Stability of Nontransitive Trybuła Triplets under Taking Sums and Maxima

We consider two nontransitive Trybuła triplets. The first one features the maximum possible nontransitivity strength for a nontransitive cycle of three random variables. The second one is the parametric triplet with equal expectations and variances. For both triplets, we study the stability of nontransitivity under taking the sum and the maximum of two independent copies of random variables. We show that the first one is stable under taking the sums (in the opposite direction of stochastic precedence) and ceases to be nontransitive under taking the maximum. For the second triplet, we show that the nontransitivity is preserved on a certain subinterval of the original interval of values of the parameter $\varepsilon$ (under taking the sums, also in the opposite direction of stochastic precedence). For both transformations, we obtain polynomial equations, whose roots $\varepsilon_{\mathrm{cr}}$ define the boundaries of the stability intervals.