Having a rather basic understanding of probabilities I would like to ask you what exactly means the following notation. I am looking at the Gardiner's handbook for stochastic methods and am interested in understanding what the following notation means. On page 32 he used $\langle\rangle$ brackets which I have not come across before:
"If $\textbf{s}$ is the vector $(s_1,s_2,...,s_n)$ and $\textbf{X}$ the vector of random variables $(X_1,X_2,...,X_n)$ then the characteristic function is defined by $\phi(\textbf{s})= \langle\text{exp}\left(i\textbf{s} \dot{} \textbf{X}\right)\rangle$"
What do those brackets there mean? Also, what role exactly, vector $\textbf{s}$ plays here?
He then uses square brackets, which I also do not understand:
"If the moments $\langle\Pi_i X_i^{m_{i}}\rangle = \left[ \Pi_i \left( -i \frac{\partial}{\partial s_i}\right)^{m_i} \phi(\textbf{s}) \right]_{s=0}$"
What does $s=0$ on the RHS of the square bracket mean. Also $m_i$ appears nowhere before this, what does it mean?
As Slug Pue said, the notation $\langle Y\rangle$ means the expectation of $Y$ (an other notation is $\mathbb EY$). Here, $\mathbf s$ is a $n$-dimensional vector; we take the Euclidian inner product with $X$ (in the one dimensional case, it was the usual product; in the infinite dimensional case, we use continuous linear functionals).
The notation $\left[g(\mathbf s)\right]_{\mathbf s=0}$ is a notation for $g(\mathbf 0)$, that is, "the function between brackets evaluated at $\mathbf 0$". The point is that the function $g$ may be hard to express, especially when partial derivatives are involved.
The $m_i$ are non-negative integers. We compute the expectation of the product of powers of the components of the random vector. Since the number $i$ is involved, it would be better to use the index $j$, or $k$, or $l$.