f*********1
发帖数: 117 | 1 1. convergence in distribution;
2. convergence in probability;
1 doesn't imply 2.
但俺想不出converge in distribution 但是不converge in probability的例子了
谁能给列举几个？
多谢！ |
f****e
发帖数: 590 | 2 a sequence of iid random variable
【在 f*********1 的大作中提到】
: 1. convergence in distribution;
: 2. convergence in probability;
: 1 doesn't imply 2.
: 但俺想不出converge in distribution 但是不converge in probability的例子了
: 谁能给列举几个？
: 多谢！
|
g****y
发帖数: 71 | 3 convergence in probability usually refers to the law of large numbers. e.g.
average of iid Xi should converge in p to its expectation.
convergence in distribution usually refers to the central limit theorem.
e.g. square root of n times (average of iid Xi minus its expectation) will
behave like a normal distribution. it converges in distribution to a normal
random variable. but indeed it's not a normal random variable even in the
limit. |
f*********1
发帖数: 117 | 4 谢谢findle!
你的意思是the average of a sequence of iid RVs?
a sequence of iid random variable
【在 f****e 的大作中提到】
: a sequence of iid random variable
|
f****e
发帖数: 590 | 5 不是，就是iid
convergence in distribution就是说他们的cumulative function收敛，但是这些rv有
没有关系就不管了
convergence almost sure, in probability包括in Lp，都要求这些rv's之间要有联系
【在 f*********1 的大作中提到】
: 谢谢findle!
: 你的意思是the average of a sequence of iid RVs?
:
: a sequence of iid random variable
|
f*********1
发帖数: 117 | 6 thank u, gatsby!
.
normal
【在 g****y 的大作中提到】
: convergence in probability usually refers to the law of large numbers. e.g.
: average of iid Xi should converge in p to its expectation.
: convergence in distribution usually refers to the central limit theorem.
: e.g. square root of n times (average of iid Xi minus its expectation) will
: behave like a normal distribution. it converges in distribution to a normal
: random variable. but indeed it's not a normal random variable even in the
: limit.
|
n******r
发帖数: 1247 | 7 this is not right
a sequence of iid random variables converges with probability 1
an example for 1 doesn't imply 2 can be
toss a fair coin
X=1 if head, 0 if tail
Y=1 if tail, 0 if head
X Y have the same cumulative distribution, but for e<1,P(|X(w)-Y(w)|>e)=1
therefore no convergence in probability
【在 f****e 的大作中提到】
: a sequence of iid random variable
|
A***d
发帖数: 25 | 8
The following is an example:
Suppose Xn follows exponential distribution with df n/(n+1)exp{-nx/(n+1)}, X
follows exponential
distribution with df exp(-x). Further assume that Xn and X are independent.
Obviously, Xn converges in law to
X. However, by using the convolution formula, we can compute that P(|Xn-X|>=
e)->exp(-e)>0 for any e>0.
【在 f*********1 的大作中提到】
: 1. convergence in distribution;
: 2. convergence in probability;
: 1 doesn't imply 2.
: 但俺想不出converge in distribution 但是不converge in probability的例子了
: 谁能给列举几个？
: 多谢！
|
k*******d
发帖数: 1340 | 9 X_n are N(0,1)
X_0 is -X_n
clearly X_n -> X_0 in distribution since they have the same distribution;
However
P(|X_n-X_0|>\eps) = P(|2X_0| > \eps), where X_0 is N(0,1), clearly this prob
does not go to zero.
还有，converge in distribution不要求随机变量在同一个probability space上，
converge in probablity 要求在同一个prob space上，除非X_0是常数
Note that if X_n -->(in distr) X_0, where P(X_0 = c) = 1, c is a constant
then X_n --> X_0 in prob |
