Almost Sure

YouTube video by Almost Sure What does Riemann Zeta have to do with Brownian Motion?

New YouTube video uploaded on connections between Riemann zeta and Brownian motion!

What does Riemann Zeta have to do with Brownian Motion?

youtu.be/YTQKbgxbtiw

Moments of random variable equal 2 xi

It's a while since my last YT video upload. Nothing is happening, I am busy working on the next one. Just taking longer than expected (been very busy recent weekends).

Will be on connections between Riemann zeta and Brownian motion.

ETA a few days to a week.

I posted this as a YouTube short, so I’ll refer to my description there

Helter skelter

YouTube video by Almost Sure Wild Expectations

New YouTube video: Wild Expectations!

This looks at weird properties of conditional expectations, such as two random variables being bigger than each other 'on average'

youtu.be/4ZwRXVVepj8?...

Absolutely continuous rvs summing to singular rv

Proof of singular rv

Proof of absolute continuity

Whatever you do, 𝙛𝙤𝙧 𝙩𝙝𝙚 𝙡𝙤𝙫𝙚 𝙤𝙛 𝙂𝙤𝙙, do not ask random variables to be Lebesgue measurable!

If you were so stupid to do that, then 𝙮𝙤𝙪 𝙬𝙤𝙪𝙡𝙙 𝙣𝙤𝙩 𝙚𝙫𝙚𝙣 𝙗𝙚 𝙖𝙗𝙡𝙚 𝙩𝙤 𝙖𝙙𝙙 𝙧𝙖𝙣𝙙𝙤𝙢 𝙫𝙖𝙧𝙞𝙖𝙗𝙡𝙚𝙨 𝙩𝙤𝙜𝙚𝙩𝙝𝙚𝙧.

initial->unitial (autocorrect!)

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

interesting. I'm only familiar with commutative & free independence

YouTube video by Almost Sure The Algebra of Mixed Quantum States

New video released on mixing quantum and classical probabilities.

The Algebra of Mixed Quantum States

youtu.be/K6h62Gr0nwg

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?

new-ish…actually a few weeks old now. I forgot to post here when I initially launched it on YouTube

YouTube video by Almost Sure This unexpected proof shocked mathematicians

New video released on the Gaussian correlation inequality.

This unexpected proof shocked mathematicians!

youtu.be/WJGR1oc6Gxo?...

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...

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)

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?

bsky.app/profile/almo...

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.

another martingale surface

scaled gamma(1) process

scaled gamma(40) process

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

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.

Probability fact:

A sequence X_0,X_1,X_2,…,X_i,… of Gamma(a_i) rv’s has independent increments

X_1-X_0,X_2-X_1,…

if and only if it has independent ratios

X_0/X_1,X_1/X_2,…

in which case a_{i+1}>=a_i and,

X_{i+1}-X_i~Gamma(a_{i+1}-a_i)
X_i/X_{i+1}~Beta(a_i,a_{i+1})

Either way, it’s lightning fast using my new Schrödinger’s Cat8 cables

distribution calculation

Probability fact:

If X,Y are independent Gamma(a), Gamma(b) random variables then

X/(X+Y), X+Y

are independent Beta(a,b), Gamma(a+b) rvs.

Equivalently: if X,Y are independent Beta(a,b), Gamma(a+b) random variables then

XY, (1-X)Y

are independent Gamma(a), Gamma(b) rvs.

My extension of Hermite-Hadamard:

If c = (pa+qb)/(p+q) for p,q > 0 then

f(c)≤M(t)≤(pf(a)+qf(b))/(p+q)

where M(t) is the average value of f under Beta(qt,pt) distribution scaled to interval [a,b].

M(t) is ctsly decreasing from

M(0)=(pf(a)+qf(b))/(p+q)
to
M(∞) = f(c)
pbs.twimg.com/media/GoSUpd...

Ok, I managed to prove this!

So Beta(at,bt) is decreasing in the convex order and can find reverse martingale (continuous Itô diffusion)

X(t)=E[X(s) | {X(u),u >=t}]
(all s < t)

with marginals

X(t)~Beta(at,bt)

Question: fixed a, b > 0, are distributions Beta(at,bt) decreasing in convex order over t > 0?

Equivalent to existence of a reverse-time martingale X_t ~ Beta(at,bt).

Equivalently:

-(d/dt)E[(x-X_t)_+] >=0

for all 0 < x < 1.
Plots suggest so: parameterised as s=a+b,mean=a/s

YouTube video by Almost Sure Brownian Motion - A Beautiful Monster

New YouTube video posted

"Brownian motion - A Beautiful Monster"

youtu.be/IgMmsnzye1s?...

YouTube video by Almost Sure Brownian Motion - A Beautiful Monster

New YouTube video posted

"Brownian motion - A Beautiful Monster"

youtu.be/IgMmsnzye1s?...