Let $(X\_1,X\_2,\ldots,X\_k)$ be distributed according to a multinomial distribution with parameters $(n;p\_1,p\_2,\ldots, p\_k),$ i.e. 


$$P(X\_1=n\_1,\ldots,X\_k=n\_k) = {n\choose n\_1,n\_2,\ldots,n\_k} p\_1^{n\_1}\ldots p\_k^{n\_k}~,$$


for $n\_i\geq 0, \sum\_{i=1}^k n\_i = n.$


The multivariate central limit theorem tells us that 


$$\frac{1}{\sqrt{n}}(X\_1-np\_1,\ldots,X\_k-np\_k)\stackrel{d}{\to} (Z\_1,\ldots Z\_k)\sim \mathcal{N}(0,\Sigma),$$


where $\Sigma\_{ij} = \begin{cases} p\_i(1-p\_i) & i=j \\ -p\_ip\_j & i\neq j\end{cases}.$


Uniform integrability tells us that 


$$\frac{1}{\sqrt{n}}\mathbb{E}\left[\left(\sum\_{i=1}^k (X\_i-np\_i)^2\right)^{\frac 12}\right]\to \mathbb{E}\left[\left(\sum\_{i=1}^k Z\_i^2\right)^{\frac 12}\right]~.$$


I am interested in showing uniform convergence above, i.e. I want an upper bound $f(n,k)$ on the difference


$$\left|\frac{1}{\sqrt{n}}\mathbb{E}\left[\left(\sum\_{i=1}^k (X\_i-np\_i)^2\right)^{\frac 12}\right]-\mathbb{E}\left[\left(\sum\_{i=1}^k Z\_i^2\right)^{\frac 12}\right]\right|$$


that depends only on $n$ and $k$ but not on $(p\_1,\ldots,p\_k)$ such that $f(n,k)\to 0$ as $n\to\infty$ for every fixed $k.$



Uniform convergence of 2-norm of a multinomial vector
Let $(e\_1,\dots,e\_k)$ denote the standard basis in $\mathbb{R}^k$. 
Then 
$$\frac1{\sqrt n}\,(X\_1-np\_1,\dots,X\_k-np\_k)\overset{D}=\frac1{\sqrt n}\,\sum\_{j=1}^nV\_j=:V, 
$$
where $\overset{D}=$ stands for the equality in distribution, 
$V\_j:=U\_j-\mathbb{E}U\_j$, and the $U\_j$'s are iid random vectors in $\mathbb{R}^k$ such that $\mathbb{P}(U\_j=e\_\ell)=p\_\ell$ for $\ell=1,\dots,k$. 
Let $W\sim N(0,\Sigma)$, where $\Sigma$ is the covariance matrix of $V\_1$. 


We shall show more than requested: 
$$(0)\qquad\big|\,\mathbb{E}\|V\|-\mathbb{E}\|W\|\,\big|<0.99 n^{-1/6}; 
$$
the latter bound is uniform, not only in the $p\_\ell$'s, but also in $k$. 


By Corollary 2.1 and formula (1.10) in [1], 
$$(1)\qquad\big|\,\mathbb{E}\|V\|-\mathbb{E}\|W\|\,\big|
\le\sqrt{\frac2\pi}\,\int\_{\mathbb{R}^k}\gamma\_k(du)
\int\_0^\infty\frac{dt}{t^2}|\mathbb{E}e^{itu\cdot V}-\mathbb{E}e^{itu\cdot W}|,
$$
where $\gamma\_k$ is the standard Gaussian measure on $\mathbb{R}^k$. 


Using inequality (13) in [2] and writing $\int\_0^{|b|}\frac{a^2}2\,\exp\big(\frac{a^2}2\big)\,da
\le e^{b^2/2}\frac{|b|^3}6$ for real $b$, for all $u\in\mathbb{R}^k$ and $t\in\mathbb{R}$ we have 
$$(2)\qquad|\mathbb{E}e^{itu\cdot V}-\mathbb{E}e^{itu\cdot W}|
\le c|t|^3\delta\_n(u),\quad\text{where}\quad\delta\_n(u):=\frac{\mathbb{E}|u\cdot V\_1|^3}{\sqrt n} 
$$
and $c:=1/(6\sqrt{2\pi})$. 
Let $A:=\big(2/(c\delta\_n(u))\big)^{1/3}$. 
Using (2) for $t\in(0,A)$ and the trivial inequality $|\mathbb{E}e^{itu\cdot V}-\mathbb{E}e^{itu\cdot W}|\le2$ for $t\ge A$, for the integral $\int\_0^\infty$ in (1) one has 
$$\int\_0^\infty\le\int\_0^A t\,dt\,c\delta\_n(u)
+2\int\_A^\infty\frac{dt}{t^2}
=3\times2^{-1/3}(c\delta\_n(u))^{1/3}. 
$$
So, by (1), with $c\_1:=\sqrt{\frac2\pi}\,\times3\times2^{-1/3}c^{1/3}$, 
$$(3)\qquad\big|\,\mathbb{E}\|V\|-\mathbb{E}\|W\|\,\big|
\le c\_1 \int\_{\mathbb{R}^k}\gamma\_k(du)\delta\_n(u)^{1/3}
\le c\_1 \Big(\int\_{\mathbb{R}^k}\gamma\_k(du)\delta\_n(u)\Big)^{1/3}. 
$$
Next, by part (vi) of Theorem 2.3 of [3], $\mathbb{E}|u\cdot V\_1|^3\le1.32\mathbb{E}|u\cdot U\_1|^3
=1.32\sum\_{\ell=1}^k p\_\ell|u\_\ell|^3$, where $u=(u\_1,\dots,u\_k)\in\mathbb{R}^k$. 
So, 
$$\sqrt n\,\int\_{\mathbb{R}^k}\gamma\_k(du)\delta\_n(u)
=\int\_{\mathbb{R}^k}\gamma\_k(du)\mathbb{E}|u\cdot V\_1|^3
\le1.32\int\_{\mathbb{R}^k}\gamma\_k(du)\mathbb{E}|u\cdot U\_1|^3
$$
$$=1.32\sum\_{\ell=1}^k p\_\ell\int\_{\mathbb{R}^k}\gamma\_k(du)|u\_\ell|^3
=1.32\sum\_{\ell=1}^k p\_\ell\int\_{\mathbb{R}}\gamma\_1(du\_1)|u\_1|^3
=1.32\times2\sqrt{\frac2\pi}. 
$$
Now (0) follows from (3). 


Similarly one can obtain bounds on the rate of convergence for $\mathbb{E}\|V\|^p$ for $p$ other than $1$. 


[1] <http://arxiv.org/abs/1506.00537> ; a version of this is to appear In The American Mathematical Monthly. 


[2] <http://arxiv.org/abs/0912.0726>


[3] <http://arxiv.org/abs/1111.2622> ; published at <http://link.springer.com/chapter/10.1007%2F978-3-0348-0490-5_6>


