Doctoral Theses

Some External Problem in Graph Theory of Convergence to Normality With Applications.

2-28-2011

2-28-2012

Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

Degree Name

Doctor of Philosophy

Mathematics

Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

NA

Abstract (Summary of the Work)

The cuntral rela of the normal distribution in ststirstionl thecry and methodology is beyond qaestion. Apart from other sttra- ctive festures, one important reason why this diatribution has been found to be so aznful ta that 1t turna out to be the lisiting distrihution of sony vll-koun atatietiee (for eaunpla the akaple 20, oanple zomenta ate,) after Daitable stanaardisati ons under lly very moderste seeunytiona.A result of pivotal imgortance in thia reopect ls tha so called centanl limit Theoren (CL2) which saya that if x1,x2.. is a guenoe of iiare'a with common mean and common vartance o2 (o < o2 < ), then defining Sr (t)and Pn((t) = p((Sn -nu)/( no ) < t), one has ,ls tha distributton fanntion of a. 0,1) variable, one major linitation of the shove resalt io thas it doea sot anything reguzting the rote of convergence of Pn, to 1. the independent work of berry and Esseen show that if in addition E x1 3 < ,The condition (1,1,2) was later veakened by Katz (1963) to (1.1.4)E [x2g(x)]< where g(x) is an even, nonne gative, real valued, nondecreasing function with x /g(x) nondecreasing on 0, ). He could then prove that (1.1.5) t These uniform rates of convergence, though very useful are non adcquate for many purposes. For instance, if it is kmown that 8 ic a function such th t Eg(T|< , where T is N(0,1), then in view of the fact that (s -n )/(VÃ±a) , T, one might intuitively expect that E g((s-n )/ha|) --> E g(T) as n -> Or, one might be interested in knowing whe ther a Ln-version of the Berry Esseen The orem holds, 1.e., whe ther(1.1.6) |sup 1 Fn(t)-(t) Il, - Cg-1 %3! as n -> 0 Questions of this type cannot be answered from the uniform rates of convergence given in (1,1.3) or (1.1,5).The main reason why bounds of the type (1.1.3) or (1.1.5) are inadequate for the above purposes is that they do not reflect the role oft t in the rate of convergence. The following result of Magaev(1965) gives a nonuniform rate of convergence (t) to (1), let X, X2, and EIX|o(t)Theorem 1.1,.1let be in with E X1,EX21 0, 1 Then, |F, (t31) - Fn(t)| 3 Cn-1/2 (1 â€¢ t3 ,-1, (1,1.7)where C is a universal constantRecently Michel (1976) has provided an interesting approaci in the study of nonuniform rates of convergence of P,(t) to (t). His way to tackle the problen .a to break up ne positive axis into tuo regiona, and obtain two different bounds for the difference |Pn(t) - 0(t) | depending on the region where t2 belongs. Thia idea waa possibly implict in Esseen (1945), but was explored very effectively by Michel (1976.

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