An improved result in almost sure local central limit theorem for the partial sums

作者：Xu, Feng; Zeng, Zhen^*

来源：PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES, 2022, 132(1): 14.

DOI：10.1007/s12044-022-00662-x

• Almost sure local central limit theorem
• partial sum
• weight sequences

The almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let {X-k, k >= 1} be a sequence of independent and identically distributed random variables. Under a fairly general condition, an universal result in almost sure local limit theorem for the partial sums S-k = Sigma(k)(i=1) X-i is established on the weight d(k) = k(-1) exp(log(beta) k), 0 <= beta < 1/2: @@@ lim(n ->infinity) 1/D-n Sigma(n)(k=1) d(k) 1(a(k) <= S-k < b(k))/P(a(k) <= S-k < b(k)) = 1 a.s., @@@ where D-n = Sigma(n)(k=1) d(k), -infinity <= a(k) <= 0 <= b(k) <= infinity, k = 1, 2, .... This result extends previous results in the almost sure local central limit theorems from d(k) = 1/k to d(k) = k(-1) exp(log(beta) k), 0 <= beta < 1/2.

• 单位
  桂林理工大学