Can Characteristic Functions Determine CDF Without Defined Moments?

[EngWiPy]

Suppose I have a random variable whose moments are not defined, can I still use the characteristic function to find the CDF of that random variable?

 

[Stephen Tashi]

Suppose I have a random variable whose moments are not defined,

Distinguish between "not defined" and "not known".

Do you have a random variable known to be from a family of distributions (e.g. Cauchy) whose moments do not exist?

-or do you have a random variable whose moments exist but are not known numerical values?

 

[EngWiPy]

Distinguish between "not defined" and "not known".

Do you have a random variable known to be from a family of distributions (e.g. Cauchy) whose moments do not exist?

-or do you have a random variable whose moments exist but are not known numerical values?

The random variable in question has a distribution is similar to Cauchy distribution, but not exactly the same. Its CDF and PDF are given by

$$F_X(x)=1-\frac{1}{1+x}\\f_X(x)=\frac{1}{(1+x)^2}$$

respectively, for $0\leq x<\infty$ I searched the table of integral for the integration

$$\int_0^{\infty}\frac{x}{(1+x)^2}\,dx$$

but the conditions to evaluate the integral are not met in my case.

 

[Stephen Tashi]

Since you know the CDF, I don't understand your question in post #1 about "finding" the CDF.

 

[EngWiPy]

Since you know the CDF, I don't understand your question in post #1 about "finding" the CDF.

Right, I need to find the CDF of the summation of such random variables

$$Y=\sum_{k=1}^KX_k$$

where $\{X_k\}$ are i.i.d. random variables with CDF and PDF as given previously.

 

[Stephen Tashi]

can I still use the characteristic function to find the CDF of that random variable?

So what you mean to ask is "If I have a random variable X whose moments do not exist, can I use its characterisic function to find the CDF of Y = X1 + X2 + ...XN where each Xi is an independent realization of X?"

The characteristic function of X exists even if its moments do not. Y has a characteristic function that is the product of N copies of the characteristic function of X. Whether these facts suggest a practical way to find the CDF of Y in your specific problem, I don't know.

 

[EngWiPy]

So what you mean to ask is "If I have a random variable X whose moments do not exist, can I use its characterisic function to find the CDF of Y = X1 + X2 + ...XN where each Xi is an independent realization of X?"

The characteristic function of X exists even if its moments do not. Y has a characteristic function that is the product of N copies of the characteristic function of X. Whether these facts suggest a practical way to find the CDF of Y in your specific problem, I don't know.

Yes, I just want to find the CDF of the sum of the random variables, and yes it will be the product of the characteristic functions of the individual random variables because they are independent. I wanted to know if this is a valid way to find the CDF, given that the individual random variables' moments don't exist.