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 Mat. Zametki, 2008, Volume 84, Issue 1, Pages 69–98 (Mi mz5195)

On the Distribution of Integer Random Variables Satisfying Two Linear Relations

V. P. Maslova, V. E. Nazaikinskiib

a M. V. Lomonosov Moscow State University
b A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

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
$$\sum_{j=l_0}^\infty jN_{j}\le M, \qquad \sum_{j=l_0}^\infty N_j=N,$$
where $l_0,M,N$ are natural numbers. If $M,N\to\infty$ 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 $d\ge2$ 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 $\sum_{j=l}^{\infty} N_j$ from the corresponding cumulative integrals for the limit distribution. In an earlier paper (see [6]), we studied the case $d=3$.

Keywords: Bose–Einstein distribution, multiplicative measure, cumulative distribution, cumulative integral, Bose particles

DOI: https://doi.org/10.4213/mzm5195

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English version:
Mathematical Notes, 2008, 84:1, 73–99

Bibliographic databases:

UDC: 519.2+531.19

Citation: V. P. Maslov, V. E. Nazaikinskii, “On the Distribution of Integer Random Variables Satisfying Two Linear Relations”, Mat. Zametki, 84:1 (2008), 69–98; Math. Notes, 84:1 (2008), 73–99

Citation in format AMSBIB
\Bibitem{MasNaz08} \by V.~P.~Maslov, V.~E.~Nazaikinskii \paper On the Distribution of Integer Random Variables Satisfying Two Linear Relations \jour Mat. Zametki \yr 2008 \vol 84 \issue 1 \pages 69--98 \mathnet{http://mi.mathnet.ru/mz5195} \crossref{https://doi.org/10.4213/mzm5195} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2451886} \elib{http://elibrary.ru/item.asp?id=13595066} \transl \jour Math. Notes \yr 2008 \vol 84 \issue 1 \pages 73--99 \crossref{https://doi.org/10.1134/S0001434608070079} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000258855600007} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-50849125696} 

• http://mi.mathnet.ru/eng/mz5195
• https://doi.org/10.4213/mzm5195
• http://mi.mathnet.ru/eng/mz/v84/i1/p69

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. P. Maslov, “Thermodynamics of Nanostructures”, Math. Notes, 84:4 (2008), 592–595
2. V. P. Maslov, “Transition to the Condensate State for Classical Gases and Clusterization”, Math. Notes, 84:6 (2008), 785–813
3. V. P. Maslov, “New distribution formulas for classical gas, clusters, and phase transitions”, Theoret. and Math. Phys., 157:2 (2008), 1577–1594
4. Maslov V. P., “New Look at Thermodynamics of Gas and at Clusterization”, Russ. J. Math. Phys., 15:4 (2008), 493–510
5. V. P. Maslov, “On the new distribution generalizing the Gibbs, Bose–Einstein, and Pareto distributions”, Math. Notes, 85:5-6 (2009), 613–622
6. Maslov V. P., “Threshold levels in economics and time series”, Math. Notes, 85:3-4 (2009), 305–321
7. Maslov V. P., “Theorems on the debt crisis and the occurrence of inflation”, Math. Notes, 85:1-2 (2009), 146–150
8. Maslov V. P., “On the boundedness law for the number of words in an overabundant dictionary”, Math. Notes, 85:1-2 (2009), 296–301
9. Maslov V. P., “Theory of chaos and its application to the crisis of debts and the origin of inflation”, Russ. J. Math. Phys., 16:1 (2009), 103–120
10. Maslov V., “Dequantization, Statistical Mechanics and Econophysics”, Tropical and Idempotent Mathematics, Contemporary Mathematics, 495, ed. Litvinov G. Sergeev S., Amer Mathematical Soc, 2009, 239–279
11. Maslov V.P., “New global distributions in number theory and their applications”, J. Fixed Point Theory Appl., 8:1 (2010), 81–111
12. V. P. Maslov, “Mathematical Solution of the Gibbs Paradox”, Math. Notes, 89:2 (2011), 266–276
13. Maslov V.P., “Mixture of New Ideal Gases and the Solution of the Gibbs and Einstein Paradoxes”, Russ. J. Math. Phys., 18:1 (2011), 83–101
14. Maslov V.P., “The Relationship Between the Van-der-Waals Model and the Undistinguishing Statistics of Objectively Distinguishable Objects. the New Parastatistics”, Russ. J. Math. Phys., 21:1 (2014), 99–111
15. Maslov V.P., “Large negative numbers in number theory, thermodynamics, information theory, and human thermodynamics”, Russ. J. Math. Phys., 23:4 (2016), 510–528
16. Maslov V.P., “Negative energy, debts, and disinformation from the viewpoint of analytic number theory”, Russ. J. Math. Phys., 23:3 (2016), 355–368
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