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An invariance principle for strongly multiplicative sequences

An invariance principle for strongly multiplicative sequences Acta Mathematiea Academiae Seientiarum Hungaricae Tomus 26 (1--2), (1975), 81--85. AN INVARIANCE PRINCIPLE FOR STRONGLY MULTIPLICATIVE SEQUENCES By D. L. MCLEISH (Downsview) Introduction. Closely associated with most (though not all) central limit theorems is an invariance principle, also called Donsker's Theorem or a "functional central limit theorem". BROWN [2], and Lov~es [4], prove an invariance principle for mar- tingales and reverse martingales respectively, and invariance principles for dependent as well as independent variables appear in BILLINGSLEY [l] and PARTHASARATHY [5]. Here, we show an invariance principle complementary to several of the "strongly multiplicative" central limit theorems investigated recently, primarily by the Hungar- ian school. Let {X~; i-1, 2, 3, ...} be a sequence of random variables from an arbitrary probability triple (f2, ~, P) and suppose E(Xi)=0, E(X~)=G~ for all i= 1, 2, 3 ..... For convenience of notation, we will put S,,,~,= --~ X~ and Vs _ ~ ~r~,2 i=m+l i=m+l S,=So,, and V~= V02,,. By "Z', we shall mean "converges weakly to", usually in 9 the space of real numbers (in which case it is equivalent to convergence in distribu- tion) or in the space of right continuous functions on [0, 1] with no discontinuities of the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

An invariance principle for strongly multiplicative sequences

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01895950
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Abstract

Acta Mathematiea Academiae Seientiarum Hungaricae Tomus 26 (1--2), (1975), 81--85. AN INVARIANCE PRINCIPLE FOR STRONGLY MULTIPLICATIVE SEQUENCES By D. L. MCLEISH (Downsview) Introduction. Closely associated with most (though not all) central limit theorems is an invariance principle, also called Donsker's Theorem or a "functional central limit theorem". BROWN [2], and Lov~es [4], prove an invariance principle for mar- tingales and reverse martingales respectively, and invariance principles for dependent as well as independent variables appear in BILLINGSLEY [l] and PARTHASARATHY [5]. Here, we show an invariance principle complementary to several of the "strongly multiplicative" central limit theorems investigated recently, primarily by the Hungar- ian school. Let {X~; i-1, 2, 3, ...} be a sequence of random variables from an arbitrary probability triple (f2, ~, P) and suppose E(Xi)=0, E(X~)=G~ for all i= 1, 2, 3 ..... For convenience of notation, we will put S,,,~,= --~ X~ and Vs _ ~ ~r~,2 i=m+l i=m+l S,=So,, and V~= V02,,. By "Z', we shall mean "converges weakly to", usually in 9 the space of real numbers (in which case it is equivalent to convergence in distribu- tion) or in the space of right continuous functions on [0, 1] with no discontinuities of the

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: May 21, 2016

References

  • Convergence of Probability Measures
    Billingsley, P.
  • Martingale central limit theorems
    Brown, B. M.
  • A central limit theorem for strongly multiplicative systems
    Komlós, J.
  • The central limit theorem for backward martingales
    Loynes
  • Probability Measures on Metric Spaces
    Parthasarathy, K. R.