Tight L2 bound on moments approximation and reference

Question

Consider $f\in L^2(I)$, where $I$ is the unit interval and $L^2$ is w.r.t. Lebesgue measure, and consider an approximation of $f$ denoted by $\tilde{f}\in L^2$.

The error in approximated the moments of $f$ by those of $\tilde{f}$ can be readily bounded by $\|f-\tilde{f}\|_2$, e.g., denoting the respective expectancies as $\mu$ and $\tilde{\mu}$, one has that

$$\left| \mu - \tilde{\mu} \right| \leq \|f-\tilde{f} \|_2 \, , $$

and if we define the variance in the usual way, as ${\rm Var}(f) = \mathbb{E}[(f-\mu)^2]$, then

$$\left| {\rm Var}(f) - {\rm Var}(\tilde{f}) \right| \leq \left(\mu +\tilde{\mu} + \|f\|_2 + \|\tilde{f}\|_2 \right) \|f-\tilde{f}\|_2 \, .$$

Questions:

1. Are these bounds tight?
2. Is there a known formula for the error in $\mathbb{E}[(f-\mu)^n]$ for all $n\in \mathbb{N}$?
3. Is there a known textbook reference for these inequalities? It seems like I'm hardly the first to have discovered them.

Remark: The above bounds are obtained using mainly the triangle and Cauchy-Schwarz inequalities.

Answer 1

Let $X:=f$ and $Y:=\tilde f$. Let $\mu:=EX$ and $\nu:=EY$. Let $\|\cdot\|:=\|\cdot\|_2$. Let $D:=Y-X$. For any random variable $V$, let $\tilde V:=V-EV$.

Then the first displayed inequality can be rewritten as $ED\le\|D\|$. The upper bound here is attained if e.g. $D=1$. So, this bound is tight.

The second displayed inequality is in general false. E.g., suppose that $\mu=\nu<0$, $\tilde X=0$, and $P(\tilde Y=\pm1)=1/2$. Then the left-hand side of that inequality is $1$, whereas its right-hand side is $2\mu+|\mu|+\sqrt{\mu^2+1}<1$.

However, one can write \begin{multline*} \text{Var}\; Y-\text{Var}\; X=E(\tilde Y^2-\tilde X^2)=E\tilde D(\tilde X+\tilde Y) \\ \le\|\tilde D\|\,(\|\tilde X\|+\|\tilde Y\|) \le\|D\|\,(\|X\|+\|Y\|), \end{multline*} by the Cauchy-Schwarz and Minkowski inequalities. So, \begin{equation*} |\text{Var}\; Y-\text{Var}\; X| \le\|\tilde D\|\,(\|\tilde X\|+\|\tilde Y\|) \le\|D\|\,(\|X\|+\|Y\|). \tag{1} \end{equation*} The penultimate bound on $|\text{Var}\; Y-\text{Var}\; X|$ is attained e.g. when $Y=cX$ for a real constant $c>0$; if, in addition, $EX=0$, then the latter bound is attained as well. So, the latter two bounds are both tight, in their terms.

One can also adopt this approach to bound the "error" for the central moments $E(X-\mu)^n=E\tilde X^n$ of higher orders $n$ by writing $a^n-b^n=(a-b)\sum_{j=0}^{n-1} a^jb^{n-1-j}$ for $a=\tilde X$ and $b=\tilde Y$, and then using the H\"older and Minkowski inequalities, e.g., as follows: \begin{align*} |E\tilde X^n-E\tilde Y^n| &\le\|\tilde X-\tilde Y\|_n\;\sum_{j=0}^{n-1} \|\tilde X^j\tilde Y^{n-1-j}\|_{n/(n-1)} \\ &\le\|\tilde X-\tilde Y\|_n\;\sum_{j=0}^{n-1} \|\tilde X\|_n^j \|\tilde Y\|_n^{n-1-j} \tag{2} \\ &=\|\tilde X-\tilde Y\|_n\; \frac{\|\tilde X\|_n^n-\|\tilde Y\|_n^n}{\|\tilde X\|_n-\|\tilde Y\|_n}, \end{align*} the latter equality holding if $\|\tilde X\|_n\ne\|\tilde Y\|_n$. For the second inequality in the latter display, we use the obvious identity $\|f^m\|_p=\|f\|_{mp}^m$ for positive $m$ and $p$.

Clearly, the bound in (2) is a generalization of the first bound in (1).

For $p\ne2$, the inequality $\|\tilde X\|_p\le\|X\|_p$ will not hold in general. The optimal constant factor $c_p$ in the inequality $\|\tilde X\|_p\le c_p\|X\|_p$ for real $p>1$ was given in in Optimal Re-centering Bounds ... / arXiv. In particular, $c_3=\frac13\,(17+7\sqrt7)^{1/3}=1.0957\dots$.