takes the form
$$f(t)=e^{-t^2/2}\Sigma^{m+1}_{k=0}\theta_k(it)^k+R_m,\,(9)$$
where
$$|R_m|\leq a_{m+1}\cdot|t|^{m+1}+\|\theta_m\|\cdot|b_2|\cdot|t|^{m+2}+(a_{m+1}\cdot|b_2|+\|\theta_{m-1}\|\cdot b_4)\cdot|t|^{m+3}.\, (10)$$
Asymptotic expansion (9) coincides with expansions from (7), which are obtained under the assumption that distribution P has the absolute moments $\beta_{m+2}$ of order $m + 2$ but have a different estimate of the approximation accuracy (9)
$$|R_m|\leq\beta_{m+2}\cdot|t|^{m+2}+\|\theta_{m+1}\|\cdot|b_2|\cdot|t|^{m+3}+\|\theta_m\|\cdot b_4\cdot|t|^{m+4}.\,(11)$$
A comparison of the last two bounds leads to the question of the real accuracy of the expansion (4) (see [2]).