#### Date of Submission

2-22-1979

#### Date of Award

2-22-1980

#### Institute Name (Publisher)

Indian Statistical Institute

#### Document Type

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

#### Subject Name

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Ghosh, Malay

#### Abstract (Summary of the Work)

We obtain non-uniform rates of convergence to normality of the partial sums in a triangular array of random variables, where variables in each array are independently distributed. Section 2 of this chapter generalizes the results of Michel (1976) mainly in the direction of considering a triangular array of rand om variables. A slight generality in the moment assumptions is also made. The later extension is quite in spirit with Katzs (1963) extension of the classical Berry-Esseen theorem. Since by Tomkins theorem (see Tomkins (1971) or Stout (1974)) the laws of the iterated logarithm are directly related to the zone where 1-fn(tn)~ Ñ„ (-tn)âˆž where as a corollary of our theorem (2.2.6) (see in particular (2,2.52)) we are able to show that laws of iterated lo, arithm for sn holds if for some e > 0, which is incidents best known result for independent case (soe Stout again P 275). As other applications o these non-uniform rates we prove moment type convergences and a non-uniform L. version of Berry-Esseen theorem extending tho results of Erickson (1973).In section 3 of Chapter 2 we consider the case when all the finite moments of the underlying random variables exist but the m.g.f need not necessarily exist. Consideration of this situation helps us weaken the assumptions of the known results of the last four decades. As a consequence of non-uniform bounds in this case we prove CramÃ©rs(1938) results on normal approximation zone and on large deviations under milder conditions. We also prove Bahadur (1960) type upper class results on excessively large deviation (i.e., deviation of the form n a, a > 0) for random variables in a triangular array, sharpening his results with an estimate of 6 (n, e) (see Bahadur, 1960 paper) in i.i.d case, for both upper and lower class estimates. Other applications of these nonuniform rates includes moment type convergence and a stronger non-uniform Lp version of the Berry-Esseen theorem.In Chapter 2 we obtain non-uniform rates of convergence to normality of the partial sums in a triangular array of random variables, where variables in each array are independently distributed. Section 2 of this chapter generalizes the results of Michel (1976) mainly in the direction of considering a triangular array of rand om variables. A slight generality in the moment assumptions is also made. The later extension is quite in spirit with Katzs (1963) extension of the classical Berry-Esseen theorem. Since by Tomkins theorem (see Tomkins (1971) or Stout (1974)) the laws of the iterated logarithm are directly related to the zone where 1-fn(tn)~ Ñ„ (-tn)âˆž where as a corollary of our theorem (2.2.6) (see in particular (2,2.52)) we are able to show that laws of iterated lo, arithm for sn holds if for some e > 0, which is incidents best known result for independent case (soe Stout again P 275). As other applications o these non-uniform rates we prove moment type convergences and a non-uniform L. version of Berry-Esseen theorem extending tho results of Erickson (1973).In section 3 of Chapter 2 we consider the case when all the finite moments of the underlying random variables exist but the m.g.f need not necessarily exist. Consideration of this situation helps us weaken the assumptions of the known results of the last four decades. As a consequence of non-uniform bounds in this case we prove CramÃ©r's(1938) results on normal approximation zone and on large deviations under milder conditions. We also prove Bahadur (1960) type upper class results on excessively large deviation (i.e., deviation of the form n a, a > 0) for random variables in a triangular array, sharpening his results with an estimate of 6 (n, e) (see Bahadur, 1960 paper) in i.i.d case, for both upper and lower class estimates. Other applications of these nonuniform rates includes moment type convergence and a stronger non-uniform L. version of the Berry-Esseen theorem. In section 4 of Chapter 2 we partially cover the extreme case viz, when m.g.f necessarily exists but the rand om variables are not necessarily bounded. Since it is known (see Peller 1969) that so far normal approximation zone and large deviation zones are concerned boundlessness of the random variables is not of much help compared to the milder condition of existence of moment generating function, we content ourselves with a relatively stronger form compared to section 3, of L versions of the Berry-Esseen theorem and moment type convergences.

#### Control Number

ISILib-TH28

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

#### DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

#### Recommended Citation

Dasgupta, Ratan Dr., "On Some Non Uniform Rates of Convergence to Normality with Applications." (1980). *Doctoral Theses*. 174.

https://digitalcommons.isical.ac.in/doctoral-theses/174

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842951