Thank you for the video. I don’t know if I can get an hint to my question... Let’s say I manage to use the the Strong law of large numbers on an integer sequence (Γp(n))p∈N (n is a fixed integer), to demonstrate that the sequence Γ converges to a constant limit γ (that doesn’t depend on n), what can we say about the set of integers n∈N that doesn’t fullfill such statement (set of integers n for which Γ doesn’t tend to γ ) 1) can we say it’s a “null set” of N 2) if it is a null set, can it be an infinite set? (I am studying probabilities on my own without teacher, on my spare time, so your answers are precious... ). Thank you in advance
I am not sure I understand your question. Is Gamma_p a sequence of random variables and n an element of the sample space? In this case, a more conventional notation would be X_n (omega) so X instead of Gamma, n instead of p, and omega instead of n. Or do you mean that Gamma_n is a sequence of random variables and Gamma_{p (n)} is a subsequence?
Thanks for the consideration. Sorry it’s not clear. I tried to be concise to avoid making you spend much time... but it seems that I failed :) ... I hope it‘s clearer (I explained the context below). If not clear yet, no need to spend your time trying (I never studied probabilities... I lately spent few days/weeks learning on my own... so all is fresh in my head, my expressions are not always rigorous/understandable). I am working on a number theory problem, using a recursive function with n∈N as a start point (Γ_0=n) and p is the number of iterations of the recursive function. Γp is the value reached after p iteration on n. When I compute the general expression of Γ, I find that it practically depends only on p, and some average values a_i=(x1+x2+...+x_i)/i ; i=1...p, and x_j are order of divisibility by 2 of the number reached after j iterations (for example if after j iteration starting at Γ_0=6, I reach Γj=48 then x_j=4, since 48 is divisible by 2^4=16 and not divisible by 2^5 ... similarly x_0=1 as 6 is divisible by 2^1 only). The values of (x_j)j∈N seem to appear randomly (x_i+1 doesn’t seem to depend on the value of x_i: demonstrating that would be the hardest part!), while the average (a_i)i∈N seems to converge to 2 all the time (regardless of choice of the starting point n), which also implies that Γ also converges to a constant number which doesn’t depend on n... the problem consists of proving this convergence for any n∈N. So even if this is more an analytical problem (part of Analysis branch of math) and not a probabilistic problem... I see many similarities... (randomness of (x_i), and the deterministic end of the average (a_i) ... that seems just saying « in average even numbers are divisible by 4») so I tried to see if there could be a bridge between the 2 domains (probability/analysis) for this problem (which would be beautiful, as it means we might use probabilities to demonstrate convergence in the analytical sense). The strong law of large numbers seemed to be the most promising law to provide such a bridge. In that case (x_i) could be seen as the realisation of a sequence of random variables (Xi) applied on Γ_i(n) ... that evaluates its order of divisibility by 2. it’s a long way to try to show that that is right (at a first glance it doesn’t seem right as all variables depend at least on the choice of n their independence could be a dream)... but I just wanted to see if it happens to be feasible, right, how helpful probabilities can be on this problem, If I suppose the above is true and I compute the mean E(X) of the variables Xi I find that it is equal to the infinite sum: Σ k*P(X=k)=Σ k/(2^(k+1) k∈N which is indeed equal to 2. Which mean that if we can use the strong law of large numbers then the average sequence (ai) shall tend to 2 (which is exactly what I wanted to prove), « almost surely », so the set of cases in which this is wrong is P-negligible. I can’t translate the last sentence into something tangible... which set we are taking about here ? Set of n ? or all is so wrong that it doesn’t make sense? And if it’s n would that mean that the negligible set of numbers for which Γ does not converge to γ is a finite set? So that’s the origin of the question. PS. I already know that if this set is finite, then it is empty (because if for a given n, Γ doesn’t converge ... then based on n, I can construct an infinite set of numbers for which Γ doesn’t converge to γ.... contradiction!) that why I am interested in characterising this negligible set. PS2. I think I will delete these comments afterwards... (too bad it's not intuitive on TH-cam how to send private messages!).
@@AdamAbdelmalkOUCHATTI-ECLX Dear Adam. This is a difficult problem. I can't solve it just like that. But here are some thoughts. It seems that your sequence is chaotic more than random. Not sure this would help to solve the problem. Now, the way you define the sequence using iterations makes me think of the ergodic theorem, which is a stronger result than the strong law of large numbers. In fact, I already recorded a video (that I will post next week) with a complete proof of the law of large numbers but also the statement (with no proof) of the ergodic theorem. There is not much in my video but you can look online. I think the best reference about ergodic theory in general is the book of Karl Petersen. This does not solve the problem but this is definitely in this direction that I would go: use ergodic theory, which is in fact at the intersection of probability and number theory. Good luck! Nicolas.
@@theprobabilitychannel-prof8089 thanks for the consideration again! I will take a look at Ergodic theorem... looking forward to the video you mentioned too!! It’s fanny you talked about “Chao”, because just a week ago I bought 2 books, 1 is about P-adic numbers... and the second one is about Chaos Theory (& some fractals)! :) (surely I was influenced by the chaotic behaviour of this function... and the fractals I also seem to see in the sequence (x_i)). Thank you!
This video is part of the playlist Advanced Probability th-cam.com/video/qGsHiHwgInU/w-d-xo.html.
