Lindeberg functions and the law of the iterated logarithm

Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if...
Ausführliche Beschreibung

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Autor*in:	Tomkins, R. J. [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	1983

Schlagwörter:	Independent Random Variable Iterate Logarithm Real Sequence Finite Variance Martingale Difference

Anmerkung:	© Springer-Verlag 1983

Übergeordnetes Werk:	Enthalten in: Probability theory and related fields - Berlin : Springer, 1992, 65(1983), 1 vom: Nov., Seite 135-143
Übergeordnetes Werk:	volume:65 ; year:1983 ; number:1 ; month:11 ; pages:135-143

Links:	Volltext

DOI / URN:	10.1007/BF00535000

Katalog-ID:	SPR005996422

Internformat


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041			\|a eng
100	1		\|a Tomkins, R. J. \|e verfasserin \|4 aut
245	1	0	\|a Lindeberg functions and the law of the iterated logarithm
264		1	\|c 1983
336			\|a Text \|b txt \|2 rdacontent
337			\|a Computermedien \|b c \|2 rdamedia
338			\|a Online-Ressource \|b cr \|2 rdacarrier
500			\|a © Springer-Verlag 1983
520			\|a Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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}%$
650		4	\|a Independent Random Variable \|7 (dpeaa)DE-He213
650		4	\|a Iterate Logarithm \|7 (dpeaa)DE-He213
650		4	\|a Real Sequence \|7 (dpeaa)DE-He213
650		4	\|a Finite Variance \|7 (dpeaa)DE-He213
650		4	\|a Martingale Difference \|7 (dpeaa)DE-He213
773	0	8	\|i Enthalten in \|t Probability theory and related fields \|d Berlin : Springer, 1992 \|g 65(1983), 1 vom: Nov., Seite 135-143 \|w (DE-627)254638791 \|w (DE-600)1462994-X \|x 1432-2064 \|7 nnns
773	1	8	\|g volume:65 \|g year:1983 \|g number:1 \|g month:11 \|g pages:135-143
856	4	0	\|u https://dx.doi.org/10.1007/BF00535000 \|z lizenzpflichtig \|3 Volltext
912			\|a GBV_USEFLAG_A
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912			\|a GBV_ILN_20
912			\|a GBV_ILN_22
912			\|a GBV_ILN_23
912			\|a GBV_ILN_24
912			\|a GBV_ILN_31
912			\|a GBV_ILN_32
912			\|a GBV_ILN_39
912			\|a GBV_ILN_40
912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_121
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_206
912			\|a GBV_ILN_224
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_374
912			\|a GBV_ILN_602
912			\|a GBV_ILN_647
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2018
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2043
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2116
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2119
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
912			\|a GBV_ILN_2148
912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2158
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2193
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_2808
912			\|a GBV_ILN_4012
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
912			\|a GBV_ILN_4046
912			\|a GBV_ILN_4112
912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4126
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4277
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
912			\|a GBV_ILN_4307
912			\|a GBV_ILN_4313
912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
912			\|a GBV_ILN_4325
912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4328
912			\|a GBV_ILN_4333
912			\|a GBV_ILN_4334
912			\|a GBV_ILN_4335
912			\|a GBV_ILN_4336
912			\|a GBV_ILN_4338
912			\|a GBV_ILN_4346
912			\|a GBV_ILN_4367
912			\|a GBV_ILN_4393
912			\|a GBV_ILN_4700
912			\|a GBV_ILN_4753
951			\|a AR
952			\|d 65 \|j 1983 \|e 1 \|c 11 \|h 135-143

