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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IntrodiirtiorrThe random variables (r.vs.) X , . i E S : = ( 1 , 2 ...I, to he dealt uith are measurable mappings from a probability space (9. 91, P ) into a measure space (B, %), R being a RANACH space with a countable hasis (e,),,,, and ' 3 the o-algebra of 13orel sets of B. The type of convergence to be mainly considered is the convergence in distribution of the B-valued partial sums Sn=c Xi, thus the weak convergence o f the distrihutions PAa of S, to that of a limiting r.v.Z, denoted hy Pz. This means that n i -1 for each f

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