s*****e
发帖数: 20 | 1 Let S_0=0 and for n \in N define
T_n = inf { t >= S_{n-1} | B_t >= 1} and S_n = inf{ t>= T_n | B_t <= -1}
Show that P(lim n -> infinity T_n = infinity) = 1 |
c**********e
发帖数: 2007 | 2 This is simple. Notice T_n-T_{n-1} is i.i.d. This makes T_n=X1+X2+...+Xn
were Xi's iid.
Since Xi's have mean infinity, we can use Y_i=max(X_i,1). Then apply the law
of big numbers. |
C***m
发帖数: 120 | 3 用LLN需要var有限，借这个帖子，再问一个，
定义 \tau = inf { t >= 0 | B_t >= 1}
求E(\tau)和var(\tau)，谢谢。
其实想算这个
\tau = inf { t >= 0 | B_t >= a or B_t <= -b}
的mean 和var. a,b>0 |
a****h
发帖数: 126 | 4 what does inf mean?
【在 s*****e 的大作中提到】
: Let S_0=0 and for n \in N define
: T_n = inf { t >= S_{n-1} | B_t >= 1} and S_n = inf{ t>= T_n | B_t <= -1}
: Show that P(lim n -> infinity T_n = infinity) = 1
|
c**********e
发帖数: 2007 | 5
For the original question, var(X_i)=infty, but var(max(X_i,1))
So there is no need to worry about it.
Both infinity.
E\tau = ab. I do not remember the var. I'll check it late.
【在 C***m 的大作中提到】
: 用LLN需要var有限，借这个帖子，再问一个，
: 定义 \tau = inf { t >= 0 | B_t >= 1}
: 求E(\tau)和var(\tau)，谢谢。
: 其实想算这个
: \tau = inf { t >= 0 | B_t >= a or B_t <= -b}
: 的mean 和var. a,b>0
|
C***m
发帖数: 120 | 6 Thank you for your comments. I was trying to see why var(max(X_i,1))
For the var, I was thinking E(W_t^4-3t^2)=0, so maybe E(W_\tau^4-3\tau^2)=0
as well. It seems doesn't work cause this method may lead a negative var...
【在 c**********e 的大作中提到】
:
: For the original question, var(X_i)=infty, but var(max(X_i,1)): So there is no need to worry about it.
: Both infinity.
: E\tau = ab. I do not remember the var. I'll check it late.
|
c**********e
发帖数: 2007 | 7
Faint. max(X_i,1)<=1. Of course its var
0
I tried. This one might not have a nice close form solution unless special
case such as a=b or a=2b.
【在 C***m 的大作中提到】
: Thank you for your comments. I was trying to see why var(max(X_i,1)): For the var, I was thinking E(W_t^4-3t^2)=0, so maybe E(W_\tau^4-3\tau^2)=0
: as well. It seems doesn't work cause this method may lead a negative var...
|
k*****y
发帖数: 744 | 8 貌似T_1 是第一次到达1的时间；S_1是T_1之后第一次到达-1的时间；T_2是S_1之后第
一次到达1的时间...
【在 a****h 的大作中提到】
: what does inf mean?
|
r****t
发帖数: 10904 | 9 其实 T_1 和后来的都不是 iid 的
【在 k*****y 的大作中提到】
: 貌似T_1 是第一次到达1的时间；S_1是T_1之后第一次到达-1的时间；T_2是S_1之后第
: 一次到达1的时间...
|
c**********e
发帖数: 2007 | 10 Does it matter?
【在 r****t 的大作中提到】
: 其实 T_1 和后来的都不是 iid 的
|
|
|
C***m
发帖数: 120 | 11 看起来对的。
可是，当 a很大（10）,b比较小（1） 最后算出来E(\tau^2)<0。
还有这个max(X_i,1)<=1，是笔误吧，还是有什么原因我没看出来。。。
idea
【在 k*****y 的大作中提到】
: 貌似T_1 是第一次到达1的时间；S_1是T_1之后第一次到达-1的时间；T_2是S_1之后第
: 一次到达1的时间...
|
k*****y
发帖数: 744 | 12 Thanks for pointing that out. I made a few mistakes before.
Here is the updated version. It looks more symmetric and consistent. Hope I
got it right this time.
【在 C***m 的大作中提到】
: 看起来对的。
: 可是，当 a很大（10）,b比较小（1） 最后算出来E(\tau^2)<0。
: 还有这个max(X_i,1)<=1，是笔误吧，还是有什么原因我没看出来。。。
:
: idea
|
C***m
发帖数: 120 | 13 厉害，这个应该对了，谢谢。
I
【在 k*****y 的大作中提到】
: Thanks for pointing that out. I made a few mistakes before.
