Central limit theorem and weak law of large numbers with rates for martingales in Banach spaces

Butzer, Paul L.; Hahn, L.; Roeckerath, M.Th.

doi:10.1016/0047-259x(83)90027-1

Cited by 19 publications

(7 citation statements)

References 19 publications

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“…Basu (1976), Dvoretzky (1970), Bolthausen (1982), Butzer et al (1983), Häussler (1988), Rackauskas (1990)). This and related conditions have been assumed in several papers on the martingale convergence theorem (cf.…”

Section: Lemma 8111 Let the Regular Conditional Distributionsmentioning

confidence: 99%

Mass Transportation Problems

1998

Probability and Its Applications

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Section: Lemma 8111 Let the Regular Conditional Distributionsmentioning

confidence: 99%

Mass Transportation Problems

1998

Probability and Its Applications

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“…The latter. were generalized to the case of not necessarily independent random variables which form martingale difference sequences or al-rays in ' BuJTZER, HAHN and ROECKERATH [10,11], and to more general types of dependent random variables in BUTZER and ScHuLz [13][14][15], 'as well as to arbitrary sequences for the particular case of identically distributed random variables in BUTZER and KmscIEFINK [12]. As faras the authors are aware, there are no comparable general limit ,theorems with rates in the literature.…”

Section: Il€irmentioning

confidence: 99%

General Limit Theorems with $o$-Rates and Markov Processes under Pseudo-Moment Conditions

Butzer¹,

Kirschfink²

1988

Z. Anal. Anwend.

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Making use of the Dvoretzky extension of the Trotter-operator method, a general convergence theorem with o -rates for dependent, real-valued random variables satisfying a pseudo-moment condition is established. Applications are to general convergence theorems, a central limit theorem as well as a weak law of large numbers for Markov processes with discrete time parameter. Further, pseudo-moment conditions are discussed in detail and the results are compared with the theory of probability-metrics.

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“…For details concerning the relationship between the concepts of q'-dpe'oniposability V and infinite divisibility see [14].…”

Section: 6)mentioning

confidence: 99%

“…3.6, 141, the . rates in [4] being established for R"-valued r.vs., and in [38,14] for Banachspace valued r.vs. )…”

Section: Introduction and Historymentioning

confidence: 99%

General Random Sum Limit Theorems for Martingales with large $\mathcal O$-Rates

Butzer¹,

Schulz²

1983

Z. Anal. Anwend.

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Die vorliegende Arbeit beschäftigt sich mit groB-O Fehlerabschatzungenfur die Konvergenz in Verteilung von zufalligen Summen nicht notwendig unabhangiger Zufallsvariablen. Als Anwendungen einés aligemeinen Satzes werden iowohl Versionen des zentralen Grenzwertsatzes als auch des schwachen Gesetzes der.grol3en Zahien für Marti ngaldifferenzenfolgen im Falle der ufailigen Summation durch spezielle \ahl der Grenzzufallsvariablen hergeleitet. Beide Sätze werden mit 9-Konvergenzraten versehen. Pa6oTa nocBnu4eHa 19701ACHRaNt nOrpeliluOCTu Jrn CXOHMOCTII B pacnpeeiieiiuit CJ1y4aflHau CYMM 113 61ya1l1Hx Beiwiiiii, KOTOUC HO O6H3aTeJThIIO He3aBItCmlMbI. llpnMeHeH item HeRo-Topoft o6ueft TeopeMbl BHBOHTCR BHHTH IeHTpaJm1I0ft npeLenbHofl Teope'lM it cia6oro 3alcoua 601btmtx q uceji jni paaHocTHoI'o pa mapTHHranOB B ciy'iae ciyathioro cyMMupo--BaHUH [1OCOJCTB0M qacTuoro nai6opa npeenbHoft C 4aftHoft nepeMemlilon. B o6eiix reopeiax XaIOTCH 0-0UeHHH jjJIF1 CHOOCTIi cxouMpcTu.• This paper is concerned with large-9 error estimates for convergence in distribution of random sums of not necessarily independent random variables. As application s of a general theorem one obtains the-random-sum versions of the central limit theorem and of the weak law of large numbers for martingale difference sequences by specializing the limiting random variable. Both theorems are equipped with 0-rates.Dedicated to the memory of WOLFGANG RICHTER , a scholar of the theory of randomly indexed random variables. -7*

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Central limit theorem and weak law of large numbers with rates for martingales in Banach spaces

Cited by 19 publications

References 19 publications

Mass Transportation Problems

Mass Transportation Problems

General Limit Theorems with $o$-Rates and Markov Processes under Pseudo-Moment Conditions

General Random Sum Limit Theorems for Martingales with large $\mathcal O$-Rates

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