a.s. convergence uniform distribution
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
I am having troubles with proving almost surely convergence for the following problem:
Let $U_j$ be IID $U(0,1)$ distributed and define $A_n$ to be:
$A_n=sum_k=1^n prod_j=1^k U_j$ for $nin mathbbN$.
for $nrightarrow infty$, I want to prove that $A_n$ converges almost surely to some $A$.
I tried with LLN, but this did not give anything.
probability-theory convergence random-variables uniform-distribution
edited Aug 17 at 8:15
saz
73.1k553113
asked Aug 17 at 6:33
Jonathan Kiersch
699
add a comment |Â
up vote
2
down vote
favorite
I am having troubles with proving almost surely convergence for the following problem:
Let $U_j$ be IID $U(0,1)$ distributed and define $A_n$ to be:
$A_n=sum_k=1^n prod_j=1^k U_j$ for $nin mathbbN$.
for $nrightarrow infty$, I want to prove that $A_n$ converges almost surely to some $A$.
I tried with LLN, but this did not give anything.
probability-theory convergence random-variables uniform-distribution
edited Aug 17 at 8:15
saz
73.1k553113
asked Aug 17 at 6:33
Jonathan Kiersch
699
I have edited the question making two small changes. I hope I didn't ruin anything.
– Kavi Rama Murthy
Aug 17 at 7:21
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am having troubles with proving almost surely convergence for the following problem:
Let $U_j$ be IID $U(0,1)$ distributed and define $A_n$ to be:
$A_n=sum_k=1^n prod_j=1^k U_j$ for $nin mathbbN$.
for $nrightarrow infty$, I want to prove that $A_n$ converges almost surely to some $A$.
I tried with LLN, but this did not give anything.
probability-theory convergence random-variables uniform-distribution
edited Aug 17 at 8:15
saz
73.1k553113
asked Aug 17 at 6:33
Jonathan Kiersch
699
I am having troubles with proving almost surely convergence for the following problem:
Let $U_j$ be IID $U(0,1)$ distributed and define $A_n$ to be:
$A_n=sum_k=1^n prod_j=1^k U_j$ for $nin mathbbN$.
for $nrightarrow infty$, I want to prove that $A_n$ converges almost surely to some $A$.
I tried with LLN, but this did not give anything.
probability-theory convergence random-variables uniform-distribution
edited Aug 17 at 8:15
saz
73.1k553113
asked Aug 17 at 6:33
Jonathan Kiersch
699
edited Aug 17 at 8:15
saz
73.1k553113
edited Aug 17 at 8:15
saz
73.1k553113
edited Aug 17 at 8:15
saz
73.1k553113
73.1k553113
asked Aug 17 at 6:33
Jonathan Kiersch
699
asked Aug 17 at 6:33
Jonathan Kiersch
699
asked Aug 17 at 6:33
Jonathan Kiersch
699
699
I have edited the question making two small changes. I hope I didn't ruin anything.
– Kavi Rama Murthy
Aug 17 at 7:21
add a comment |Â
I have edited the question making two small changes. I hope I didn't ruin anything.
– Kavi Rama Murthy
Aug 17 at 7:21
I have edited the question making two small changes. I hope I didn't ruin anything.
– Kavi Rama Murthy
Aug 17 at 7:21
I have edited the question making two small changes. I hope I didn't ruin anything.
– Kavi Rama Murthy
Aug 17 at 7:21
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
5
down vote
accepted
Note that $A_n$ is an increasing sequence of random variables, and therefore the pointwise limit equals $$A=sup_n in mathbbN A_n=lim_n to infty A_n.$$ In order to show that this limit is well-defined (in the sense $A<infty$ a.s.) we note that by the monotone convergence theorem
$$mathbbE left( sup_n in mathbbN A_n right) = sup_n in mathbbN mathbbE(A_n) = sup_n in mathbbN sum_k=1^n frac12^k < infty,$$
and so $A in L^1(mathbbP)$. This implies, in particular, $A<infty$ almost surely.
answered Aug 17 at 6:57
saz
73.1k553113
Everything makes sense, thank you! But what is the space L^1(P)? And is there a "general technique" to prove almost surely convergence, because I am often struggling with that. Or just some tips to consider?