: Here is the updated version. It looks more symmetric and consistent. Hope I
: got it right this time.
|
c**********e
发帖数: 2007 | 14 The answer is correct. This is great!
I
【在 k*****y 的大作中提到】
: Thanks for pointing that out. I made a few mistakes before.
: Here is the updated version. It looks more symmetric and consistent. Hope I
: got it right this time.
|
Q***5
发帖数: 994 | 15 Here is a more primitive proof:
There exists a>0 such that prob(X_i>a) = b>0
Now, forn any constant D,
Prob(T_n
^(n-[D/a]-1)*b^([D/a]+1) goes to 0 as n goes to infinity
law
【在 c**********e 的大作中提到】
: This is simple. Notice T_n-T_{n-1} is i.i.d. This makes T_n=X1+X2+...+Xn
: were Xi's iid.
: Since Xi's have mean infinity, we can use Y_i=max(X_i,1). Then apply the law
: of big numbers.
|
c**********e
发帖数: 2007 | 16 Your idea is good. But your proof is for convergence in probability
while the original problem is to prove convergence a.s..
b)
【在 Q***5 的大作中提到】
: Here is a more primitive proof:
: There exists a>0 such that prob(X_i>a) = b>0
: Now, forn any constant D,
: Prob(T_n: ^(n-[D/a]-1)*b^([D/a]+1) goes to 0 as n goes to infinity
:
: law
|
Q***5
发帖数: 994 | 17 Good point, but here convergence is trivial because T_n is monotonically
increasing.
More strictly, assume the conclusion does not hold, then there exists a>0
such that P(lim_n T_n 0
Since T_n is non-decreasing, P(T_N=P(lim_n T_n
contradicts the conclusion above.
【在 c**********e 的大作中提到】
: Your idea is good. But your proof is for convergence in probability
: while the original problem is to prove convergence a.s..
:
: b)
|
c**********e
发帖数: 2007 | 18 You are right. As Tn-T1 is the sum of some iid positive difference,
convergence in probability implies convergence a.s..
【在 Q***5 的大作中提到】
: Good point, but here convergence is trivial because T_n is monotonically
: increasing.
: More strictly, assume the conclusion does not hold, then there exists a>0
: such that P(lim_n T_n 0
: Since T_n is non-decreasing, P(T_N=P(lim_n T_n : contradicts the conclusion above.
|
s*****e
发帖数: 20 | 19 Let S_0=0 and for n \in N define
T_n = inf { t >= S_{n-1} | B_t >= 1} and S_n = inf{ t>= T_n | B_t <= -1}
Show that P(lim n -> infinity T_n = infinity) = 1 |
c**********e
发帖数: 2007 | 20 This is simple. Notice T_n-T_{n-1} is i.i.d. This makes T_n=X1+X2+...+Xn
were Xi's iid.
Since Xi's have mean infinity, we can use Y_i=max(X_i,1). Then apply the law
of big numbers. |
|
|
C***m
发帖数: 120 | 21 用LLN需要var有限，借这个帖子，再问一个，
定义 \tau = inf { t >= 0 | B_t >= 1}
求E(\tau)和var(\tau)，谢谢。
其实想算这个
\tau = inf { t >= 0 | B_t >= a or B_t <= -b}
的mean 和var. a,b>0 |
a****h
发帖数: 126 | 22 what does inf mean?
【在 s*****e 的大作中提到】
: Let S_0=0 and for n \in N define
: T_n = inf { t >= S_{n-1} | B_t >= 1} and S_n = inf{ t>= T_n | B_t <= -1}
: Show that P(lim n -> infinity T_n = infinity) = 1
|
c**********e
发帖数: 2007 | 23
For the original question, var(X_i)=infty, but var(max(X_i,1))
So there is no need to worry about it.
Both infinity.
E\tau = ab. I do not remember the var. I'll check it late.
【在 C***m 的大作中提到】
: 用LLN需要var有限，借这个帖子，再问一个，
: 定义 \tau = inf { t >= 0 | B_t >= 1}
: 求E(\tau)和var(\tau)，谢谢。
: 其实想算这个
: \tau = inf { t >= 0 | B_t >= a or B_t <= -b}
: 的mean 和var. a,b>0
|
C***m
发帖数: 120 | 24 Thank you for your comments. I was trying to see why var(max(X_i,1))
For the var, I was thinking E(W_t^4-3t^2)=0, so maybe E(W_\tau^4-3\tau^2)=0
as well. It seems doesn't work cause this method may lead a negative var...
【在 c**********e 的大作中提到】
:
: For the original question, var(X_i)=infty, but var(max(X_i,1)): So there is no need to worry about it.
: Both infinity.
