Abstract
Poisson processes are generalizations of the Poisson distribution which are often used to describe the random behavior of some counting random quantities such as the number of arrivals to a queue, the number of hits to a webpage etc.
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Notes
- 1.
The geometric distribution has the property of abscence of memory if we request it to be satisfied only for 0, 1, ...
- 2.
Other books call this the memoryless property.
- 3.
Another way of saying the same thing is to say that \(\delta _1\) is a probability measure so that \(\delta _1(A)=1\) if \(1\in A\) and zero otherwise.
- 4.
This measure is essentially unique, although we have not yet discussed its uniqueness. This will follow because the exponential function is a generating family. Recall the discussion after (1.1) and before Exercise 1.1.11. That is, exponential functions generate indicators and therefore the corresponding measures have to be equal.
- 5.
In some advanced texts this definition is considered in greater generality, without the condition that there are a finite number of counted events in any finite interval.
- 6.
Given n independent random variables \(U_1,\cdots , U_n\) each with the uniform distribution in [0, t], the order statistic distribution is the n-dimensional distribution of the n random variables once they have been ordered.
- 7.
Recall that independence of random process means that the \(\sigma \)-fields generated by these process are independent.
- 8.
Recall results related with the law of large numbers.
- 9.
This is an exercise to test your understanding.
- 10.
Recall that o(1) stands for any function that converges to zero as the related parameter (which should be clear from the statement) approaches a certain limit. In this case, the parameter is h.
- 11.
A stochastic process is a family of random variables \(\{N_t\}_{t\in [0,\infty )}\) such that \(N:\varOmega \times [0,\infty )\rightarrow \mathbb {R}\) is jointly measurable. I suppose that you interpreted this in a similar way in Definition 2.1.17.
- 12.
Recall that o(h) is any term such that \( \lim _{h\rightarrow 0}\frac{o(h)}{h}=0 \).
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Kohatsu-Higa, A., Takeuchi, A. (2019). Simple Poisson Process and Its Corresponding SDEs. In: Jump SDEs and the Study of Their Densities. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-32-9741-8_2
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