Abstract
In this chapter many mathematical details or proofs are not given so we refer the reader to the appropriate references in basic probability theory. See for example [10, 60].
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- 1.
We assume as usual that these \(\sigma \)-fields are complete.
- 2.
The notation \( \lim \) denotes limits for functions or in the case of random variables this denotes limits in the (a.s.) sense. This may also be denoted using the symbol \(\rightarrow \).
- 3.
This is somewhat equivalent to replacing Laplace transforms by Fourier transforms.
- 4.
If you have more experience in analysis, maybe you have learned this concept as the so called Fourier transform. Although there is a slight variation here as in most books, one starts with periodic functions.
- 5.
See e.g. Section 30 in [10].
- 6.
That is, \(\sum _{i, j=0}^n\varphi (\theta _i-\theta _j)z_i\bar{z}_j\ge 0\) for any sequence of real numbers \(\theta _i\in \mathbb {R}\) and \(z_i\in \mathbb {C}\), \(i=1,..., n\).
- 7.
Note that we are already assuming that equality as functions means equality a.e. Hint: If you want more information you can check also Proposition 6.1.1.
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Kohatsu-Higa, A., Takeuchi, A. (2019). Review of Some Basic Concepts of Probability Theory. In: Jump SDEs and the Study of Their Densities. Universitext. Springer, Singapore. https://doi.org/10.1007/978-981-32-9741-8_1
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