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### Infinitely Divisible Density Continuous?

Posted: **June 8th, 2012, 8:36 am**

by **AdareIre**

Intuitively it seems that an infinitely divisible distribution should be continuous but I don't see how it comes into play in the definition of a Levy process or in the Levy-Khinthcine formula?I am particularly concerned with continuity at the origin? Or whether the density function should even be differentiable?Thanks in advance

### Infinitely Divisible Density Continuous?

Posted: **June 8th, 2012, 9:05 am**

by **ACD**

Poisson distribution is infinitely divisible and is a discrete distribution (Poisson process is a Levy process), there are several others (negative binomial for example) so your intuition is off the mark.

### Infinitely Divisible Density Continuous?

Posted: **June 8th, 2012, 9:41 am**

by **AdareIre**

Sorry for ambiguity in my previous post.I understand that discrete distributions can be infinitely divisible.Consider a strictly positive density function f(x) defined for all x\in(-\infty,\infty). Can f be infinitely divisible if it has a discontinuity at a certain point ? e.g. say a discontinuity at x = 0 such thatMust f also be differentiable for x\in(-\infty,\infty) to be infinitely divisible?

### Infinitely Divisible Density Continuous?

Posted: **June 8th, 2012, 10:33 am**

by **eh**

The density of the marginal law of Levy process does not necessarily exist.I don't see anything intuitive about a density being continuous. K. Sato's book discusses distribution properties of the marginal laws of Levy processes. Look at that.

### Infinitely Divisible Density Continuous?

Posted: **June 8th, 2012, 11:45 am**

by **AdareIre**

Thanks eh for your comments,My intuition was coming from the perspective that if an inf div density f exists which has a discontinuity at some point. Then there exist a density function f_n such that an n-fold convolution of f_n with itself gives f. I find it hard to see how an n-fold convolution of any function could produce a function with a discontinuity. It seems to me that the resultant density would be smooth.Analogously, suppose the random variable X has a density f and the random variable X_n has a density function f_n as described above. Again it seems strange that the sum of n X_n random variables would have a density function which is not smooth or continuous at least.Unfortunately there is not a copy of Sato in the Library....

### Infinitely Divisible Density Continuous?

Posted: **June 8th, 2012, 1:17 pm**

by **Alan**

QuoteOriginally posted by: AdareIreThanks eh for your comments,My intuition was coming from the perspective that if an inf div density f exists which has a discontinuity at some point. Then there exist a density function f_n such that an n-fold convolution of f_n with itself gives f. I find it hard to see how an n-fold convolution of any function could produce a function with a discontinuity. It seems to me that the resultant density would be smooth.Suppose the support of f is a lattice (say +/- n, n an integer). Then, so will be the convolutions ...

### Infinitely Divisible Density Continuous?

Posted: **June 8th, 2012, 2:32 pm**

by **AdareIre**

I have worked through some examples with a discontinuity at zero and indeed it does seem to be okay!Suppose f is an infinitely divisible 2-EPT function with no pointmass at zero but is discontinuous at zero. Let f_n be the density function corresponding to the n^th root of the characteristic function of f. As f is discontinuous at zero f_n will have a pointmass at zero.I have no proof of this but it seems to work and make intuitive sense!! Thanks