Deleting o(n) terms (for the laws of large numbers) or o(√n) terms (for the central limit theorem) from an i.i.d. sample leaves the classical limit theorems valid, provided the deleted indices are chosen without using the data.
Deleting Items and Disturbing Mesh Theorems for Riemann Definite Integral and Their Applications
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abstract
Based on the definition of Riemann definite integral,deleting items and disturbing mesh theorems on Riemann sums are given. After deleting some items or disturbing the mesh of partition, the limit of Riemann sums still converges to Riemann definite integral under specific conditions. These theorems can deal with a class of complicate limitation of sum and product of series of a function, and demonstrate that the geometric intuition of Riemann definite integral is more profound than ordinary thinking of area of curved trapezoid.
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Extension of Limit Theory with Deleting Items Partial Sum of Random Variable Sequence
Deleting o(n) terms (for the laws of large numbers) or o(√n) terms (for the central limit theorem) from an i.i.d. sample leaves the classical limit theorems valid, provided the deleted indices are chosen without using the data.