– Jonathan Kiersch
Aug 17 at 7:51
@JonathanKiersch $A in L^1(mathbbP)$ means simply that $A$ has finite expectation, i.e. $mathbbE(|A|)<infty$. Re your 2nd question: Well, no, unfortunately there is no general technique. Often it helps to analyze the structure of the sequence; for instance if $A_n = sum_j=1^n Y_j$ then you can use different techniques depending whether the random variables $Y_j$ are independent and/or identically distributed. For independent random variables it's worth trying Kolmogorov's series theorem or Lévy's theorem ... if they are additionally identically distributed, then the strong law of
– saz
Aug 17 at 8:09
large numbers is a good idea. The most difficult case is that the random variables $(Y_j)$ are not independent or $(A_n)$ doesn't have this particular structure...then, for instance, sometimes the Borel-Cantelli lemma is useful. In the end, the only thing which helps is to keep solving such exercises... it certainly helps to get a better intuition.
– saz
Aug 17 at 8:12
I see, Thank you. Very helpful!
– Jonathan Kiersch
Aug 17 at 9:43
@JonathanKiersch You are welcome.
– saz
Aug 17 at 10:10
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
Note that $A_n$ is an increasing sequence of random variables, and therefore the pointwise limit equals $$A=sup_n in mathbbN A_n=lim_n to infty A_n.$$ In order to show that this limit is well-defined (in the sense $A<infty$ a.s.) we note that by the monotone convergence theorem
$$mathbbE left( sup_n in mathbbN A_n right) = sup_n in mathbbN mathbbE(A_n) = sup_n in mathbbN sum_k=1^n frac12^k < infty,$$
and so $A in L^1(mathbbP)$. This implies, in particular, $A<infty$ almost surely.
answered Aug 17 at 6:57
saz
73.1k553113
Everything makes sense, thank you! But what is the space L^1(P)? And is there a "general technique" to prove almost surely convergence, because I am often struggling with that. Or just some tips to consider?
– Jonathan Kiersch
Aug 17 at 7:51
@JonathanKiersch $A in L^1(mathbbP)$ means simply that $A$ has finite expectation, i.e. $mathbbE(|A|)<infty$. Re your 2nd question: Well, no, unfortunately there is no general technique. Often it helps to analyze the structure of the sequence; for instance if $A_n = sum_j=1^n Y_j$ then you can use different techniques depending whether the random variables $Y_j$ are independent and/or identically distributed. For independent random variables it's worth trying Kolmogorov's series theorem or Lévy's theorem ... if they are additionally identically distributed, then the strong law of
– saz
Aug 17 at 8:09
large numbers is a good idea. The most difficult case is that the random variables $(Y_j)$ are not independent or $(A_n)$ doesn't have this particular structure...then, for instance, sometimes the Borel-Cantelli lemma is useful. In the end, the only thing which helps is to keep solving such exercises... it certainly helps to get a better intuition.
– saz
Aug 17 at 8:12
I see, Thank you. Very helpful!
– Jonathan Kiersch
Aug 17 at 9:43
@JonathanKiersch You are welcome.
– saz
Aug 17 at 10:10
add a comment |Â
up vote
5
down vote
accepted
Note that $A_n$ is an increasing sequence of random variables, and therefore the pointwise limit equals $$A=sup_n in mathbbN A_n=lim_n to infty A_n.$$ In order to show that this limit is well-defined (in the sense $A<infty$ a.s.) we note that by the monotone convergence theorem
$$mathbbE left( sup_n in mathbbN A_n right) = sup_n in mathbbN mathbbE(A_n) = sup_n in mathbbN sum_k=1^n frac12^k < infty,$$
and so $A in L^1(mathbbP)$. This implies, in particular, $A<infty$ almost surely.
answered Aug 17 at 6:57
saz
73.1k553113
Everything makes sense, thank you! But what is the space L^1(P)? And is there a "general technique" to prove almost surely convergence, because I am often struggling with that. Or just some tips to consider?
