Convergence in distribution of sum of random, independent variables.
up vote
1
down vote
favorite
There is a sequence $(X_n)_{n≥1}$ of independent random variables, where for n ≥ 1 the distribution for $X_n$ is given
$$P (X_n = 0) = frac{1}{n}$$ $$P(X_n = 2n) = 1 −frac{1}{n}$$
Examine if the sum below is convergent in distribution, if so, find the desired distribution
$$S_n=frac{X_1 + X_2 + . . . + X_n}{n}-n$$ So here is the deal, the expected value: $Efrac{X_k}{n}=frac{2k-2}{n}$, I was going to use Central Limit Theorem, then it would converge to N(0,2) -1, but the Linder erg condition is not satisfied.
Any hint is appricieted and also different approaches, as I must say I think I am in deep dark in here.
probability probability-distributions weak-convergence
asked Dec 2 at 20:44
ryszard eggink
303110
add a comment |
up vote
1
down vote
favorite
There is a sequence $(X_n)_{n≥1}$ of independent random variables, where for n ≥ 1 the distribution for $X_n$ is given
$$P (X_n = 0) = frac{1}{n}$$ $$P(X_n = 2n) = 1 −frac{1}{n}$$
Examine if the sum below is convergent in distribution, if so, find the desired distribution
$$S_n=frac{X_1 + X_2 + . . . + X_n}{n}-n$$ So here is the deal, the expected value: $Efrac{X_k}{n}=frac{2k-2}{n}$, I was going to use Central Limit Theorem, then it would converge to N(0,2) -1, but the Linder erg condition is not satisfied.
Any hint is appricieted and also different approaches, as I must say I think I am in deep dark in here.
probability probability-distributions weak-convergence
asked Dec 2 at 20:44
ryszard eggink
303110
" as Sn takes only positive values" Not true.
– Did
Dec 2 at 21:32
You are right obviousely, but I guess the limit still holds as n goes to infinity, but I have no idea how to prove it
– ryszard eggink
Dec 2 at 22:06
The distribution functions $(F_n)$ of $(S_n)$ do not seem like they are tight.
– parsiad
Dec 3 at 0:39
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
There is a sequence $(X_n)_{n≥1}$ of independent random variables, where for n ≥ 1 the distribution for $X_n$ is given
$$P (X_n = 0) = frac{1}{n}$$ $$P(X_n = 2n) = 1 −frac{1}{n}$$
Examine if the sum below is convergent in distribution, if so, find the desired distribution
$$S_n=frac{X_1 + X_2 + . . . + X_n}{n}-n$$ So here is the deal, the expected value: $Efrac{X_k}{n}=frac{2k-2}{n}$, I was going to use Central Limit Theorem, then it would converge to N(0,2) -1, but the Linder erg condition is not satisfied.
Any hint is appricieted and also different approaches, as I must say I think I am in deep dark in here.
probability probability-distributions weak-convergence
asked Dec 2 at 20:44
ryszard eggink
303110
There is a sequence $(X_n)_{n≥1}$ of independent random variables, where for n ≥ 1 the distribution for $X_n$ is given
$$P (X_n = 0) = frac{1}{n}$$ $$P(X_n = 2n) = 1 −frac{1}{n}$$
Examine if the sum below is convergent in distribution, if so, find the desired distribution
$$S_n=frac{X_1 + X_2 + . . . + X_n}{n}-n$$ So here is the deal, the expected value: $Efrac{X_k}{n}=frac{2k-2}{n}$, I was going to use Central Limit Theorem, then it would converge to N(0,2) -1, but the Linder erg condition is not satisfied.
Any hint is appricieted and also different approaches, as I must say I think I am in deep dark in here.
probability probability-distributions weak-convergence
probability probability-distributions weak-convergence
asked Dec 2 at 20:44
ryszard eggink
303110
asked Dec 2 at 20:44
ryszard eggink
303110
edited Dec 3 at 22:13
asked Dec 2 at 20:44
ryszard eggink
303110
asked Dec 2 at 20:44
ryszard eggink
303110
asked Dec 2 at 20:44
ryszard eggink
303110
303110
" as Sn takes only positive values" Not true.
– Did
Dec 2 at 21:32
You are right obviousely, but I guess the limit still holds as n goes to infinity, but I have no idea how to prove it
– ryszard eggink
Dec 2 at 22:06
The distribution functions $(F_n)$ of $(S_n)$ do not seem like they are tight.
– parsiad
Dec 3 at 0:39
add a comment |
" as Sn takes only positive values" Not true.
– Did
Dec 2 at 21:32
You are right obviousely, but I guess the limit still holds as n goes to infinity, but I have no idea how to prove it
– ryszard eggink
Dec 2 at 22:06
The distribution functions $(F_n)$ of $(S_n)$ do not seem like they are tight.
– parsiad
Dec 3 at 0:39
" as Sn takes only positive values" Not true.
– Did
Dec 2 at 21:32
" as Sn takes only positive values" Not true.
– Did
Dec 2 at 21:32
You are right obviousely, but I guess the limit still holds as n goes to infinity, but I have no idea how to prove it
– ryszard eggink
Dec 2 at 22:06
You are right obviousely, but I guess the limit still holds as n goes to infinity, but I have no idea how to prove it
– ryszard eggink
Dec 2 at 22:06
The distribution functions $(F_n)$ of $(S_n)$ do not seem like they are tight.
– parsiad
Dec 3 at 0:39
The distribution functions $(F_n)$ of $(S_n)$ do not seem like they are tight.
– parsiad
Dec 3 at 0:39
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3023152%2fconvergence-in-distribution-of-sum-of-random-independent-variables%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
" as Sn takes only positive values" Not true.
– Did
Dec 2 at 21:32
You are right obviousely, but I guess the limit still holds as n goes to infinity, but I have no idea how to prove it
– ryszard eggink
Dec 2 at 22:06
The distribution functions $(F_n)$ of $(S_n)$ do not seem like they are tight.
– parsiad
Dec 3 at 0:39