Tapi

general recent conversations

$Mathematics$ Mathematics

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Tapi

Feb 16 11:10

https://math.stackexchange.com/questions/5035857/proof-of-measurability

can someone help with this?

Tapi

Jan 17 19:03

Tapi

Jan 17 19:03

This is a part of the proof where they show that X is continuous a.s. at 0. Can someone explain this?

Tapi

Jan 17 19:00

Tapi

Jan 17 19:00

I'm reading the following theorem from the book on Brownian Motion by Mortars, Peres.

Tapi

Jan 10 09:02

@Jakobian you around?
Is it true that Xn is a measurable map from a finite probability space to C[0,1] trivially?

Tapi

Jan 8 20:04

I figured

Tapi

Jan 8 19:59

Is the set of sequences with finitely many rational non zero entries (rest are 0) not in a bijection with the power set of rational numbers?

Tapi

Jan 8 13:29

Why is this measurable?

Tapi

Jan 8 13:28

@Jakobian Let B be a borel set in R; preimage of B= $\{ f \in C[0,1] | f(t) \in B) \}$

Tapi

Jan 8 13:19

@Jakobian why is $X_n(\cdot, \omega)$ measurable?

Tapi

Jan 8 13:17

okay

Tapi

Jan 8 13:15

by using a polynomial approximation

Tapi

Jan 8 13:15

oh because we can approximate the function at irrational points using rational points since f is continuous

Tapi

Jan 8 13:13

it's continuous that's why?

Tapi

Jan 8 13:13

@Jakobian this is true though

Tapi

Jan 8 13:12

Tapi

Jan 8 13:11

got it

Tapi

Jan 8 13:08

okay

Tapi

Jan 8 13:03

generating set that forms a pi system

Tapi

Jan 8 13:02

so the idea i was going with is finding a generating set and showing the inverse image thing for that set

Tapi

Jan 8 13:02

right

Tapi

Jan 8 13:01

I basically have to show inverse image of a set from sigma(C[0,1]) under Xn would belong to \mathcal{A}

Tapi

Jan 8 12:58

typically when it's R we say it's a random variable, when it's R^k we say it's a random vector and when C[0,1] we say it's a random continuous function

Tapi

Jan 8 12:57

@Jakobian yes

Tapi

Jan 8 12:57

@Jakobian I wanted to show that the map Xn is measurable

Tapi

Jan 8 12:50

Tapi

Jan 8 12:49

It's about constructing random elements which are functions

Tapi

Jan 8 12:48

@Jakobian I was doing an exercise and I thought this was the way to go

Tapi

Jan 8 12:46

Just came back

Tapi

Jan 8 12:46

I had a class

Tapi

Jan 8 10:38

this can be written as countable intersection union of sets of the form f| ev_t < 1-1/n

Tapi

Jan 8 10:33

Tapi

Jan 8 10:29

I see

Tapi

Jan 8 10:28

why convergence is relevant

Tapi

Jan 8 10:28

i was tryin to show $\sigma(ev_t : t \in [0,1]) = \mathcal{B}(C[0,1])$

Tapi

Jan 8 10:26

i don't get what you're saying

Tapi

Jan 8 10:26

Tapi

Jan 8 10:25

@Jakobian no im not sure if we are taking ptwise convergence

Tapi

Jan 8 10:21

evaluation maps are continuous (using sup metric)

Tapi

Jan 8 10:19

yes

Tapi

Jan 8 10:18

@Jakobian this function: ev_t :C[0,1] to R, takes f to f(t) , (vary t) doesn't generate borel (C[0,1])?

Tapi

Jan 8 09:56

how to show that Evaluation Maps Generate Borel-sigma(C[0,1])?

Tapi

Nov 13, 2024 21:04

Let (Ω, A, P) be a probability space, Q be a probability with Q ≪ P. Suppose A is countably generated. Show that there is a sequence Fn of finite fields which increase to A. Let An(ω) be the Fn-atom containing ω. Define Xn(ω) = Q(An(ω))1[P(An(ω))>0]/P(An(ω))

Show that {Xn, Fn} is a u.i. martingale. Deduce that Q = X · P for some
integrable random variable X.

can someone help with this?

Tapi

Oct 15, 2024 20:12

can someone help with this?

Tapi

Oct 15, 2024 20:12

Tapi

Sep 15, 2024 22:20

I liked discrete probability, and I'm currently enjoying percolation theory

Tapi

Sep 15, 2024 22:16

hmm it has a different flavour

Tapi

Sep 15, 2024 22:13

I've met people who barely had an introductory course on Probability

Ten fold

CrossValidated's general room for gossip, grumbles, and idle c...

Tapi

Jan 10 06:15

does anyone know how a 3 component PCG sampler can be viewed as a 2 component one? is it simply by clubbing the first two steps?