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

Abstract

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*

The Conditional Lindeberg–Trotter Operator in the Resolution of Limit Theorems with Rates for Dependent Random Variables. Applications to Markovian Processes

“…Remark: As already mentioned in the introduction, Theorem 2 and the Corollary are the most general theorems known to us in the matter. They are generalizations of the comparable results for independent random, variables [7] and those for Martingale difference arrays [11]. Possible applications are indicated in the introduction.…”

“…-Proof: The theorem follows by Theorem 2, noting that X* is -deconiposable (see e.g. [111) with P. = ")Z" where the decomposition components 7,,. are, normally distributd random variables-with mean zero and variance ak2 ; they may, without loss of generality (see [11), be chosen to. be independent , amongst themselves as well as of the random variables X k • Let us now forinulatesonie handy versions of the central limit theorem for dependent randohi variables.…”

“…-This Lemma is o be found implicitly in ZOLOTAREV [30] (see also [311), and formulated explicitly in [11]. Let us now consider the general convergenbe theorem for the strong cônvergenée, as mentioned above.…”

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

“…As faras the authors are aware, there are no comparable general limit ,theorems with rates in the literature. • The type of convergence considered in this paper is more general than that in BUTZER, HAHN and ROECRERATH [10], or in BUTZER and SCHULZ [13] where the dependency structure is of type of martingale difference sequences. In fact; the convergence considered in (1.1) is that for the difference f /(x + u) dFT,,(x) -f /(x ± u) dFz(x)…”

“…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.…”