Well, if you have a point process that you try modeling as a Poisson process, and find it has heavy tails, there are several possibilities. What are the key assumptions for a Poisson Process:
-There is a constant rate function
-Events are memoryless, that is P(E in (t,t+d)) is independent of t and when other events are.
-The waiting time until the next event is exponentially distributed (kinda what the previous two are saying)
So, how can you violate these assumptions to get heavy tails?
-Non-constant rate function. If the rate function switches between, say, two values, you'll have too many short wait-times, and too many long wait-times, given the overall rate function. This can show itself as having heavy tails.
-The waiting time is not exponentially distributed. In which case, you don't have a Poisson process. You have some other sort of point process.
Note that in the extreme case, any point process can be modeled by a NHPP - put a delta function at each event, and set the rate to 0 elsewhere. I think we can all agree that this is a poor model, having little predictive power. So if you are interested in a NHPP, you'll want to think a bit about whether that is the right model, or whether you are overly-adjusting a model to fit your data.
I guess your "this is a very newbie question" refers to this of your many questions:
"...but conceptually, would the point of a power law be violated if there
are some zeroes in the data?"**
No. The concept remains valid as the same class of distributions may be applied to data
with or without zeros. You may be interested in reading more about Tweedie class of distributions here and then here.
For example, the well-known Taylor’s law says that the variance is proportional to a power of the mean. Taylor’s law is mathematically identical to the variance-to-mean power law that characterizes the Tweedie distributions, that is for any random variable that obeys a Tweedie distribution, the variance relates to the mean by the power law. Since that "any random variable" can be discrete, continuous or a combination of both, the concept of the power law
may equally apply to data that are counts (Poisson), reals (Normal), positive reals (Gamma), or positive reals with the added positive mass at zero (compound Poisson–gamma).
Given your "there are some zeros in the data" and your comment "yes, my values are counts", simple Poisson may work. If not, e.g. zeros are too few or too many, you may try Neyman Type A distribution (this R package manual mentions it the context of the Tweedie class of distributions).
I hope some of the above helps.
Best Answer
Wikipedia is often a reasonable start point for basic definitions. In this case there is an entry for heavy-tailed distributions.
for all $\lambda>0$. This can be interpreted as: the tails decay slower than the exponential and this has implications on the existence of moments (see the same wikipedia entry).
Then,
(1). To the exponential or compared to an exponential-type behaviour.
(2). This can be empirically checked using a normal QQ-plot, for example,
x <- rt(1000,3)
qqnorm(x)
qqline(x,col="red")
As you can see, the lack of linear fit is observed in both tails.
(3). An immediate implication of the use of heavy-tailed distribution is that you observe more extreme observations or that the model can capture this sort of behaviour. This is, values far from the shoulders of the distribution. This is reflected in the summary statistics, for example compare the statistics of a normal sample and those of a $t$ sample with $3$ degrees of freedom
summary(rnorm(1000))
summary(rt(1000,3))