Abstract
Let X1 and N ≥ 0 be integer-valued power-law random variables. For a randomly stopped sum SN = X1+⋯+XN of independent and identically distributed copies of X1, we establish a first-order asymptotics of the local probabilities P(SN = t) as t → +1 ∞. Using this result, we show the scaling k-δ, 0 ≤ δ ≤ 1, of the local clustering coefficient (of a randomly selected vertex of degree k) in a power-law affiliation network.
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A. Aleškevičienė, R. Leipus, and J. Šiaulys, Tail behavior of random sums under consistent variation with applications to the compound renewal risk model, Extremes,11:261–279, 2008.
S. Asmussen, S. Foss, and D. Korshunov, Asymptotics for sums of random variables with local subexponential behaviour, J. Theor. Probab., 16:489–518, 2003.
A. Baltrūnas and J. Šiaulys, Second order asymptotic behaviour of subordinated sequences with longtailed subordinator, J. Math. Anal. Appl., 332:22–31, 2007.
M. Bloznelis, Degree and clustering coefficient in sparse random intersection graphs, Ann. Appl. Probab., 23:1254–1289, 2013.
M. Bloznelis, Degree-degree distribution in a power law random intersection graph with clustering, Internet Math., 2017.
M. Bloznelis, E. Godehardt, J. Jaworski, V. Kurauskas, and K. Rybarczyk, Recent progress in complex network analysis: Models of random intersection graphs, in B. Lausen, S. Krolak-Schwerdt, and M. Böhmer (Eds.), Data Science, Learning by Latent Structures, and Knowledge Discovery, Springer, Berlin, Heidelberg, 2015, pp. 69–78.
M. Bloznelis and J. Petuchovas, Correlation between clustering and degree in affiliation networks, in A. Bonato, F. Chung Graham, and P. Prałat (Eds.), Algorithms and Models for the Web Graph. 14th International Workshop, WAW 2017, Toronto, ON, Canada, June 15–16, 2017. Revised Selected Papers, Theoretical Computer Science and General Issues, Vol. 10519, Springer, 2017, pp. 90–104.
A.A. Borovkov and K.A. Borovkov, Asymptotic Analysis of Random Walks. Heavy-tailed Distributions, Encycl. Math. Appl., Vol. 118, Cambridge Univ. Press, Cambridge, 2008.
D. Denisov, S. Foss, and D. Korshunov, Asymptotics of randomly stopped sums in the presence of heavy tails, Bernoulli, 16:971–994, 2010.
R.A. Doney, A large deviation local limit theorem, Math. Proc. Camb. Philos. Soc., 105:575–577, 1989.
S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes, Pseudofractal scale-free web, Phys. Rev. E, 65:066122, 2002.
P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events, Springer, New York, 1997.
P. Embrechts, M. Maejima, and E. Omey, A renewal theorem of Blackwell type, Ann. Probab., 12:561–570, 1984.
P. Erdős, W. Feller, and H. Pollard, A property of power series with positive coefficients, Bull. Am. Math. Soc., 55:201–204, 1949.
H. Federer, Geometric Measure Theory, Springer, New York, 1969.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, New York, 1968.
S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd ed., Springer, New York, 2013.
I. Foudalis, K. Jain, C. Papadimitriou, and M. Sideri, Modeling social networks through user background and behavior, in A. Frieze, P. Horn, and P. Prałat (Eds.), Algorithms and Models for the Web Graph. 8th International Workshop,WAW2011, Atlanta, GA, USA, May 27–29, 2011, Proceedings, Theoretical Computer Science and General Issues, Vol. 6732, Springer, Heidelberg, 2011, pp. 85–102.
B.V. Gnedenko and A.N. Kolmogorov, Limit distributions for sums of independent random variables, Addison-Wesley, Cambridge, 1954.
C.C. Heyde, A nonuniform bound on convergence to normality, Ann. Probab., 3:903–907, 1975.
T. Hilberdink, On the Taylor coefficients of the composition of two analytic functions, Ann. Acad. Sci. Fenn., Math., 21:189–204, 1996.
T. Hilberdink, Asymptotic expansions for Taylor coefficients of the composition of two functions, Asymptotic Anal., 63(3):125–142, 2009.
I. A. Ibragimov and Yu V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971.
A.A. Mogulskii, An integro-local theorem that is applicable on the whole half-axis for sums of random variables with regularly varying distributions, Sib. Math. J., 49:669–683, 2008.
M.E.J. Newman, S.H. Strogatz, and D.J. Watts, Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E, 64:026118, 2002.
K.W. Ng and Q. Tang, Asymptotic behavior of tail and local probabilities for sums of subexponential random variables, J. Appl. Probab., 41:108–116, 2004.
V.V. Petrov, Sums of Independent Random Variables, Springer, New York, Heidelberg, 1975.
L. Ravasz and A.L. Barabási, Hierarchical organization in complex networks, Phys. Rev. E, 67:026112, 2003.
A. Vázquez, R. Pastor-Satorras, and A. Vespignani, Large-scale topological and dynamical properties of Internet, Phys. Rev. E, 65:066130, 2002.
C. Yu, Y. Wang, and Y. Yang, The closure of the convolution equivalent distribution class under convolution roots with applications to random sums, Stat. Probab. Lett., 80:462–472, 2010.
A.Yu. Zaigraev, A.V. Nagaev, and A. Jakubowski, Large deviation probabilities for sums of lattice random vectors with heavy tailed distribution, Discrete Math. Appl., 7:313–326, 1997.
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Dedicated to Professor Vygantas Paulauskas on the occasion of his 75th birthday
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Bloznelis, M. Local probabilities of randomly stopped sums of power-law lattice random variables. Lith Math J 59, 437–468 (2019). https://doi.org/10.1007/s10986-019-09462-9
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DOI: https://doi.org/10.1007/s10986-019-09462-9