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
th partial sum of the sequence
*
+
. The partial sum sequence is also an infinite sequence of
functions.
2. The Three Modes of Probabilistic Convergence
( )
( )
( )
Special Notations:
→
where
should be replaced by one of the following symbols or abbreviations:
mode notation
meaning
mathematical definition
(sure) convergence
→
almost surely, with probability 1
.{
}/
in probability
,
 
→
in law, in distribution, or weakly
( )
( )
Convergence in Probability to a constant and Convergence in Distribution to a constant (singleton)
for some constant
.
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 populationsample language of statistics,
the
collection
*
+
is called a
sampling frame
of
size
;
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the realized values of the collection, denoted as
*
+
, is called a
sample
;
the common distribution
that all
’s follow is called the
population
.
The fundamental problem of statistics is to infer properties of the unknown population
by investigating the
properties of the many samples
*
+
obtained under the sampling frame
*
+
.
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 Spring '08
 SMSLee
 Statistics, Central Limit Theorem, Probability, Probability theory, STATISTICS AND ACTUARIAL SCIENCE

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