Summary
For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern −1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.
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References
Ahmad, I. A. (1979). A note on rates of convergence in the multidimensional CLT for maximum partial sums,J. Multivariate Anal.,9, 214–222.
Ahmad, I. A. (1980). On concentration functions of random sums of independent random variables,J. London Math. Soc.,21, 189–192.
Petrov, V. V. (1970). On an estimation of the concentration function of a sum of independent random variables,Theory Prob. Appl.,15, 701–703.
Von Bahr, B. and Esseen, C. G. (1965). Inequalities for ther-th absolute moment of a sum of random variables, 1≦r≦2,Ann. Math. Statist.,39, 299–303.
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Ahmad, I.A. On the deviation between the distributions of sums and maximum sums of IID random vectors. Ann Inst Stat Math 34, 351–358 (1982). https://doi.org/10.1007/BF02481034
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DOI: https://doi.org/10.1007/BF02481034