Date of Submission


Date of Award


Institute Name (Publisher)

Indian Statistical Institute

Document Type

Doctoral Thesis

Degree Name

Doctor of Philosophy

Subject Name



Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)


Rao, B. V. (TSMU-Kolkata; ISI)

Abstract (Summary of the Work)

In thia theoia ve study the locel behaviour or semi mertingalea. Civen e cant!nuoue sent rartineale and an i-tetval (n,t), ve define a new pracees which airrara the bereufour of the originel procese in the interval (a, b). Thie neu procees tur ns cut to he a semi-nartingale uhose junpa during [J,t: ere closely reinted to the number of croosinge of (o,t) during [0,t ].Jur sterting point is indned P, Levy's rortingale cherocterizotion of around an motion : If (xt) and (x2t - t) ore cantinuOun lacol nar- tingales then (xt) must be a Srownion mation. Let now Hn(x,t) denote the n-th Hornito polyremiai (oefined in Section 1,2). Suppose thet (xt) and (Hn(xt,t)) ara continus locel mor tingolca. Lovy charac- terization says thet for n = 2, this imaldios that (Xt) muat be a Brounian mation. For n > 2, this is not in gonorol true. Houav er ty Impoaing on (nnalytical) condi tion (Theorem 1.2.1, A.2), ve ehou that (xt) muet be e Brownien motion. Roughly speaking the condition aaya thet the sojourn tima of the candi de te in o certein eet ahould have zero lebeegue meos ure. In tre next sectior (Section 1,3) we atudy the sojouen tima of a continuous rartinçole in or interval (c,b). Je denari be the expectad mojourn timo of the orocoes in (a,b) during i0,t). in terme of the axpocted numbor of croseings of (o,b) during (D,t ond terma uhich are amall - af tha or der of (b - a)2 (Thoorem 1,3.1), Tue Corollarias are Irmediate, Firstly,uo çot on expectad vereion of Lowy doun crossing therem. Secondly, the ratlo of tha expected aojaurn timo in (o,b) during (0,t) to the expeotad number of scoesings of (a,b during [0,t] con- verços to (b- a)22 an t->a. The idea of an1-nertingales Baso- oietad uith ar Oneinge fallawa naturn:ly fron Theoren 1.3.1 (oee the prefacing remarka in Section 1,4, Chapter 1). In Soction 1.4 we prove the existonce af such a Bori-mer tingele by tera-hand techniqueet deve- lopad in the couree of the preaf af Thaerom 1,3.1.


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