Conditions on random variable to satisfy limit property

In summary, the problem is to find sufficient and preferably also necessary conditions on random variable X such that its characteristic function g(x) satisfies the limit property:\lim_{t\to0}\frac{1-g(\lambda t)}{1-g(t)}=\lambda^2The necessary and sufficient condition is that 0 < var(X) < infinity and g'(0) = 0, which follows from the assumption that X is symmetric around 0 and has an expected value of 0.
  • #1
jjhyun90
8
0

Homework Statement


The problem is to find sufficient and preferably also necessary conditions on random variable X such that its characteristic function g(x) satisfies the limit property:
[itex]\lim_{t\to0}\frac{1-g(\lambda t)}{1-g(t)}=\lambda^2[/itex]
I may assume X is symmetric around 0, so the characteristic function is real and even.

Homework Equations


[itex]g(t)=\int_{-\infty}^{\infty} e^{itx}f_{X}(x) dx[/itex]


The Attempt at a Solution



I'm stuck immediately after trying to apply l'Hopital's rule. Any suggestions would be helpful.
Thank you.
 
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  • #2
What is the value of [itex]g(0)[/itex]?
 
  • #3
[itex]g(0)=1[/itex] since it is a characteristic function.
 
  • #4
So, what can you say about your limit? Can you evaluate it?
 
  • #5
Using l'Hopital's rule and chain rule, the equivalent condition I need will be
[itex]\lim_{t\to0}\frac{g'(\lambda t)}{g'(t)} = \lambda [/itex].
 
  • #6
jjhyun90 said:
Using l'Hopital's rule and chain rule, the equivalent condition I need will be
[itex]\lim_{t\to0}\frac{g'(\lambda t)}{g'(t)} = \lambda [/itex].

Not really, evaluate these derivatives explicitly:

[tex]
\frac{d}{d t} \left(1 - g(\lambda t)\right)
[/tex]

[tex]
\frac{d}{d t} \left(1 - g(t)\right)
[/tex]
 
  • #7
I must be confused. If I'm not mistaken, the first evaluates to [itex]-\lambda g'(\lambda t)[/itex] and the second [itex]-g'(t)[/itex].
 
  • #8
True, so your limit becomes:

[tex]
\lim_{t \rightarrow 0}{\frac{-\lambda g'(\lambda t)}{-g'(t)}} = \lambda \, \lim_{t \rightarrow}{\frac{g'(\lambda t)}{g'(t)}}
[/tex]

Before you run in evaluating this limit, you need to consider two cases:

1) [itex]g'(0) \neq 0[/itex]. Then the limit is simply 1 and you get the result you posted in post #5. However, this is not what you have in the condition of the problem.

2) [itex]g'(0) = 0[/itex] What is the limit in this case?

BTW, what does [itex]g'(0)[/itex] mean?
 
  • #9
The condition I posted on #5 is what I desire to have, rather than what will follow from the assumption. Sorry for the confusion.
By the assumption that X is symmetric around 0 and thus its expected value is 0, it follows g'(0)=0, so it is necessary to use l'Hopital's rule again.
Thus it seems a sufficient condition is that 0 < var(X) < infinity. Please let me know if I am wrong, but otherwise thank you for your help!
 
  • #10
Yes, you are correct.
 

1. What is the limit property for a random variable?

The limit property for a random variable is a condition that states that as the sample size increases, the probability of the random variable approaching a specific value also increases.

2. What are the necessary conditions for a random variable to satisfy the limit property?

The random variable must be independent and identically distributed (i.i.d.) and the sample size must be large enough for the limit property to hold.

3. Can a random variable that does not satisfy the limit property still be considered a random variable?

Yes, a random variable that does not satisfy the limit property can still be considered a random variable, but it may not accurately represent the underlying distribution of the data.

4. How does the central limit theorem relate to the limit property for a random variable?

The central limit theorem states that as the sample size increases, the sampling distribution of the sample mean will approach a normal distribution. This is closely related to the limit property as it also involves the convergence of a random variable to a specific value as the sample size increases.

5. Are there any practical implications of the limit property for a random variable?

Yes, the limit property is important in statistical analysis as it allows us to make inferences about a population based on a sample. It also helps to ensure the accuracy and reliability of statistical models and predictions.

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