## 3 Method
### Method 1: Central Limit Theorem  
Consider the sum of random variables \(S_{n}\equiv\sum_{i=1}^{n}X_{i}\). Then, in particular  
\[E[S_{n}]=m_{n}=\sum_{i=1}^{n}\mu_{i},\quad D[S_{n}]=\sigma^{2}=\sigma^{2}(n) \equiv\sum_{i=1}^{n}\sigma_{i}^{2}. \tag{9}\]  
One of the classical formulations of the central limit theorem is contained, for example, in [54] (p. 383, Theorem 27.2). **[PERSON]'s Central Limit Theorem. **_Suppose that sequence\(X_{i}\), \(i=1,\ldots,\ n\) is independent and satisfies (9). If [PERSON]'s condition_  
\[\underset{n\rightarrow\infty}{\sum}_{i=1}^{n}\frac{1}{\sigma^{2}}\underset{| X_{i}-\mu_{i}|\geq\sigma^{0}}{\int}\left(X_{i}-\mu_{i}\right)^{2}d\mathrm{P}=0, \tag{10}\]  
_holds for all positive, then \(\frac{S_{n}-m_{n}}{\sigma}\to N_{0,1}\) at \(n\rightarrow\infty\)._  
Here  
\[\frac{S_{n}-m_{n}}{\sigma}=\frac{(X_{1}-\mu_{1})+\ldots+(X_{n}-\mu_{n})}{\sqrt {\sum_{i=1}^{n}\sigma_{i}^{2}}},\]  
and \(N_{0,1}\equiv N(0;1)\) denotes the standard Gaussian random variable, i.e., having zero mathematical expectation and variance equal to one. The density of the standard normal distribution is  
\[f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}.\]  
This theorem follows from the more general [PERSON] theorem [54] (p. 385, Theorem 27.3). While the classical central limit theorem requires finite moments of order 2, [PERSON]'s central limit theorem requires finite moments of order \(2+\delta\) for some \(\delta>0\). [PERSON]'s Central Limit Theorem. : _Suppose that sequence \(X_{i}\), \(i=1,\dots,\ n\) is independent and satisfies (9). If [PERSON]' condition_  
\[\lim_{n\to\infty}\frac{1}{\sigma^{2+\delta}}{\sum}_{i=1}^{n}E\Big{[}|X_{i}-\mu_ {i}|^{2+\delta}\Big{]}=0, \tag{11}\]  
_holds for some \(\delta>0\), then \((S_{n}-m_{n})/\sigma\to N_{0,1}\) at \(n\to\infty\)._  
As shown in [54] (p. 385), the Lindenberg condition (10) follows from the [PERSON] condition (11), but not vice versa. Note that fulfilling the conditions of the [PERSON] and [PERSON] theorems is only sufficient to satisfy the requirement of independence and not the identical distribution of random variables. A special case of [PERSON]'s theorem is the Bates theorems [55] on the continuous probability distribution of the mean of independent uniformly distributed random variables on the unit interval and the Irwin-Hall theorems [56; 57] on the continuous probability distribution for the sum of independent and identically distributed random variables. These theorems also tend to have a normal distribution. Applying [PERSON]'s theorem to the sum \(S_{n}\equiv\sum_{i=1}^{n}X_{i}\), we find that \(S_{n}\) will approach the normal distribution. The summation is performed per each element of \(X_{i}\), and \(S_{n}\) will also have 10,242 elements. Due to the trend to the normal distribution, the maximum values at the right tail of the distribution (as well as the minimum values in the left tail) will have a relatively low probability. For example, if the extremum is greater than \(3\sigma\), where \(\sigma\) is the variance of this normal distribution, then the probability of encountering these values will be 0.27%. In other words, if we have a large sample from the population of all sums \(S_{n}\), then the number of extremes in the sample above \(3\sigma\) will be approximately 0.27%. There are 10,242 points in our set. This means that 99.73% of values will be expected to be less than \(3\sigma\) in both directions, and approximately no more than 30 values are greater than \(3\sigma\). The matching algorithm will use these 30 longest vectors (and the corresponding pyramids--triplets of non-coplanar vectors in steps A5 and A6).