Abstract:
This paper investigates two nontransitive triplets originally proposed by S.Trybula. The first triplet realizes the maximal attainable probability for nontransitive cycles of three random variables. The second is a parametric triplet of random variables with equal means and variances. We investigate the stability of nontransitivity for both triplets under summation and under the maximum of two independent copies. It is shown that the first triplet preserves nontransitivity under summation, with the pairwise stochastic precedence relations reversed, but loses it under the maximum operation. For the second triplet, nontransitivity is preserved on a restricted subinterval of the parameter $\varepsilon$, again with reversed stochastic precedence under summation. For both operations, we derive polynomial equations whose roots determine the endpoints $\varepsilon_{\mathrm{cr}}$ of the stability intervals.
Keywords:nontransitive triplets, nontransitive tuples, stability of nontransitivity, stochastic precedence, Trybula triplets, sums of random variables, maximum of random variables.
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