On the rate of approximation in the random sum CLT for dependent variables

Adhir Kumar Basu

Aplikace matematiky (1987)

  • Volume: 32, Issue: 3, page 169-176
  • ISSN: 0862-7940

Abstract

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Capital " O " and lower-case " o " approximations of the expected value of a class of smooth functions ( f C r ( R ) ) of the normalized random partial sums of dependent random variables by the expectation of the corresponding functions of Gaussian random variables are established. The same types of approximation are also obtained for dependent random vectors. This generalizes and improves previous results of the author (1980) and Rychlik and Szynal (1979).

How to cite

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Basu, Adhir Kumar. "On the rate of approximation in the random sum CLT for dependent variables." Aplikace matematiky 32.3 (1987): 169-176. <http://eudml.org/doc/15490>.

@article{Basu1987,
abstract = {Capital $"O"$ and lower-case $"o"$ approximations of the expected value of a class of smooth functions $(f\in C^r(R))$ of the normalized random partial sums of dependent random variables by the expectation of the corresponding functions of Gaussian random variables are established. The same types of approximation are also obtained for dependent random vectors. This generalizes and improves previous results of the author (1980) and Rychlik and Szynal (1979).},
author = {Basu, Adhir Kumar},
journal = {Aplikace matematiky},
keywords = {random sums; central limit theorem; approximation theorems; random vectors; random sums; central limit theorem; approximation theorems},
language = {eng},
number = {3},
pages = {169-176},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the rate of approximation in the random sum CLT for dependent variables},
url = {http://eudml.org/doc/15490},
volume = {32},
year = {1987},
}

TY - JOUR
AU - Basu, Adhir Kumar
TI - On the rate of approximation in the random sum CLT for dependent variables
JO - Aplikace matematiky
PY - 1987
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 32
IS - 3
SP - 169
EP - 176
AB - Capital $"O"$ and lower-case $"o"$ approximations of the expected value of a class of smooth functions $(f\in C^r(R))$ of the normalized random partial sums of dependent random variables by the expectation of the corresponding functions of Gaussian random variables are established. The same types of approximation are also obtained for dependent random vectors. This generalizes and improves previous results of the author (1980) and Rychlik and Szynal (1979).
LA - eng
KW - random sums; central limit theorem; approximation theorems; random vectors; random sums; central limit theorem; approximation theorems
UR - http://eudml.org/doc/15490
ER -

References

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  1. A. K. Basu, 10.1016/0047-259X(80)90070-6, J. Multivariate Anal. 10, (1980), 565-578. (1980) Zbl0452.60027MR0599690DOI10.1016/0047-259X(80)90070-6
  2. P. L. Butzer L. Hahn W. Westphal, 10.1016/0021-9045(75)90042-8, J. Approx. Theory 13 (1975), 327-340. (1975) MR0394809DOI10.1016/0021-9045(75)90042-8
  3. M. Mamatov I. Nematov, On a limit theorem for sums of a random number of independent random variables, (Russian). Izv. Akad. Nauk, USSR Ser. Fiz. Mat. Nauk, 3 (1971), 18-24. (1971) MR0295419
  4. H. Robbins, 10.1090/S0002-9904-1948-09142-X, Bull. Amer. Math. Soc. 54 (1948), 1151-1161. (1948) Zbl0034.22503MR0027974DOI10.1090/S0002-9904-1948-09142-X
  5. Z. Rychlík D. Szynal, On the limit behavior of Sum of a random number of independent random variables, Coll. Math. 28 (1973), 147-159. (1973) MR0334311
  6. Z. Rychlík D. Szynal, On the rate of approximation in the random C-L.T, Theory of probability and Appl. 24 (1979), 620-625. (1979) MR0541376
  7. E. Rychlík Z. Rychlík, The generalized Anscombe Condition and its applications in random sum limit theorems, Lecture Notes in Math. Probability in Banach spaces I Springer-Verlag 828 (1980), 244-250. (1980) 
  8. V. Sakalauskas, 10.1007/BF00972282, Lithuanian Math. Jour. (Eng. Trans.) 17, 4 (1977), 567-572. (1977) MR0464370DOI10.1007/BF00972282
  9. S. Kh. Sirazhdinov G. Orazov, Generalization of a theorem of Robbins, (Russian), In Limit theorems and Statistical Inferences, Tashkent 1960, 154-162. (1960) 

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