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Given a $n\times n$ symmetric random matrix such that

  1. all diagonal elements are all fixed as $0$.
  2. all elements in upper triangle (excluding the diagonal) are i.i.d. uniform random variables over $[0,1]$. Lower triangle values are set accordingly to ensure symmetry.

The questions is:

Is there any known result for the distribution of the min/max row sum; or any suggested method to find this?

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  • $\begingroup$ Sounds hard, but, I wonder if a recursive method would work: The $2\times 2$ case is easy, and if we have a joint distribution on the rows of a $k \times k$ case then perhaps we can get it for $k+1$ which just adds an independent single number to all rows (except the last row) of the $k \times k$ matrix, and adds those same new numbers to the last row. $\endgroup$ – Michael Jan 14 '18 at 4:12
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Letting the $ij^{th}$ entry of the matrix be the random variable $U_{ij}$, the sum for the $i^{th }$ row is $$\sum_{j\neq i}U_{ij}=\sum_{j<i}U_{ji}+\sum_{i<j}U_{ij}$$ due to the symmetric condition. Note that $\{U_{j,i}~|~j<i\}$ and $\{U_{i,j}~|~i<j\}$ both belong to the upper triangular part.

Now its about finding the joint distribution of the random variables, and from the joint distribution calculating the distribution for the highest order statistic, for which you can consult this paper. Only bear in mind that the row sums, written as $\sum_{j<i}U_{ji}+\sum_{i<j}U_{ij}$ for distinct $i's$, are not independent.

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