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}) \cr a^{0} &=& (1,0,\ldots,0) \cr \hat{a} &=& (\hat{a_{1}},\hat{a_{2}},\ldots,\hat{a_{N}}) \; \text{where } \hat{a_{j}} \geq 0 \; \forall j \;\; \text{and } \sum_{j=1}^{N} \hat{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: $$ \bar{a} \prec \hat{a} $$ $$ \hat{a} \prec a^{0} $$ 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:

$$ Pr(|\sum_{j=1}^{N} \bar{a_{j}} X_{j} - K | \geq \epsilon K) \leq Pr(|\sum_{j=1}^{N} \hat{a_{j}} X_{j} - K | \geq \epsilon K) $$ $$ Pr(|\sum_{j=1}^{N} \hat{a_{j}} X_{j} X_{j} - K | \geq \epsilon K) \leq Pr(|\sum_{j=1}^{N} a_{j}^{0} X_{j} - K | \geq \epsilon K) $$

Any ideas?

Thanks for reading