A**T
发帖数: 362 | 1 Excercise 6.8 in R. Durrett "Probability: Theory and Examples"
Let X_n be independent Poisson r.v.'s with EX_n = \lambda_n and let S_n = X_
1 + ... + X_n. Show that if \sum \lambda_n = \infty. then S_n/ES_n -> 1 a.s.
这题书上是有提示的。先用chebyshev's inequality证converges in probability. 然
后找一个subsequence such that 1). S_{n_k}/ES_{n_k} -> 1 a.s. and 2) ES_{n_{k
+1}}/ES_{n_k} -> 1 as k -> \infty. 不知道哪位大师做过这道题目, 我找不到这样
的subsequence.
多谢！ | Q***5
发帖数: 994 | 2 The hint does not seem to make sense. Can you double check or explain how do
you prove the conclusion even if you assume such a subsequence exits? | A**T
发帖数: 362 | 3 You can read the file that I attached. The idea is from Theorem 6.8, and
the problem I asked is Exercise 6.8.
do
【在 Q***5 的大作中提到】
: The hint does not seem to make sense. Can you double check or explain how do
: you prove the conclusion even if you assume such a subsequence exits?
| n******t
发帖数: 4406 | 4 This is just a standard excercise. Just copy the example and it's done.
【在 A**T 的大作中提到】
: You can read the file that I attached. The idea is from Theorem 6.8, and
: the problem I asked is Exercise 6.8.
:
: do
| Q***5
发帖数: 994 | 5 I see.
I guess you can not use the hint directly, for example when \lambda_n = 2^n
for all n, you just can not find a subsequence such that ES_{n_{k+1})/ES_{n
_k} converges to 1.
You can do the following to `expand' the sequence to make sure that each X_n
has a small \lambda_n: if \lambda_n>1, let X_n = X_{n,1}+X_{n,2}+...+X_{n,
k}, where each X_{n,i} is a Poisson r.v. with \lambda<1. You can now replace
X_n by this sequence.
Now the same trick played in proof of Th6.8 in selecting subsequence
【在 A**T 的大作中提到】
: You can read the file that I attached. The idea is from Theorem 6.8, and
: the problem I asked is Exercise 6.8.
:
: do
| A**T
发帖数: 362 | 6 Sounds reasonable. I tried hard to think where I can use the properties of
Poisson r.v's in the proof. Your solution obviously uses the property that a
poisson r.v can be splited into small poisson r.v's.
Thanks.
n
{n
_n
n,
replace
【在 Q***5 的大作中提到】
: I see.
: I guess you can not use the hint directly, for example when \lambda_n = 2^n
: for all n, you just can not find a subsequence such that ES_{n_{k+1})/ES_{n
: _k} converges to 1.
: You can do the following to `expand' the sequence to make sure that each X_n
: has a small \lambda_n: if \lambda_n>1, let X_n = X_{n,1}+X_{n,2}+...+X_{n,
: k}, where each X_{n,i} is a Poisson r.v. with \lambda<1. You can now replace
: X_n by this sequence.
: Now the same trick played in proof of Th6.8 in selecting subsequence
| n******t
发帖数: 4406 | 7 This question is effectively equal to show for a poisson process N(t) with
parameter 1,
lim N(t)/t = 1 as t-> infty, something called elementary renewal theorem.
a
【在 A**T 的大作中提到】
: Sounds reasonable. I tried hard to think where I can use the properties of
: Poisson r.v's in the proof. Your solution obviously uses the property that a
: poisson r.v can be splited into small poisson r.v's.
: Thanks.
:
: n
: {n
: _n
: n,
: replace
|
|
