General theorems on rates of convergence in distribution of random variables II. Applications to the stable limit laws and weak law of large numbers

Butzer, Paul L.; Hahn, L.

doi:10.1016/0047-259x(78)90072-6

Cited by 29 publications

(12 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Below in section Results we demonstrate by different examples, that our algorithm is more efficient than the standard Kolmogorov-Smirnov test. We note that the limit theorems which specify the rate of convergence to the stable law in one and multi-dimensional cases have been studied in [ 39 – 41 ]. The problem of convergence to a stable distribution via order statistics was considered in [ 42 ].…”

Section: Introductionmentioning

confidence: 99%

Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem

2015

View full text Add to dashboard Cite

In this paper we propose an algorithm to distinguish between light- and heavy-tailed probability laws underlying random datasets. The idea of the algorithm, which is visual and easy to implement, is to check whether the underlying law belongs to the domain of attraction of the Gaussian or non-Gaussian stable distribution by examining its rate of convergence. The method allows to discriminate between stable and various non-stable distributions. The test allows to differentiate between distributions, which appear the same according to standard Kolmogorov–Smirnov test. In particular, it helps to distinguish between stable and Student’s t probability laws as well as between the stable and tempered stable, the cases which are considered in the literature as very cumbersome. Finally, we illustrate the procedure on plasma data to identify cases with so-called L-H transition.

show abstract

Section: Introductionmentioning

confidence: 99%

Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem

2015

View full text Add to dashboard Cite

show abstract

“…RYCULK and D SZYNAL [41] on aceountof our use of K-functional methods. They are indeed just as sharp as those-of V P. L. BUTZER and L. H.&JrN [11,12] in the case of non-randorn'surnmation of independent r.vs. Returning to, the proofs again; our main theorem is based , upon a modificat ion of the Trotter operatort heoretic method to the situation of not necessarily independent r.vs.…”

mentioning

confidence: 61%

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

Butzer¹,

Schulz²

1983

Z. Anal. Anwend.

View full text Add to dashboard Cite

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*

show abstract

“…The corresponding basic asymptotic expansion assertions are Thms. 4 However, relatively little effort has been devoted to the central limit theorem for general second order stochastic processes which are not necessarily stationary nor have independent increments. Now the process (1.10) is not necessarily strictly or weakly stationary, unless v ( t ) , the function associated to the Bore1 measure V, reduces to a linear function, nor does { have independent increments.…”

Section: ) X ( T ) = C Ana(t-tn)mentioning

confidence: 99%

“…For the small-0 counterpart of Lemma 3 we need a further condition, namely a Lindeberg-type condition of order k (see [4]; this is Merent to definition in [I]). A GLSP satisfying conditions (I)-(III) and Bk,T .c coy T > 0, is said to fulfd this condition provided for Tco and each E > 0 (compare with (B2)).…”

Section: A Lindeberg-type Condition and Further Lemmasmentioning

confidence: 99%

Asymptotic expansions for central limit theorems for general linear stochastic processes. I: General theorems on rates of convergence

Butzer

Gather

Törnig³

1979

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper deals with so‐called general linear stochastic processes (GLSP), defined by T. Kawata in 1972 in generalization of work of R. Lugannani and J. B. Thomas of 1967/71. These second order processes (which are not necessarily stationary nor have independent increments) are described by rather weak requirements, so that several processes such as some random noise and pulse train processes are specific models of these GLSP. Part I is concerned with two general theorems giving asymptotic expansions (including those for the density function) in the central limit theorem for such GLSP, together with error rates. The assumptions for the corresponding θ– and o–error estimates seem rather natural: in the former, apart from assumptions on the inherent structure of such GLSP, the existence of certain moments of higher order as well as a Cramer‐type condition are assumed, in the latter in addition a Lindeberg‐type condition of higher order. Fourier analytic machinery is used for the proofs.

show abstract

General theorems on rates of convergence in distribution of random variables II. Applications to the stable limit laws and weak law of large numbers

Cited by 29 publications

References 7 publications

Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem

Discriminating between Light- and Heavy-Tailed Distributions with Limit Theorem

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

Asymptotic expansions for central limit theorems for general linear stochastic processes. I: General theorems on rates of convergence

Contact Info

Product

Resources

About