## Bernoulli

• Bernoulli
• Volume 19, Number 2 (2013), 610-632.

### Total variation error bounds for geometric approximation

#### Abstract

We develop a new formulation of Stein’s method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a coupling between the original distribution and the “discrete equilibrium” distribution from renewal theory. We illustrate the approach in four non-trivial examples: the geometric sum of independent, non-negative, integer-valued random variables having common mean, the generation size of the critical Galton–Watson process conditioned on non-extinction, the in-degree of a randomly chosen node in the uniform attachment random graph model and the total degree of both a fixed and randomly chosen node in the preferential attachment random graph model.

#### Article information

Source
Bernoulli Volume 19, Number 2 (2013), 610-632.

Dates
First available in Project Euclid: 13 March 2013

https://projecteuclid.org/euclid.bj/1363192040

Digital Object Identifier
doi:10.3150/11-BEJ406

Mathematical Reviews number (MathSciNet)
MR3037166

Zentralblatt MATH identifier
06168765

#### Citation

Peköz, Erol A.; Röllin, Adrian; Ross, Nathan. Total variation error bounds for geometric approximation. Bernoulli 19 (2013), no. 2, 610--632. doi:10.3150/11-BEJ406. https://projecteuclid.org/euclid.bj/1363192040.

#### References

• [1] Aldous, D. (1989). Probability Approximations via the Poisson Clumping Heuristic. Applied Mathematical Sciences 77. New York: Springer.
• [2] Barbour, A.D. and Čekanavičius, V. (2002). Total variation asymptotics for sums of independent integer random variables. Ann. Probab. 30 509–545.
• [3] Barbour, A.D. and Grübel, R. (1995). The first divisible sum. J. Theoret. Probab. 8 39–47.
• [4] Barbour, A.D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Studies in Probability 2. New York: Oxford Univ. Press.
• [5] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18 279–290.
• [6] Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. Ann. Probab. 18 1388–1402.
• [7] Chen, L.H.Y., Goldstein, L. and Shao, Q.M. (2011). Normal Approximation by Stein’s Method. Probability and Its Applications (New York). Heidelberg: Springer.
• [8] Daly, F. (2008). Upper bounds for Stein-type operators. Electron. J. Probab. 13 566–587.
• [9] Daly, F. (2010). Stein’s method for compound geometric approximation. J. Appl. Probab. 47 146–156.
• [10] Ford, E. (2009). Barabási–Albert random graphs, scale-free distributions and bounds for approximation through Stein’s method. Ph.D. thesis. Univ. Oxford.
• [11] Goldstein, L. (2009). Personal communication and unpublished notes. In Stein Workshop, January 2009, Singapore.
• [12] Goldstein, L. and Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.
• [13] Kalashnikov, V. (1997). Geometric Sums: Bounds for Rare Events with Applications: Risk Analysis, Reliability, Queueing. Mathematics and Its Applications 413. Dordrecht: Kluwer Academic.
• [14] Lalley, S.P. and Zheng, X. (2011). Occupation statistics of critical branching random walks in two or higher dimensions. Ann. Probab. 39 327–368.
• [15] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
• [16] Mattner, L. and Roos, B. (2007). A shorter proof of Kanter’s Bessel function concentration bound. Probab. Theory Related Fields 139 191–205.
• [17] Peköz, E.A. (1996). Stein’s method for geometric approximation. J. Appl. Probab. 33 707–713.
• [18] Peköz, E.A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 587–608.
• [19] Peköz, E., Röllin, A. and Ross, N. (2011). Degree asymptotics with rates for preferential attachment random graphs. Preprint. Available at arXiv:org/abs/1108.5236.
• [20] Phillips, M.J. and Weinberg, G.V. (2000). Non-uniform bounds for geometric approximation. Statist. Probab. Lett. 49 305–311.
• [21] Röllin, A. (2005). Approximation of sums of conditionally independent variables by the translated Poisson distribution. Bernoulli 11 1115–1128.
• [22] Röllin, A. (2008). Symmetric and centered binomial approximation of sums of locally dependent random variables. Electron. J. Probab. 13 756–776.
• [23] Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surveys 8 210–293.
• [24] Ross, S. and Peköz, E. (2007). A Second Course in Probability. Boston, MA: www.ProbabilityBookstore.com.
• [25] Yaglom, A.M. (1947). Certain limit theorems of the theory of branching random processes. Doklady Akad. Nauk SSSR (N.S.) 56 795–798.