Dtjytyk

How to run whichever function is bound to a certain key

Ultrafilters as a double dual

Is there a logarithm base for which the logarithm becomes an identity function?

Called into a meeting and told we are being made redundant (laid off) and "not to share outside". Can I tell my partner?

Is it a Cyclops number? "Nobody" knows!

A running toilet that stops itself

Prove that this is a surjection and find the kernel

Why would /etc/passwd be used every time someone executes `ls -l` command?

How can I portion out frozen cookie dough?

Can the Witch Sight warlock invocation see through the Mirror Image spell?

Can you train your ranger to master different fighting styles?

Why is there an extra space when I type "ls" on the Desktop?

What is better: yes / no radio, or simple checkbox?

3.5% Interest Student Loan or use all of my savings on Tuition?

PTIJ: Sport in the Torah

Who is the hero of my every day?

Do black holes violate the conservation of mass?

Use Mercury as quenching liquid for swords?

What would be the most expensive material to an intergalactic society?

Short story about cities being connected by a conveyor belt

Is the differential, dp, exact or not?

Can I negotiate a patent idea for a raise, under French law?

Is it appropriate to ask a former professor to order a book for me through an inter-library loan?

What does *dead* mean in *What do you mean, dead?*?

Determining if a Large Deviation Principle exists for a sequence of probability measures.

Understanding large deviation theory/principleTopological issues in large deviation principleOn proving that a certain subset is dense in a metric space of probability measures equipped with Prohorov metricDefinition of Large Deviation PrincipleLarge deviation theory--examples of irregular setsLarge deviations principle for $X^{lfloor alpha N rfloor}/N$Equivalence in the formulation of the “Large Deviation Principle”Large deviation principle for Gaussian random variables with mean $0$ and variance $1/n$.Reconciling definitions of large deviations principleLarge Deviation Principle

0

$begingroup$

Given a sequence $(P_n)_n$ of probability measures on a space satisfies a LDP with rate $r_n$ and rate function $I$ if both the following hold:

$$ limsup_{n}frac{1}{r_n}log P_n [F] le - inf_{x in F}I(x), forall F: text{ closed } $$

$$ liminf_{n}frac{1}{r_n}log P_n [G] ge - inf_{x in G}I(x), forall G : text{ open } $$

Consider the following sequences:

a) $P_n = Uniform([-n,n])$

b) $P_n = Uniform([-1/n,1/n])$

Determine if either of these satisfy the LDP with rate n, if so give the rate function $I$.

I already know that (a) doesn't satisfy the LDP but (b) does.

My biggest issue here is how to solve these problems, the way it is stated it feels like I should already have an idea of what this $I$ should be.

Do you first find a limiting probability?

In the case of (a) there is no uniform distribution on all of the real line so it seems immediate there is no LDP to be satisfied.

In (b) as $n to infty $ in the limit we get $delta_0(x) = 1_{{x = 0}}$ so it makes sense that we would have $I(x) = 0$ if $x= 0$ otherwise $I = infty$.

That approach works here but I'm not sure if that is the approach I should always take.

probability-theory probability-distributions large-deviation-theory

share|cite|improve this question

edited yesterday

all.over

asked yesterday

all.overall.over

537

$endgroup$

add a comment | 

0

$begingroup$

Given a sequence $(P_n)_n$ of probability measures on a space satisfies a LDP with rate $r_n$ and rate function $I$ if both the following hold:

$$ limsup_{n}frac{1}{r_n}log P_n [F] le - inf_{x in F}I(x), forall F: text{ closed } $$

$$ liminf_{n}frac{1}{r_n}log P_n [G] ge - inf_{x in G}I(x), forall G : text{ open } $$

Consider the following sequences:

a) $P_n = Uniform([-n,n])$

b) $P_n = Uniform([-1/n,1/n])$

Determine if either of these satisfy the LDP with rate n, if so give the rate function $I$.

I already know that (a) doesn't satisfy the LDP but (b) does.

My biggest issue here is how to solve these problems, the way it is stated it feels like I should already have an idea of what this $I$ should be.

Do you first find a limiting probability?

In the case of (a) there is no uniform distribution on all of the real line so it seems immediate there is no LDP to be satisfied.

In (b) as $n to infty $ in the limit we get $delta_0(x) = 1_{{x = 0}}$ so it makes sense that we would have $I(x) = 0$ if $x= 0$ otherwise $I = infty$.

That approach works here but I'm not sure if that is the approach I should always take.

probability-theory probability-distributions large-deviation-theory

share|cite|improve this question

edited yesterday

all.over

asked yesterday

all.overall.over

537

$endgroup$

add a comment | 

$begingroup$

Given a sequence $(P_n)_n$ of probability measures on a space satisfies a LDP with rate $r_n$ and rate function $I$ if both the following hold:

$$ limsup_{n}frac{1}{r_n}log P_n [F] le - inf_{x in F}I(x), forall F: text{ closed } $$

$$ liminf_{n}frac{1}{r_n}log P_n [G] ge - inf_{x in G}I(x), forall G : text{ open } $$

Consider the following sequences:

a) $P_n = Uniform([-n,n])$

b) $P_n = Uniform([-1/n,1/n])$

Determine if either of these satisfy the LDP with rate n, if so give the rate function $I$.

I already know that (a) doesn't satisfy the LDP but (b) does.

My biggest issue here is how to solve these problems, the way it is stated it feels like I should already have an idea of what this $I$ should be.

Do you first find a limiting probability?

In the case of (a) there is no uniform distribution on all of the real line so it seems immediate there is no LDP to be satisfied.

In (b) as $n to infty $ in the limit we get $delta_0(x) = 1_{{x = 0}}$ so it makes sense that we would have $I(x) = 0$ if $x= 0$ otherwise $I = infty$.

That approach works here but I'm not sure if that is the approach I should always take.

probability-theory probability-distributions large-deviation-theory

share|cite|improve this question

edited yesterday

all.over

asked yesterday

all.overall.over

537

$endgroup$

share|cite|improve this question

edited yesterday

all.over

asked yesterday

all.overall.over

537

share|cite|improve this question

edited yesterday

all.over

asked yesterday

all.overall.over

537

asked yesterday

all.overall.over

537

asked yesterday

all.overall.over

537