High peak values of transmission signals in wireless communication systems lead to wasteful energy consumption and degradation of several transmission performances. We continue the theoretical contributions made by Boche and Farell towards the understanding of peak value reduction, using the strategy known as tone reservation for orthogonal transmission schemes. There it was shown that for OFDM systems, the combinatorial object called arithmetic progression plays an important role in setting limitations for the applicability of the tone reservation method. In this work, we consider ourselves with the performance of the tone reservation in the probabilistic asymptotic setting. We show in particular that for a sufficiently large number N of carriers, choosing each element of that set independently with arbitrary small probability, yields in turn a set of carriers, for which the PAPR reduction problem is not solvable with
certain explicitly given threshold constants with probability 1 as N goes to infinity.