Let $ Y := Z^T \Sigma Z $. We have $ \operatorname{var}(Y) = \mathbb{E}(Y^2) - \mathbb{E}(Y)^2 $ and $ \mathbb{E}(Y) = \sum_i \sigma_{i, i} $. We thus need to compute $ \mathbb{E}(Y^2) $. For this, ...

$ \text{Dear James,} $ you can write your function as a(n infinite) sum of homogeneous symmetric functions \begin{equation*}%$ h_\ell(x_1, \dots, x_k) := \sum_{1 \leqslant i_1 \leqslant \cdots \...

$\textit{Lemma :}$ We have \begin{align*}%$ \frac{1}{d!} \sum_{\lambda\in \text{Hooks}(d)} (-1)^{\ell(\lambda)-1} \, \dim \lambda \prod_{\Box \in \lambda}(x + c(\Box) )(y + c(\Box) ) = [c_d]\...

Using my previous remark (namely the correction of the operator $ \mathcal{O} $), an expression would just involve the "skeleton" of your continuous time random walk, namely the walk with initial law $...

The answer is yes. In general, if a random variable $X$ has a Lebesgue density $ f_X : \mathbb{R}^d \to \mathbb{R}_+ $, a Stein operator is given by $ g \mapsto \frac{\mathrm{div}(g f_X) }{ f_X } $, ...

This is only a partial answer, as the computations can become quite involved. We have $$ \mathbb{E}( X_t J_t ) = \mathbb{E}\left( X_t Q \int_{ \mathbb{R}_+ \times [0, t] } d\mu \right) = \mu_Q \...

You can also have a look at definition 1.7 in this article : https://arxiv.org/pdf/1507.07765.pdf where it is called a "decoupling inequality". Only the upper bound, though, and not completely the ...

Concerning the finite sum, I am not sure you can get it (maybe for particular values of $ \nu $). Here is a possible "closed" expression for your sum. Start by writing $ \lambda := \mu + \alpha $ with ...

The (2) is just the independence of the Brownian motion $ (\beta_t)_t $ with the ``clock'' $ (H_T)_T $. This is a classical result due to Lévy (and more generally, this is the Dambis-Dubins-Schwartz ...

Thanks a lot ! This considerably helps me. I am also concerned about the deviations (moderate and large), but the range is certainly simpler to deal with than the fluctuation one. Concerning the ...