Abstract
A central limit theorem with a remainder term for a cumulative process W(t) was proved by the author in an earlier paper. Here we show that the average of maximum errors taken over all values of t is actually smaller than what one would expect if a formula for the worst possible case for each t were used. This improvement is in line with what Heyde and Leslie (1972) obtained in a case of central limit theorems for the sequences of independent random variables.
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