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Consider $N$ i.i.d random variables, $X_{1}, X_{2}, \ldots, X_{N}$ , that are chi-squared of degree $K \geq 2$. Also consider the following 3 vectors:

\begin{eqnarray*} \bar{a} &=& (\frac{1}{N},\frac{1}{N},\ldots,\frac{1}{N}) \\ a^{0} &=& (1,0,\ldots,0) \\ a^{*} &=& (a_{1}^{*},a_{2}^{*},\ldots,a_{N}^{*}) \; \text{where } a_{j}^{*} \geq 0 \; \forall j \;\; \text{and } \sum_{j=1}^{N} a^{*}_{j} = 1 \\ \end{eqnarray*} Note that they all sum to 1 (so the mean is preserved) and in terms of "majorization" we can say: \begin{eqnarray*} \bar{a} \prec a^{*} \\ a^{*} \prec a^{0} \end{eqnarray*} where $a \prec b$ means that $a$ is majorized by $b$.

Also let $0 \leq \epsilon \leq 1$ be a small. Now I am wondering under what conditions can we say the following:

\begin{eqnarray*} Pr(|\sum_{j=1}^{N} \bar{a}_{j} X_{j} - K | \geq \epsilon K) \leq Pr(|\sum_{j=1}^{N} a^{*}_{j} X_{j} - K | \geq \epsilon K) \\ Pr(|\sum_{j=1}^{N} a^{*}_{j} X_{j} - K | \geq \epsilon K) \leq Pr(|\sum_{j=1}^{N} a^{0}_{j} X_{j} - K | \geq \epsilon K) \\ \end{eqnarray*}

Any ideas?

Thanks for reading

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