– Jonathan Kiersch
Aug 17 at 7:51
@JonathanKiersch $A in L^1(mathbbP)$ means simply that $A$ has finite expectation, i.e. $mathbbE(|A|)<infty$. Re your 2nd question: Well, no, unfortunately there is no general technique. Often it helps to analyze the structure of the sequence; for instance if $A_n = sum_j=1^n Y_j$ then you can use different techniques depending whether the random variables $Y_j$ are independent and/or identically distributed. For independent random variables it's worth trying Kolmogorov's series theorem or Lévy's theorem ... if they are additionally identically distributed, then the strong law of
– saz
Aug 17 at 8:09
large numbers is a good idea. The most difficult case is that the random variables $(Y_j)$ are not independent or $(A_n)$ doesn't have this particular structure...then, for instance, sometimes the Borel-Cantelli lemma is useful. In the end, the only thing which helps is to keep solving such exercises... it certainly helps to get a better intuition.
– saz
Aug 17 at 8:12
I see, Thank you. Very helpful!
– Jonathan Kiersch
Aug 17 at 9:43
@JonathanKiersch You are welcome.
– saz
Aug 17 at 10:10
add a comment |Â
up vote
5
down vote
accepted
up vote
5
down vote
accepted
Note that $A_n$ is an increasing sequence of random variables, and therefore the pointwise limit equals $$A=sup_n in mathbbN A_n=lim_n to infty A_n.$$ In order to show that this limit is well-defined (in the sense $A<infty$ a.s.) we note that by the monotone convergence theorem
$$mathbbE left( sup_n in mathbbN A_n right) = sup_n in mathbbN mathbbE(A_n) = sup_n in mathbbN sum_k=1^n frac12^k < infty,$$
and so $A in L^1(mathbbP)$. This implies, in particular, $A<infty$ almost surely.
answered Aug 17 at 6:57
saz
73.1k553113
Note that $A_n$ is an increasing sequence of random variables, and therefore the pointwise limit equals $$A=sup_n in mathbbN A_n=lim_n to infty A_n.$$ In order to show that this limit is well-defined (in the sense $A<infty$ a.s.) we note that by the monotone convergence theorem
$$mathbbE left( sup_n in mathbbN A_n right) = sup_n in mathbbN mathbbE(A_n) = sup_n in mathbbN sum_k=1^n frac12^k < infty,$$
and so $A in L^1(mathbbP)$. This implies, in particular, $A<infty$ almost surely.
answered Aug 17 at 6:57
saz
73.1k553113
edited Aug 17 at 8:12
answered Aug 17 at 6:57
saz
73.1k553113
answered Aug 17 at 6:57
saz
73.1k553113
answered Aug 17 at 6:57
saz
73.1k553113
73.1k553113
Everything makes sense, thank you! But what is the space L^1(P)? And is there a "general technique" to prove almost surely convergence, because I am often struggling with that. Or just some tips to consider?
– Jonathan Kiersch
Aug 17 at 7:51
@JonathanKiersch $A in L^1(mathbbP)$ means simply that $A$ has finite expectation, i.e. $mathbbE(|A|)<infty$. Re your 2nd question: Well, no, unfortunately there is no general technique. Often it helps to analyze the structure of the sequence; for instance if $A_n = sum_j=1^n Y_j$ then you can use different techniques depending whether the random variables $Y_j$ are independent and/or identically distributed. For independent random variables it's worth trying Kolmogorov's series theorem or Lévy's theorem ... if they are additionally identically distributed, then the strong law of
– saz
Aug 17 at 8:09
large numbers is a good idea. The most difficult case is that the random variables $(Y_j)$ are not independent or $(A_n)$ doesn't have this particular structure...then, for instance, sometimes the Borel-Cantelli lemma is useful. In the end, the only thing which helps is to keep solving such exercises... it certainly helps to get a better intuition.
– saz
Aug 17 at 8:12
I see, Thank you. Very helpful!
– Jonathan Kiersch
Aug 17 at 9:43
@JonathanKiersch You are welcome.
– saz
Aug 17 at 10:10
add a comment |Â
Everything makes sense, thank you! But what is the space L^1(P)? And is there a "general technique" to prove almost surely convergence, because I am often struggling with that. Or just some tips to consider?