Indexfelder

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matchkey_str	article:14322064:1983----::idbrfntosnteaotetr
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publishDate	1983
allfields	10.1007/BF00535000 doi (DE-627)SPR005996422 (SPR)BF00535000-e DE-627 ger DE-627 rakwb eng Tomkins, R. J. verfasserin aut Lindeberg functions and the law of the iterated logarithm 1983 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 1983 Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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}%$ Independent Random Variable (dpeaa)DE-He213 Iterate Logarithm (dpeaa)DE-He213 Real Sequence (dpeaa)DE-He213 Finite Variance (dpeaa)DE-He213 Martingale Difference (dpeaa)DE-He213 Enthalten in Probability theory and related fields Berlin : Springer, 1992 65(1983), 1 vom: Nov., Seite 135-143 (DE-627)254638791 (DE-600)1462994-X 1432-2064 nnns volume:65 year:1983 number:1 month:11 pages:135-143 https://dx.doi.org/10.1007/BF00535000 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 65 1983 1 11 135-143
spelling	10.1007/BF00535000 doi (DE-627)SPR005996422 (SPR)BF00535000-e DE-627 ger DE-627 rakwb eng Tomkins, R. J. verfasserin aut Lindeberg functions and the law of the iterated logarithm 1983 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 1983 Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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}%$ Independent Random Variable (dpeaa)DE-He213 Iterate Logarithm (dpeaa)DE-He213 Real Sequence (dpeaa)DE-He213 Finite Variance (dpeaa)DE-He213 Martingale Difference (dpeaa)DE-He213 Enthalten in Probability theory and related fields Berlin : Springer, 1992 65(1983), 1 vom: Nov., Seite 135-143 (DE-627)254638791 (DE-600)1462994-X 1432-2064 nnns volume:65 year:1983 number:1 month:11 pages:135-143 https://dx.doi.org/10.1007/BF00535000 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 65 1983 1 11 135-143
allfields_unstemmed	10.1007/BF00535000 doi (DE-627)SPR005996422 (SPR)BF00535000-e DE-627 ger DE-627 rakwb eng Tomkins, R. J. verfasserin aut Lindeberg functions and the law of the iterated logarithm 1983 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 1983 Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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}%$ Independent Random Variable (dpeaa)DE-He213 Iterate Logarithm (dpeaa)DE-He213 Real Sequence (dpeaa)DE-He213 Finite Variance (dpeaa)DE-He213 Martingale Difference (dpeaa)DE-He213 Enthalten in Probability theory and related fields Berlin : Springer, 1992 65(1983), 1 vom: Nov., Seite 135-143 (DE-627)254638791 (DE-600)1462994-X 1432-2064 nnns volume:65 year:1983 number:1 month:11 pages:135-143 https://dx.doi.org/10.1007/BF00535000 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 65 1983 1 11 135-143
allfieldsGer	10.1007/BF00535000 doi (DE-627)SPR005996422 (SPR)BF00535000-e DE-627 ger DE-627 rakwb eng Tomkins, R. J. verfasserin aut Lindeberg functions and the law of the iterated logarithm 1983 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 1983 Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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}%$ Independent Random Variable (dpeaa)DE-He213 Iterate Logarithm (dpeaa)DE-He213 Real Sequence (dpeaa)DE-He213 Finite Variance (dpeaa)DE-He213 Martingale Difference (dpeaa)DE-He213 Enthalten in Probability theory and related fields Berlin : Springer, 1992 65(1983), 1 vom: Nov., Seite 135-143 (DE-627)254638791 (DE-600)1462994-X 1432-2064 nnns volume:65 year:1983 number:1 month:11 pages:135-143 https://dx.doi.org/10.1007/BF00535000 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 65 1983 1 11 135-143
allfieldsSound	10.1007/BF00535000 doi (DE-627)SPR005996422 (SPR)BF00535000-e DE-627 ger DE-627 rakwb eng Tomkins, R. J. verfasserin aut Lindeberg functions and the law of the iterated logarithm 1983 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 1983 Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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}%$ Independent Random Variable (dpeaa)DE-He213 Iterate Logarithm (dpeaa)DE-He213 Real Sequence (dpeaa)DE-He213 Finite Variance (dpeaa)DE-He213 Martingale Difference (dpeaa)DE-He213 Enthalten in Probability theory and related fields Berlin : Springer, 1992 65(1983), 1 vom: Nov., Seite 135-143 (DE-627)254638791 (DE-600)1462994-X 1432-2064 nnns volume:65 year:1983 number:1 month:11 pages:135-143 https://dx.doi.org/10.1007/BF00535000 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_121 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_224 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_374 GBV_ILN_602 GBV_ILN_647 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2043 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_2808 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 65 1983 1 11 135-143
language	English
source	Enthalten in Probability theory and related fields 65(1983), 1 vom: Nov., Seite 135-143 volume:65 year:1983 number:1 month:11 pages:135-143
sourceStr	Enthalten in Probability theory and related fields 65(1983), 1 vom: Nov., Seite 135-143 volume:65 year:1983 number:1 month:11 pages:135-143
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Independent Random Variable Iterate Logarithm Real Sequence Finite Variance Martingale Difference
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container_title	Probability theory and related fields
authorswithroles_txt_mv	Tomkins, R. J. @@aut@@
publishDateDaySort_date	1983-11-01T00:00:00Z
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J.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Lindeberg functions and the law of the iterated logarithm</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">1983</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Springer-Verlag 1983</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. 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author	Tomkins, R. J.
spellingShingle	Tomkins, R. J. misc Independent Random Variable misc Iterate Logarithm misc Real Sequence misc Finite Variance misc Martingale Difference Lindeberg functions and the law of the iterated logarithm
authorStr	Tomkins, R. J.
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topic_title	Lindeberg functions and the law of the iterated logarithm Independent Random Variable (dpeaa)DE-He213 Iterate Logarithm (dpeaa)DE-He213 Real Sequence (dpeaa)DE-He213 Finite Variance (dpeaa)DE-He213 Martingale Difference (dpeaa)DE-He213
topic	misc Independent Random Variable misc Iterate Logarithm misc Real Sequence misc Finite Variance misc Martingale Difference
topic_unstemmed	misc Independent Random Variable misc Iterate Logarithm misc Real Sequence misc Finite Variance misc Martingale Difference
topic_browse	misc Independent Random Variable misc Iterate Logarithm misc Real Sequence misc Finite Variance misc Martingale Difference
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title	Lindeberg functions and the law of the iterated logarithm
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title_full	Lindeberg functions and the law of the iterated logarithm
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author-letter	Tomkins, R. J.
doi_str_mv	10.1007/BF00535000
title_sort	lindeberg functions and the law of the iterated logarithm
title_auth	Lindeberg functions and the law of the iterated logarithm
abstract	Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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-Verlag 1983
abstractGer	Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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-Verlag 1983
abstract_unstemmed	Summary For a sequence of independent random variables {Xn} with zero means and finite variances, define %$S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}%$ and tn2=2 loglog sn2; assume sn→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup Sn/($ s_{n} %$ t_{n} $)=1 a.s. if tn¦$ X_{n} $¦≦ɛn$ s_{n} $ for some real sequence n→∞ $ ɛ_{n} $→0. This paper will show that, under the weaker condition tn$ X_{n} $/$ s_{n} $→0 a.s., the a.s. limiting value of lim sup Sn($ s_{n} %$ t_{n} $) depends on the limiting behaviour of the modified Lindeberg functions %$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-Verlag 1983
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title_short	Lindeberg functions and the law of the iterated logarithm
url	https://dx.doi.org/10.1007/BF00535000
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doi_str	10.1007/BF00535000
up_date	2024-07-03T20:08:49.367Z
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Lindeberg functions and the law of the iterated logarithm

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