Thank you for the video.
I don’t know if I can get an hint to my question...
Let’s say I manage to use the the Strong law of large numbers on an integer sequence (Γp(n))p∈N (n is a fixed integer), to demonstrate that the sequence Γ converges to a constant limit γ (that doesn’t depend on n), what can we say about the set of integers n∈N that doesn’t fullfill such statement (set of integers n for which Γ doesn’t tend to γ )
1) can we say it’s a “null set” of N
2) if it is a null set, can it be an infinite set?
(I am studying probabilities on my own without teacher, on my spare time, so your answers are precious... ).
Thank you in advance
I am not sure I understand your question. Is Gamma_p a sequence of random variables and n an element of the sample space? In this case, a more conventional notation would be X_n (omega) so X instead of Gamma, n instead of p, and omega instead of n. Or do you mean that Gamma_n is a sequence of random variables and Gamma_{p (n)} is a subsequence?
Thanks for the consideration. Sorry it’s not clear. I tried to be concise to avoid making you spend much time... but it seems that I failed :) ... I hope it‘s clearer (I explained the context below). If not clear yet, no need to spend your time trying (I never studied probabilities... I lately spent few days/weeks learning on my own... so all is fresh in my head, my expressions are not always rigorous/understandable).
I am working on a number theory problem, using a recursive function with n∈N as a start point (Γ_0=n) and p is the number of iterations of the recursive function. Γp is the value reached after p iteration on n.
When I compute the general expression of Γ, I find that it practically depends only on p, and some average values a_i=(x1+x2+...+x_i)/i ; i=1...p, and x_j are order of divisibility by 2 of the number reached after j iterations (for example if after j iteration starting at Γ_0=6, I reach Γj=48 then x_j=4, since 48 is divisible by 2^4=16 and not divisible by 2^5 ... similarly x_0=1 as 6 is divisible by 2^1 only).
The values of (x_j)j∈N seem to appear randomly (x_i+1 doesn’t seem to depend on the value of x_i: demonstrating that would be the hardest part!), while the average (a_i)i∈N seems to converge to 2 all the time (regardless of choice of the starting point n), which also implies that Γ also converges to a constant number which doesn’t depend on n... the problem consists of proving this convergence for any n∈N.
So even if this is more an analytical problem (part of Analysis branch of math) and not a probabilistic problem... I see many similarities... (randomness of (x_i), and the deterministic end of the average (a_i) ... that seems just saying « in average even numbers are divisible by 4»)
so I tried to see if there could be a bridge between the 2 domains (probability/analysis) for this problem (which would be beautiful, as it means we might use probabilities to demonstrate convergence in the analytical sense).
The strong law of large numbers seemed to be the most promising law to provide such a bridge.
In that case (x_i) could be seen as the realisation of a sequence of random variables (Xi) applied on Γ_i(n) ... that evaluates its order of divisibility by 2.
it’s a long way to try to show that that is right (at a first glance it doesn’t seem right as all variables depend at least on the choice of n their independence could be a dream)... but I just wanted to see if it happens to be feasible, right, how helpful probabilities can be on this problem,
If I suppose the above is true and I compute the mean E(X) of the variables Xi
I find that it is equal to the infinite sum: Σ k*P(X=k)=Σ k/(2^(k+1) k∈N which is indeed equal to 2.
Which mean that if we can use the strong law of large numbers then the average sequence (ai) shall tend to 2 (which is exactly what I wanted to prove), « almost surely », so the set of cases in which this is wrong is P-negligible.
I can’t translate the last sentence into something tangible... which set we are taking about here ? Set of n ? or all is so wrong that it doesn’t make sense?
And if it’s n would that mean that the negligible set of numbers for which Γ does not converge to γ is a finite set?
So that’s the origin of the question.
PS. I already know that if this set is finite, then it is empty (because if for a given n, Γ doesn’t converge ... then based on n, I can construct an infinite set of numbers for which Γ doesn’t converge to γ.... contradiction!) that why I am interested in characterising this negligible set.
PS2. I think I will delete these comments afterwards... (too bad it's not intuitive on TH-cam how to send private messages!).
@@AdamAbdelmalkOUCHATTI-ECLX Dear Adam. This is a difficult problem. I can't solve it just like that. But here are some thoughts. It seems that your sequence is chaotic more than random. Not sure this would help to solve the problem. Now, the way you define the sequence using iterations makes me think of the ergodic theorem, which is a stronger result than the strong law of large numbers. In fact, I already recorded a video (that I will post next week) with a complete proof of the law of large numbers but also the statement (with no proof) of the ergodic theorem. There is not much in my video but you can look online. I think the best reference about ergodic theory in general is the book of Karl Petersen. This does not solve the problem but this is definitely in this direction that I would go: use ergodic theory, which is in fact at the intersection of probability and number theory. Good luck! Nicolas.
@@theprobabilitychannel-prof8089
thanks for the consideration again!
I will take a look at Ergodic theorem... looking forward to the video you mentioned too!!
It’s fanny you talked about “Chao”, because just a week ago I bought 2 books, 1 is about P-adic numbers... and the second one is about Chaos Theory (& some fractals)! :) (surely I was influenced by the chaotic behaviour of this function... and the fractals I also seem to see in the sequence (x_i)). Thank you!