Lindeberg functions and the law of the iterated logarithm
Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦...
Ausführliche Beschreibung
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| Autor*in: |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
1983 |
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| Umfang: |
9 |
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| Reproduktion: |
Springer Online Journal Archives 1860-2002 |
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| Übergeordnetes Werk: |
in: Probability theory and related fields - 1992, 65(1983) vom: Jan., Seite 135-143 |
| Übergeordnetes Werk: |
volume:65 ; year:1983 ; month:01 ; pages:135-143 ; extent:9 |
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NLEJ205675026 |
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| 520 | |a Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ | ||
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(DE-627)NLEJ205675026 DE-627 ger DE-627 rakwb eng Lindeberg functions and the law of the iterated logarithm 1983 9 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ Springer Online Journal Archives 1860-2002 Tomkins, R. J. oth in Probability theory and related fields 1992 65(1983) vom: Jan., Seite 135-143 (DE-627)NLEJ188987940 (DE-600)1462994-x 1432-2064 nnns volume:65 year:1983 month:01 pages:135-143 extent:9 http://dx.doi.org/10.1007/BF00535000 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 65 1983 1 135-143 9 |
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(DE-627)NLEJ205675026 DE-627 ger DE-627 rakwb eng Lindeberg functions and the law of the iterated logarithm 1983 9 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ Springer Online Journal Archives 1860-2002 Tomkins, R. J. oth in Probability theory and related fields 1992 65(1983) vom: Jan., Seite 135-143 (DE-627)NLEJ188987940 (DE-600)1462994-x 1432-2064 nnns volume:65 year:1983 month:01 pages:135-143 extent:9 http://dx.doi.org/10.1007/BF00535000 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 65 1983 1 135-143 9 |
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(DE-627)NLEJ205675026 DE-627 ger DE-627 rakwb eng Lindeberg functions and the law of the iterated logarithm 1983 9 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ Springer Online Journal Archives 1860-2002 Tomkins, R. J. oth in Probability theory and related fields 1992 65(1983) vom: Jan., Seite 135-143 (DE-627)NLEJ188987940 (DE-600)1462994-x 1432-2064 nnns volume:65 year:1983 month:01 pages:135-143 extent:9 http://dx.doi.org/10.1007/BF00535000 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 65 1983 1 135-143 9 |
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(DE-627)NLEJ205675026 DE-627 ger DE-627 rakwb eng Lindeberg functions and the law of the iterated logarithm 1983 9 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ Springer Online Journal Archives 1860-2002 Tomkins, R. J. oth in Probability theory and related fields 1992 65(1983) vom: Jan., Seite 135-143 (DE-627)NLEJ188987940 (DE-600)1462994-x 1432-2064 nnns volume:65 year:1983 month:01 pages:135-143 extent:9 http://dx.doi.org/10.1007/BF00535000 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 65 1983 1 135-143 9 |
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(DE-627)NLEJ205675026 DE-627 ger DE-627 rakwb eng Lindeberg functions and the law of the iterated logarithm 1983 9 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ Springer Online Journal Archives 1860-2002 Tomkins, R. J. oth in Probability theory and related fields 1992 65(1983) vom: Jan., Seite 135-143 (DE-627)NLEJ188987940 (DE-600)1462994-x 1432-2064 nnns volume:65 year:1983 month:01 pages:135-143 extent:9 http://dx.doi.org/10.1007/BF00535000 GBV_USEFLAG_U ZDB-1-SOJ GBV_NL_ARTICLE AR 65 1983 1 135-143 9 |
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in Probability theory and related fields 65(1983) vom: Jan., Seite 135-143 volume:65 year:1983 month:01 pages:135-143 extent:9 |
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in Probability theory and related fields 65(1983) vom: Jan., Seite 135-143 volume:65 year:1983 month:01 pages:135-143 extent:9 |
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Lindeberg functions and the law of the iterated logarithm |
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Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ |
| abstractGer |
Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ |
| abstract_unstemmed |
Summary For a sequence of independent random variables {X n} with zero means and finite variances, define $$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}$$ and t n 2 =2 loglog s n 2 ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions $$s_n^{ - 2} \sum\limits_{j = 1}^n {E(X_j^2 I(|X_j | \leqq \varepsilon s_j t_j^{ - 1} )), where \varepsilon > 0}$$ |
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