: E\tau = ab. I do not remember the var. I'll check it late.
|
c**********e
发帖数: 2007 | 25
Faint. max(X_i,1)<=1. Of course its var
0
I tried. This one might not have a nice close form solution unless special
case such as a=b or a=2b.
【在 C***m 的大作中提到】
: Thank you for your comments. I was trying to see why var(max(X_i,1)): For the var, I was thinking E(W_t^4-3t^2)=0, so maybe E(W_\tau^4-3\tau^2)=0
: as well. It seems doesn't work cause this method may lead a negative var...
|
k*****y
发帖数: 744 | 26 貌似T_1 是第一次到达1的时间；S_1是T_1之后第一次到达-1的时间；T_2是S_1之后第
一次到达1的时间...
【在 a****h 的大作中提到】
: what does inf mean?
|
r****t
发帖数: 10904 | 27 其实 T_1 和后来的都不是 iid 的
【在 k*****y 的大作中提到】
: 貌似T_1 是第一次到达1的时间；S_1是T_1之后第一次到达-1的时间；T_2是S_1之后第
: 一次到达1的时间...
|
c**********e
发帖数: 2007 | 28 Does it matter?
【在 r****t 的大作中提到】
: 其实 T_1 和后来的都不是 iid 的
|
C***m
发帖数: 120 | 29 看起来对的。
可是，当 a很大（10）,b比较小（1） 最后算出来E(\tau^2)<0。
还有这个max(X_i,1)<=1，是笔误吧，还是有什么原因我没看出来。。。
idea
【在 k*****y 的大作中提到】
: 貌似T_1 是第一次到达1的时间；S_1是T_1之后第一次到达-1的时间；T_2是S_1之后第
: 一次到达1的时间...
|
k*****y
发帖数: 744 | 30 Thanks for pointing that out. I made a few mistakes before.
Here is the updated version. It looks more symmetric and consistent. Hope I
got it right this time.
【在 C***m 的大作中提到】
: 看起来对的。
: 可是，当 a很大（10）,b比较小（1） 最后算出来E(\tau^2)<0。
: 还有这个max(X_i,1)<=1，是笔误吧，还是有什么原因我没看出来。。。
:
: idea
|
|
|
C***m
发帖数: 120 | 31 厉害，这个应该对了，谢谢。
I
【在 k*****y 的大作中提到】
: Thanks for pointing that out. I made a few mistakes before.
: Here is the updated version. It looks more symmetric and consistent. Hope I
: got it right this time.
|
c**********e
发帖数: 2007 | 32 The answer is correct. This is great!
I
【在 k*****y 的大作中提到】
: Thanks for pointing that out. I made a few mistakes before.
: Here is the updated version. It looks more symmetric and consistent. Hope I
: got it right this time.
|
Q***5
发帖数: 994 | 33 Here is a more primitive proof:
There exists a>0 such that prob(X_i>a) = b>0
Now, forn any constant D,
Prob(T_n
^(n-[D/a]-1)*b^([D/a]+1) goes to 0 as n goes to infinity
law
【在 c**********e 的大作中提到】
: This is simple. Notice T_n-T_{n-1} is i.i.d. This makes T_n=X1+X2+...+Xn
: were Xi's iid.
: Since Xi's have mean infinity, we can use Y_i=max(X_i,1). Then apply the law
: of big numbers.
|
c**********e
发帖数: 2007 | 34 Your idea is good. But your proof is for convergence in probability
while the original problem is to prove convergence a.s..
b)
【在 Q***5 的大作中提到】
: Here is a more primitive proof:
: There exists a>0 such that prob(X_i>a) = b>0
: Now, forn any constant D,
: Prob(T_n: ^(n-[D/a]-1)*b^([D/a]+1) goes to 0 as n goes to infinity
:
: law
|
Q***5
发帖数: 994 | 35 Good point, but here convergence is trivial because T_n is monotonically
increasing.
More strictly, assume the conclusion does not hold, then there exists a>0
such that P(lim_n T_n 0
Since T_n is non-decreasing, P(T_N=P(lim_n T_n
contradicts the conclusion above.
【在 c**********e 的大作中提到】
: Your idea is good. But your proof is for convergence in probability
: while the original problem is to prove convergence a.s..
:
: b)
|
c**********e
发帖数: 2007 | 36 You are right. As Tn-T1 is the sum of some iid positive difference,
convergence in probability implies convergence a.s..
【在 Q***5 的大作中提到】
: Good point, but here convergence is trivial because T_n is monotonically
: increasing.
: More strictly, assume the conclusion does not hold, then there exists a>0
: such that P(lim_n T_n 0
: Since T_n is non-decreasing, P(T_N=P(lim_n T_n : contradicts the conclusion above.
|
m******2
发帖数: 564 | |
