In view of the great importance of thecentral limit theorem (CLT), we shall give a general but self-contained version due to Lindeberg.

1. 1.

   This approach has received recent attention of Terence Tao (2015, SPA Conference, Oxford) as “the most effective way to deal with local universality for non-Hermitian random matrices.” In this context, it is referred to asLindeberg’s exchange strategy, e.g., see Tao (2012). The “exchange” refers to a substitution with a normal random variable and should not be confused another technical use of related terminology for permutation invariance.

2. 2.

   Billingsley (1986), p. 373.

3. 3.

   In general, the error of approximation in the CLT is\(O(n^{-{1\over 2}})\); see the Berry-Esseen Theorem4.6. For the binomial case the approximation is best near\(p=1/2\). However, even forp near the tail such as\(p= 0.9\) or,\(p= 0.1\), the approximation is quite good with\(n \ge 500\); e.g., see the calculations in R. Bhattacharya, L. Lin and M. Majumdar (2013): Problems of ruin and survival in economics: applications of limit theorems in probability,Sankhya,75B 145–180.

4. 4.

   Infinitely divisible distributions are naturally associated with stochastic processes having independent increments. This connection is thoroughly developed in a companion text on stochastic processes.

Authors and Affiliations

1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

   Rabi Bhattacharya

2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

   Edward C. Waymire

Corresponding author

Correspondence toRabi Bhattacharya.

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