12 Example Class 10

12 Example Class 10 - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I, FALL 2010 EXAMPLE CLASS 10 Convergence Modes and Limit Theorems Elements of Theory 1. Infinite Sequence of random variables and Its Partial Sum Sequence Infinite sequence of random variables : * ¡ ¢ £ ¢ ¤ + . An infinite sequence of random variables is an infinite sequence of functions. The partial sum sequence associated with the infinite sequence : *¥ ¡ ¢ ¥ £ ¢ ¤ + where ¥ ¦ § ∑ ¨ ¦ ¨©¡ is called the ª-th partial sum of the sequence * ¡ ¢ £ ¢ ¤ + . The partial sum sequence is also an infinite sequence of functions. 2. The Three Modes of Probabilistic Convergence «¬­®¯°±¯­²¯ ³ ´µ¶¬·¸¹º»°¯¹(¼½ ¾½ )«¬­®¯°±¯­²¯ ³ «¬­®¯°±¯­²¯¹¿­¹À°¬ÁÂÁ¿µ¿¸Ã¹(Ä) ³ «¬­®¯°±¯­²¯¹¿­¹Å¿·¸°¿Á»¸¿¬­¹(Æ) Special Notations: Ç ÈÉÊË →ÌÌÍ where ÎÏÐÑ should be replaced by one of the following symbols or abbreviations: mode notation meaning mathematical definition Ò (sure) convergence ÓÔÕ ÇÖ× Ç § ؽٽ →Í almost surely, with probability 1 Ä .{ ÓÔÕ ÇÖ× Ç § }/ § Ú Ä Ö in probability ÓÔÕ ÇÖ× Ä,| Ç Û | Ü Ý- § Þ ß¹àá¹Æ¹àá¹Ê →ÌÌÌÌÌÌÍ in law, in distribution, or weakly ÓÔÕ ÇÖ× â ã ä (å) § â ã (å) Convergence in Probability to a constant and Convergence in Distribution to a constant (singleton) Ç Ä Ö æ ç Ç Æ Ö æ for some constant æ . 3. Independent and Identically Distributed Sequence and Basic Statistical Notions If the ¨ ’s are independent and identically distributed as a same distribution (known or unknown), then * ¡ ¢ £ ¢ ¤ + is called an independent and identically distributed (i.i.d.) sequence of random variables. An iid collection of random variables is the prime device of statistics. In the population-sample language of statistics, the collection * ¡ ¢ £ ¢ ¤ ¢ Ç + is called a sampling frame of size è ; the realized values of the collection, denoted as * ¡ ¢ £ ¢ ¤ ¢ ¥ + , is called a sample ; the common distribution ¦ that all § ¨ ’s follow is called the population ....
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This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.

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12 Example Class 10 - THE UNIVERSITY OF HONG KONG...

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