Does bounded almost surely imply bounded in Lp?

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In summary, bounded almost surely means that for a random variable X(\omega), the probability that X is less than or equal to a constant K is equal to 1. This means that X is almost surely bounded by a constant, and this holds true for both a single random variable and a sequence of random variables. Additionally, if a random variable is bounded almost surely, it is also bounded in L^{p} for sets in Lebesgue measure.
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wayneckm
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Hello all,

I am a bit confused by the concept of "bounded almost surely".

If a random variable [tex]X(\omega)[/tex] is bounded a.s., so this means (i) [tex] X \leq K [/tex] for some constant [tex] K [/tex] ? or some [tex] K(\omega) [/tex]?

Also, if it is bounded almost surely, does that mean it is also bounded in [tex] L^{p} [/tex]? Apparently if case (i) is true, then it should be also bounded in [tex] L^{p} [/tex]?

Thanks.

Wayne
 
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wayneckm said:
Hello all,

I am a bit confused by the concept of "bounded almost surely".

[tex] Pr(|X|\leq M)=1[/tex].

Almost surely=almost everywhere which excludes sets of zero measure.

If L means sets in Lebesgue measure then sets of zero measure would be excluded, so I believe it would be bounded in L if it's bounded in M.

[tex] K\leq M [/tex]
 
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  • #3
wayneckm said:
Hello all,

I am a bit confused by the concept of "bounded almost surely".

If a random variable [tex]X(\omega)[/tex] is bounded a.s., so this means (i) [tex] X \leq K [/tex] for some constant [tex] K [/tex] ? or some [tex] K(\omega) [/tex]?

The [tex]K(\omega)[/tex] version would be worthless for a single random variable [tex]X[/itex], just take [tex]K(\omega) = |X(\omega)|[/tex]. Now "bounded almost surely" where you talk about a sequence of random variables is another question.
 

FAQ: Does bounded almost surely imply bounded in Lp?

What is the definition of "bounded almost surely"?

Bounded almost surely refers to a random variable that is bounded with probability 1. This means that the variable will take on values within a certain range with almost complete certainty.

What does "bounded almost surely" imply?

If a random variable is bounded almost surely, it implies that the variable is also bounded in the Lp space. This means that the variable has a finite integral over its probability space.

What is the significance of being "bounded in Lp"?

Being bounded in Lp is significant because it ensures that the variable has a finite expectation value and is therefore well-behaved mathematically. This is important in many statistical and probabilistic calculations.

Can a random variable be bounded almost surely but not bounded in Lp?

Yes, it is possible for a random variable to be bounded almost surely but not bounded in Lp. This can occur if the variable has a finite expectation value, but its higher moments are not finite.

What are some examples of random variables that are bounded almost surely and in Lp?

Some examples of random variables that are bounded almost surely and in Lp include the normal distribution, uniform distribution, and exponential distribution. These distributions have finite expectation values and all of their higher moments are also finite, making them well-behaved in Lp.

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