For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of ...For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent;（ii） For all constants λ ∈ [0, ∞）,lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.展开更多
Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such thatan↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞Set Sn=∑i...Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such thatan↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.展开更多
Health related quality of life(HRQOL) is increasingly recognized as an important clinical parameter and research endpoint in patients with hepatocellular carcinoma(HCC). HRQOL in HCC patients is multifaceted and affec...Health related quality of life(HRQOL) is increasingly recognized as an important clinical parameter and research endpoint in patients with hepatocellular carcinoma(HCC). HRQOL in HCC patients is multifaceted and affected by medical factor which encompasses HCC and its complications, oncological and palliative treatment for HCC, underlying liver disease, as well as the psychological, social or spiritual reaction to the disease. Many patients presented late with advanced disease and limited survival, plagued with multiple symptoms, rendering QOL a very important aspect in their general well being. Various instruments have been developed and validated to measure and report HRQOL in HCC patients, these included general HRQOL instruments, e.g., Short form(SF)-36, SF-12, Euro Qo L-5D, World Health Organization Quality of Life Assessment 100(WHOQOL-100), World Health Organization Quality of Life Assessment abbreviated version; general cancer HRQOL instruments, e.g., the European Organisation for Research and Treatment of Cancer(EORTC) QLQ-C30, Functional Assessment of Cancer Therapy(FACT)-General, Spitzer Quality of Life Index; and liver-cancer specific HRQOL instruments, e.g., EORTC QLQ-HCC18, FACT-Hepatobiliary(FACT-Hep), FACT-Hep Symptom Index, Trial Outcome Index. Important utilization of HRQOL in HCC patients included description of symptomatology and HRQOL of patients, treatment endpoint in clinical trial, prognostication of survival, benchmarking of palliative care service and health care valuation. In this review, difficulties regarding the use of HRQOL data in research and clinical practice, including choosing a suitable instrument, problems of missing data, data interpretation, analysis and presentation are examined. Potential solutions are also discussed.展开更多
Adopting the observational distributions of star formation rates and half-light radii of Lyman break galaxies （LBGs） in the rest frame UV, we investigate empirically the predicted stellar mass function for LBGs. It ...Adopting the observational distributions of star formation rates and half-light radii of Lyman break galaxies （LBGs） in the rest frame UV, we investigate empirically the predicted stellar mass function for LBGs. It is found that a peak exists at mass around 1.5 × 10^10M⊙ for their stellar mass function and there have been significant amount of LBGs with stellar masses higher than 10^11M⊙ at z ～ 3. There should be two families for LBG population observed by the Infrared Array Camera （IRAC） （which is a key instrument in the infrared space telescope Spitzer）, one is red and the other is blue, which are consistent with recent works. Moreover, the predicted comoving correlation lengths for LBGs with different stellar masses, i.e., with different 8 μm magnitudes, are also discussed. The red LBGs should be clustered in space more strongly than the blue LBGs. These predictions can and will be tested by the observations of IRAC, and will provide strong constraints on the current hierarchical galaxy formation model.展开更多
基金the Natural Sciences and Engineering Research Council of Canada
文摘For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent;（ii） For all constants λ ∈ [0, ∞）,lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.
基金a grant from the Natural Sciences and Engineering Research Council of Canada
文摘Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such thatan↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.
文摘Health related quality of life(HRQOL) is increasingly recognized as an important clinical parameter and research endpoint in patients with hepatocellular carcinoma(HCC). HRQOL in HCC patients is multifaceted and affected by medical factor which encompasses HCC and its complications, oncological and palliative treatment for HCC, underlying liver disease, as well as the psychological, social or spiritual reaction to the disease. Many patients presented late with advanced disease and limited survival, plagued with multiple symptoms, rendering QOL a very important aspect in their general well being. Various instruments have been developed and validated to measure and report HRQOL in HCC patients, these included general HRQOL instruments, e.g., Short form(SF)-36, SF-12, Euro Qo L-5D, World Health Organization Quality of Life Assessment 100(WHOQOL-100), World Health Organization Quality of Life Assessment abbreviated version; general cancer HRQOL instruments, e.g., the European Organisation for Research and Treatment of Cancer(EORTC) QLQ-C30, Functional Assessment of Cancer Therapy(FACT)-General, Spitzer Quality of Life Index; and liver-cancer specific HRQOL instruments, e.g., EORTC QLQ-HCC18, FACT-Hepatobiliary(FACT-Hep), FACT-Hep Symptom Index, Trial Outcome Index. Important utilization of HRQOL in HCC patients included description of symptomatology and HRQOL of patients, treatment endpoint in clinical trial, prognostication of survival, benchmarking of palliative care service and health care valuation. In this review, difficulties regarding the use of HRQOL data in research and clinical practice, including choosing a suitable instrument, problems of missing data, data interpretation, analysis and presentation are examined. Potential solutions are also discussed.
基金Supported by the National Natural Science Foundation of China under Grant No 10333020 and 10528307, and the National Key Basic Research Special Foundation （NKBRSF） of China under Grant No 1999075406, and the Shanghai Science Foundation under Grant Nos 03XD14014, 04dz05905 and 05DZ22314.
文摘Adopting the observational distributions of star formation rates and half-light radii of Lyman break galaxies （LBGs） in the rest frame UV, we investigate empirically the predicted stellar mass function for LBGs. It is found that a peak exists at mass around 1.5 × 10^10M⊙ for their stellar mass function and there have been significant amount of LBGs with stellar masses higher than 10^11M⊙ at z ～ 3. There should be two families for LBG population observed by the Infrared Array Camera （IRAC） （which is a key instrument in the infrared space telescope Spitzer）, one is red and the other is blue, which are consistent with recent works. Moreover, the predicted comoving correlation lengths for LBGs with different stellar masses, i.e., with different 8 μm magnitudes, are also discussed. The red LBGs should be clustered in space more strongly than the blue LBGs. These predictions can and will be tested by the observations of IRAC, and will provide strong constraints on the current hierarchical galaxy formation model.
