Almost Sure
@almostsure.bsky.social
George Lowther. Author of Almost Sure blog on maths, probability and stochastic calculus
https://almostsuremath.com
Also on YouTube: https://www.youtube.com/@almostsure
https://almostsuremath.com
Also on YouTube: https://www.youtube.com/@almostsure
I posted this as a YouTube short, so I’ll refer to my description there
October 5, 2025 at 10:31 PM
I posted this as a YouTube short, so I’ll refer to my description there
initial->unitial (autocorrect!)
August 26, 2025 at 1:44 PM
initial->unitial (autocorrect!)
where unit element of each sub-algebra is not necessarily unit of the full algebra (just a projection in general)
August 26, 2025 at 1:43 PM
where unit element of each sub-algebra is not necessarily unit of the full algebra (just a projection in general)
There's the question of if the freely generated product exists, according to the different types of independence.
It does for commutative & free products, as products of initial algebras.
Doesn't look like it does for boolean independence. Not as until algebras though.
Maybe as non-initial ones
It does for commutative & free products, as products of initial algebras.
Doesn't look like it does for boolean independence. Not as until algebras though.
Maybe as non-initial ones
August 26, 2025 at 1:43 PM
There's the question of if the freely generated product exists, according to the different types of independence.
It does for commutative & free products, as products of initial algebras.
Doesn't look like it does for boolean independence. Not as until algebras though.
Maybe as non-initial ones
It does for commutative & free products, as products of initial algebras.
Doesn't look like it does for boolean independence. Not as until algebras though.
Maybe as non-initial ones
interesting. I'm only familiar with commutative & free independence
August 26, 2025 at 1:37 PM
interesting. I'm only familiar with commutative & free independence
also, how can they tell it was due to a tyre blowing out? It was at night and the car was wrecked and burned out. And, I heard that these cars have run-flat tyres. I think it’s a simple case of losing control at speed
July 3, 2025 at 10:53 PM
also, how can they tell it was due to a tyre blowing out? It was at night and the car was wrecked and burned out. And, I heard that these cars have run-flat tyres. I think it’s a simple case of losing control at speed
I’ve been wondering what happened. I had a tyre blow out on the motorway before, no big deal - but they were ‘run-flat’ so I could keep driving almost normally.
Would the same thing with normal tyres result in loss of control, or was this a result of reckless driving?
Would the same thing with normal tyres result in loss of control, or was this a result of reckless driving?
July 3, 2025 at 6:10 PM
I’ve been wondering what happened. I had a tyre blow out on the motorway before, no big deal - but they were ‘run-flat’ so I could keep driving almost normally.
Would the same thing with normal tyres result in loss of control, or was this a result of reckless driving?
Would the same thing with normal tyres result in loss of control, or was this a result of reckless driving?
new-ish…actually a few weeks old now. I forgot to post here when I initially launched it on YouTube
July 3, 2025 at 12:48 PM
new-ish…actually a few weeks old now. I forgot to post here when I initially launched it on YouTube
Unfortunately its not as nice as I first thought. And I made a mistake in the writeup - when you multiply diffusions then what you get need not be Markov.
Maybe there is a more natural way of fitting martingales.
bsky.app/profile/almo...
Maybe there is a more natural way of fitting martingales.
bsky.app/profile/almo...
I am not sure if the distribution is symmetric under
X(mu,t)->1-X(1-mu,t).
It is for individual times, as it matches Dirichlet distribution, but probably not for the entire paths wrt t. Which is disappointing. Maybe it can be modified?
X(mu,t)->1-X(1-mu,t).
It is for individual times, as it matches Dirichlet distribution, but probably not for the entire paths wrt t. Which is disappointing. Maybe it can be modified?
Simulating the martingales X(μ,t) which are beta distributed and martingale wrt t.
X(μ,t) ~ Beta(μ(1-t)/t, (1-μ)(1-t)/t)
The (1-t)/t scaling is so that on range 0<t<1 we cover entire set of Beta distributions.
For each t, X(μ_{i+1},t)-X(μ_i,t) have Dirichlet distribution.
X(μ,t) ~ Beta(μ(1-t)/t, (1-μ)(1-t)/t)
The (1-t)/t scaling is so that on range 0<t<1 we cover entire set of Beta distributions.
For each t, X(μ_{i+1},t)-X(μ_i,t) have Dirichlet distribution.
April 17, 2025 at 9:18 AM
Unfortunately its not as nice as I first thought. And I made a mistake in the writeup - when you multiply diffusions then what you get need not be Markov.
Maybe there is a more natural way of fitting martingales.
bsky.app/profile/almo...
Maybe there is a more natural way of fitting martingales.
bsky.app/profile/almo...
Also I think X(mu,t) is not markov for individual mu (it is for mu as a whole). Rather, it is the ratios of the jumps wrt mu: X(mu-,t)/X(mu+,t) which are markov (i.e., diffusions)
April 17, 2025 at 9:14 AM
Also I think X(mu,t) is not markov for individual mu (it is for mu as a whole). Rather, it is the ratios of the jumps wrt mu: X(mu-,t)/X(mu+,t) which are markov (i.e., diffusions)
Here's the method of simulation, and also shows that the joint distribution of X(μ,t) is uniquely determined if we impose independent ratios property wrt μ.
April 16, 2025 at 2:03 AM
here's another plot (more 'mu' points, fewer time points.
And, gamma(1) process scaled to hit 1 at time 1 (time parameter mu to compare). Corresponds to t=0.5 in the surface plot. You can see its dominated by a few large jumps.
gamma(40) process is shown in the 3rd plot, corresponds with t=0.024
And, gamma(1) process scaled to hit 1 at time 1 (time parameter mu to compare). Corresponds to t=0.5 in the surface plot. You can see its dominated by a few large jumps.
gamma(40) process is shown in the 3rd plot, corresponds with t=0.024
April 15, 2025 at 7:48 PM
here's another plot (more 'mu' points, fewer time points.
And, gamma(1) process scaled to hit 1 at time 1 (time parameter mu to compare). Corresponds to t=0.5 in the surface plot. You can see its dominated by a few large jumps.
gamma(40) process is shown in the 3rd plot, corresponds with t=0.024
And, gamma(1) process scaled to hit 1 at time 1 (time parameter mu to compare). Corresponds to t=0.5 in the surface plot. You can see its dominated by a few large jumps.
gamma(40) process is shown in the 3rd plot, corresponds with t=0.024
Either way, it’s lightning fast using my new Schrödinger’s Cat8 cables
April 13, 2025 at 4:30 PM
Either way, it’s lightning fast using my new Schrödinger’s Cat8 cables