Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-14T15:20:28.804Z Has data issue: false hasContentIssue false
  • English
  • Français

Tauberian theorems for Jakimovski and Karamata-Stirling methods

Published online by Cambridge University Press:  26 February 2010

N. H. Bingham
N. H. Bingham
Affiliation:
Department of Mathematics, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey, TW20 0EX
Get access

Extract

We shall deal throughout with regular matrix methods of summability defined by a stochastic matrix A = (ank). Thus

means

Information

Type
Research Article
Information
Mathematika , Volume 35 , Issue 2 , December 1988, pp. 216 - 224
Copyright
Copyright © University College London 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Agnew, R. P.. The Lototsky method of evaluation of series. Michigan Math. J., 4 (1957), 105–128.CrossRefGoogle Scholar
2.Agnew, R. P.. Relations among the Lototsky, Borel and other methods for evaluation of series. Michigan Math. J., 6 (1959), 363–371.CrossRefGoogle Scholar
3.Andersen, E. Sparre. On the fluctuations of sums of random variables II. Math. Scand., 2 (1954), 195–223.Google Scholar
4.Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D.. Statistical inference under order restrictions (Wiley, 1972).Google Scholar
5.Barton, D. E. and Mallows, C. L.. Some aspects of the random sequence. Ann. Math. Statist., 36 (1965), 236–260.CrossRefGoogle Scholar
6.Bender, E. A.. Central and local limit theorems applied to asymptotic enumeration. J. Combinatorial Theory A, 15 (1973), 91–111.CrossRefGoogle Scholar
7.Bingham, N. H.. Tauberian theorems and the central limit theorem. Ann. Probab., 9 (1981), 221–231.CrossRefGoogle Scholar
8.Bingham, N. H.. On Euler and Borel summability. J. London Math. Soc. (2), 29 (1984), 141–146.CrossRefGoogle Scholar
9.Bingham, N. H.. Tauberian theorems for summability methods of random-walk type. J. London Math. Soc. (2), 30 (1984), 281–287.CrossRefGoogle Scholar
10.Bingham, N. H. and Maejima, M.. Summabiiity methods and almost sure convergence. Z. Wahrschein., 68 (1985), 383–392.CrossRefGoogle Scholar
11.Bingham, N. H. and Stadtmiiller, U.. Jakimovski methods and almost sure convergence.Google Scholar
12.Feller, W.. An introduction to probability theory and its applications. Volume I, 3rd ed. (Wiley, 1968).Google Scholar
13.Fridy, J. A. and Powell, R. A.. Tauberian theorems for matrices generated by analytic functions. Pacific J. Math., 92 (1981), 79–85.CrossRefGoogle Scholar
14.Gaier, D.. Der allgemeine Luckenumkehrsatz fur das Borel-Verfahren. Math. Z., 88 (1965), 410–417.CrossRefGoogle Scholar
15.Gaier, D.. On the coefficients and the growth of gap power series. SIAM J. Numerical Analysis, 31 (1966), 248–265.CrossRefGoogle Scholar
16.Goldie, C. M.. Record times, permutations and greatest convex minorants. Preprint (University of Sussex, 1988).Google Scholar
17.Hardy, G. H.. Divergent series (Oxford Univ. Press, 1949).Google Scholar
18.Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (Cambridge Univ. Press, 1964).Google Scholar
19.Harper, L. H.. Stirling behaviour is asymptotically normal. Ann. Math. Statist., 38 (1967). 410–414.CrossRefGoogle Scholar
20.Ibragimov, I. A. and Linnik, Yu. V.. Independent and stationary sequences of random variables (Wolters-Noordhoff, 1971).Google Scholar
21.Imhof, J.-P.. Stirling numbers and records. J. Combinatorial Theory A, 34 (1983), 252–254.CrossRefGoogle Scholar
22.Ingham, A. E.. On the high-indices theorem for Borel summability. Number theory and analysis (ed. Turán, P.) (Plenum Press, 1969), 121–135 (E. Landau Memorial Volume).Google Scholar
23.Jakimovski, A.. A generalisation of the Lototsky method of summability. Michigan Math. J., 6 (1959), 277–290.CrossRefGoogle Scholar
24.Jurkat, W. B.. Ein funktiontheoretischer Beweis fur O-Taubersätze bei den Verfahren von Borel and Euler-Knopp. Arch. Math. (Basel), 7 (1956), 278–283.CrossRefGoogle Scholar
25.Karamata, J.. Théorèmes sur la sommabilité exponentielle et d'autres sommabilites s'y rattachant. Mathematica (Cluj), 9 (1935), 164–178.Google Scholar
26.Knopp, K.. Über das Eulersche Summierungsverfahren, I, II. Math. Z., 15 (1922), 226–253, 18 (1923), 125–156.CrossRefGoogle Scholar
27.Lieb, E. H.. Convexity properties and a generating function for Stirling numbers. J. Combinatorial Theory, 5 (1968), 203–206.CrossRefGoogle Scholar
28.Martić, B.. The application of Karamata-Stirling methods to analytic continuation and relation with Borel's method of summation. Glasnik Mat.-Fiz. Astron., 16 (1961), 171–175.Google Scholar
29.Meyer-König, W.. Untersuchungen uber einige verwandte Limitierungsverfahren. Math. Z., 52 (1949), 257–304.CrossRefGoogle Scholar
30.Meyer-König, W. and Zeller, K.. Lückenumkehrsätze und Lückenperfektheit. Math. Z., 66 (1956), 203–224.CrossRefGoogle Scholar
31.Meyer-König, W. and Zeller, K.. Funktionanalytische Behandlung des Taylorschen Summierungsverfahren. Colloque sur la théorie des suites (CBRM, Paris-Louvain, 1958), 32–52.Google Scholar
32.Meyer-König, W. and Zeller, K.. On Borel's method of summability. Proc. Amer. Math. Soc., 11 (1960), 307–314.Google Scholar
33.Meyer-König, W. and Zeller, K.. Abschnittskonvergenz und Umkehrsatze beim Euler-Verfahren. Math. Z., 161 (1978), 147–154.CrossRefGoogle Scholar
34.Meyer-König, W. and Zeller, K.. Tauber-Satze und M-Perfektheit. Math. Z., 177 (1981), 257–266.CrossRefGoogle Scholar
35.Rényi, A.. On the extreme elements of observations. Hungarian, 1962. English translation: Selected Papers of Alfred Rényi, Vol. 3 (1976), 50–65.Google Scholar
36.Stam, A. J.. Cycles of random permutations. Ars Combinatoria, 16 (1983), 43–48.Google Scholar
37.Stieglitz, M.. Die allgemeine Form der O-Tauber-Bedingung fiir das Euler-Knopp und Borel Verfahren. J. reine ang. Math., 246 (1971), 172–179.Google Scholar
38.Turán, P.. On a new method of analysis and its applications, 2nd ed. (Wiley, 1984).Google Scholar
39.Vučković, V.. The mutual inclusion of Karamata-Stirling methods of summation. Michigan Math. J., 6 (1959), 291–297.CrossRefGoogle Scholar
40.Zeller, K. and Beekmann, W.. Theorie der Limitierungsverfahren, 2nd ed. (Springer, 1970).CrossRefGoogle Scholar