The author tries to derive a non-uniform concentration inequality for a random permutation sum \(W_n= \sum_{i=1}^nY(i, \pi(i))\) where \(\pi= (\pi(1), \pi(2), \dotsc,\pi(n))\) is a uniformly distributed random permutation of the numbers \(1,2,\dotsc,n\) and \(Y(i,j)\), \(i,j=1,2,\dotsc,n\), are random variables such that they and \(\pi\) are stochastically independent under the condition of finiteness of their third moments.
Reviewer’s remarks: The paper has unsubstantiated arguments in several places, too many to be mentioned here. Suppose \(Y(i,j), i,j=1,2,\dots,n\) are identically distributed with finite third moment \( \operatorname{E}|Y(i,j)|^3=1\). Then \(\delta_3= n\). Hence conditions such as \(\delta_3\leq \frac{1}{11}\) and \(n \geq 32\) in Lemma 2.1 do not hold for such sequences.