– Jonathan Kiersch
Aug 17 at 7:51
@JonathanKiersch $A in L^1(mathbbP)$ means simply that $A$ has finite expectation, i.e. $mathbbE(|A|)<infty$. Re your 2nd question: Well, no, unfortunately there is no general technique. Often it helps to analyze the structure of the sequence; for instance if $A_n = sum_j=1^n Y_j$ then you can use different techniques depending whether the random variables $Y_j$ are independent and/or identically distributed. For independent random variables it's worth trying Kolmogorov's series theorem or Lévy's theorem ... if they are additionally identically distributed, then the strong law of
– saz
Aug 17 at 8:09
large numbers is a good idea. The most difficult case is that the random variables $(Y_j)$ are not independent or $(A_n)$ doesn't have this particular structure...then, for instance, sometimes the Borel-Cantelli lemma is useful. In the end, the only thing which helps is to keep solving such exercises... it certainly helps to get a better intuition.
– saz
Aug 17 at 8:12
I see, Thank you. Very helpful!
– Jonathan Kiersch
Aug 17 at 9:43
@JonathanKiersch You are welcome.
– saz
Aug 17 at 10:10
Everything makes sense, thank you! But what is the space L^1(P)? And is there a "general technique" to prove almost surely convergence, because I am often struggling with that. Or just some tips to consider?
– Jonathan Kiersch
Aug 17 at 7:51
Everything makes sense, thank you! But what is the space L^1(P)? And is there a "general technique" to prove almost surely convergence, because I am often struggling with that. Or just some tips to consider?
– Jonathan Kiersch
Aug 17 at 7:51
@JonathanKiersch $A in L^1(mathbbP)$ means simply that $A$ has finite expectation, i.e. $mathbbE(|A|)<infty$. Re your 2nd question: Well, no, unfortunately there is no general technique. Often it helps to analyze the structure of the sequence; for instance if $A_n = sum_j=1^n Y_j$ then you can use different techniques depending whether the random variables $Y_j$ are independent and/or identically distributed. For independent random variables it's worth trying Kolmogorov's series theorem or Lévy's theorem ... if they are additionally identically distributed, then the strong law of
– saz
Aug 17 at 8:09
@JonathanKiersch $A in L^1(mathbbP)$ means simply that $A$ has finite expectation, i.e. $mathbbE(|A|)<infty$. Re your 2nd question: Well, no, unfortunately there is no general technique. Often it helps to analyze the structure of the sequence; for instance if $A_n = sum_j=1^n Y_j$ then you can use different techniques depending whether the random variables $Y_j$ are independent and/or identically distributed. For independent random variables it's worth trying Kolmogorov's series theorem or Lévy's theorem ... if they are additionally identically distributed, then the strong law of
– saz
Aug 17 at 8:09
large numbers is a good idea. The most difficult case is that the random variables $(Y_j)$ are not independent or $(A_n)$ doesn't have this particular structure...then, for instance, sometimes the Borel-Cantelli lemma is useful. In the end, the only thing which helps is to keep solving such exercises... it certainly helps to get a better intuition.
– saz
Aug 17 at 8:12
large numbers is a good idea. The most difficult case is that the random variables $(Y_j)$ are not independent or $(A_n)$ doesn't have this particular structure...then, for instance, sometimes the Borel-Cantelli lemma is useful. In the end, the only thing which helps is to keep solving such exercises... it certainly helps to get a better intuition.
– saz
Aug 17 at 8:12
I see, Thank you. Very helpful!
– Jonathan Kiersch
Aug 17 at 9:43
I see, Thank you. Very helpful!
– Jonathan Kiersch
Aug 17 at 9:43
@JonathanKiersch You are welcome.
– saz
Aug 17 at 10:10
@JonathanKiersch You are welcome.
– saz
Aug 17 at 10:10
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2885460%2fa-s-convergence-uniform-distribution%23new-answer', 'question_page');
);
Post as a guest
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
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
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
I have edited the question making two small changes. I hope I didn't ruin anything.
– Kavi Rama Murthy
Aug 17 at 7:21