Abstract
In this paper, we prove large and moderate deviation results for normalized sums of multivariate random variables of suitable triangular arrays having Bernoulli distributed components. Moreover, motivated by possible connections with the probabilistic method in number theory, we apply the univariate versions of these results to Newman polynomials with random coefficients.
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References
N. Alon and J.H. Spencer, The Probabilistic Method, 3rd ed., John Wiley & Sons, Hoboken, NJ, 2008.
K.S. Berenhaut and F. Saidak, A note on the maximal coefficients of squares of Newman polynomials, J. Number Theory, 125(2):285–288, 2007.
P. Borwein, Computational Excursions in Analysis and Number Theory, Springer-Verlag, New York, 2002.
W. Bryc, A remark on the connection between the large deviation principle and the central limit theorem, Stat. Probab. Lett., 18(4):253–256, 1993.
J. Cilleruelo, Maximal coefficients of squares of Newman polynomials, Report, Departamento de Matemáticas, Universidad Autónoma de Madrid, Spain, 2008.
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Springer, New York, 1998.
A. Dubickas, Heights of powers of Newman and Littlewood polynomials, Acta Arith., 128(2):167–176, 2007.
A. Dubickas and J. Jankauskas, The maximal value of polynomials with restricted coefficients, J. Korean Math. Soc., 46(1):41–49, 2009.
A. Dubickas and J. Jankauskas, On Newman polynomials which divide no Littlewood polynomial, Math. Comput., 78(265):327–344, 2009.
P. Erdös and P. Tetali, Representations of integers as the sum of k terms, Random Struct. Algorithms, 1(3):245–261, 1990.
M.N. Kolountzakis, Some applications of probability to additive number theory and harmonic analysis, in D.V. Chudnovsky, G.V. Chudnovsky, and M.B. Nathanson (Eds.), Number Theory (Papers from the Seminars Held at the City University of New York, New York, 1991–1995), Springer, New York, 1996, pp. 229–251.
I. Mercer, Newman polynomials, reducibility, and roots on the unit circle, Integers, 12(4):503–519, 2012.
D.J. Newman, An L1 extremal problem for polynomials, Proc. Am. Math. Soc., 16:1287–1290, 1965.
A.M. Odlyzko and B. Poonen, Zeros of polynomials with 0, 1 coefficients, Enseign. Math. (2), 39(3–4):317–348, 1993.
T. Tao and V. Vu, Additive Combinatorics, Cambridge University Press, Cambridge, 2006.
G. Yu, An upper bound for B 2[g] sets, J. Number Theory, 122(1):211–220, 2007.
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* The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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Giuliano, R., Macci, C. Asymptotic results for a class of triangular arrays of multivariate random variables with Bernoulli distributed components∗ . Lith Math J 56, 298–317 (2016). https://doi.org/10.1007/s10986-016-9320-5
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DOI: https://doi.org/10.1007/s10986-016-9320-5