The central limit theorem for triangular arrays is well-known (see Durrett, Probability: Theory and examples, 4th ed., Theorem 3.4.5). Is there a local form of this theorem for integer-valued random variables? A local central limit theorem for sums of i.i.d. integer-valued random variables is provided by Gnedenko and Kolmogorov (see Gnedenko and Kolmogorov, Limit distributions for sums of independent variables, Chapter 49). Has the latter been generalized to triangular arrays? I am pretty sure that this has been done.
I got the answer! The problem is addressed in
- McDonald, D.R. (1980), On local limit theorem for integer-valued random variables. Theory Probab Appl., 3, 613-619
- Mukhin, A.B. (1984), Local limit theorems for distributions of sums of independent random vectors. Theory Probab Appl. 29, 369-375
- Mukhin, A.B. (1991), Local limit theorems for lattice random variables. Theory Prob. Appl. 35, 698-713
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