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On the distribution of integer random variables satisfying two linear relations

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Abstract

We consider the multiplicative (in the sense of Vershik) probability measure corresponding to an arbitrary real dimension d on the set of all collections {N j } of integer nonnegative numbers N j , j = l 0, l 0 + 1, ..., satisfying the conditions

$$ \sum\limits_{j = l_0 }^\infty {jN_j \leqslant M,} \sum\limits_{j = l_0 }^\infty {N_j = N} $$

, where l 0, M, N are natural numbers. If M, N → ∞ and the rates of growth of these parameters satisfy a certain relation depending on d, and l 0 depends on them in a special way (for d ≥ 2 we can take l 0 = 1), then, in the limit, the “majority” of collections (with respect to the measure indicated above) concentrates near the limit distribution described by the Bose-Einstein formulas. We study the probabilities of the deviations of the sums Σ j=l N j from the corresponding cumulative integrals for the limit distribution. In an earlier paper (see [6]), we studied the case d = 3.

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Correspondence to V. P. Maslov.

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Original Russian Text © V. P. Maslov, V. E. Nazaikinskii, 2008, published in Matematicheskie Zametki, 2008, Vol. 84, No. 1, pp. 69–98.

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Maslov, V.P., Nazaikinskii, V.E. On the distribution of integer random variables satisfying two linear relations. Math Notes 84, 73–99 (2008). https://doi.org/10.1134/S0001434608070079

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