Bonferroni Equality in Gaussian Process Theory

Definition

Let ${X(t), t \in T}$ be a Gaussian process defined on an index set $T$. For a given threshold $u \in \mathbb{R}$, we define the excursion set as:

$$A_u = {t \in T : X(t) > u}$$

Bonferroni Inequality

The Bonferroni inequality states that:

$$P(\sup_{t \in T} X(t) > u) \leq \sum_{t \in T} P(X(t) > u)$$

Equality Case

The Bonferroni equality occurs when the above inequality becomes an equality:

$$P(\sup_{t \in T} X(t) > u) = \sum_{t \in T} P(X(t) > u)$$

This equality holds under specific conditions:

1. When the events ${X(t) > u}$ are mutually exclusive for different $t$.
2. In the limit as $u \to \infty$ for certain classes of Gaussian processes.

Mathematical Derivation

Let $I_A$ denote the indicator function of event $A$. Then:

$$\begin{align*} P(\sup_{t \in T} X(t) > u) &= P(\bigcup_{t \in T} {X(t) > u}) \ &= E[I_{{\sup_{t \in T} X(t) > u}}] \ &= E[\sup_{t \in T} I_{{X(t) > u}}] \ &\leq E[\sum_{t \in T} I_{{X(t) > u}}] \ &= \sum_{t \in T} E[I_{{X(t) > u}}] \ &= \sum_{t \in T} P(X(t) > u) \end{align*}$$

Relation to Gaussian Process Theory

In Gaussian process theory, the Bonferroni equality is particularly relevant when studying:

1. Excursion sets and their properties
2. Level crossings of Gaussian processes
3. Extreme value behavior of Gaussian fields

For a stationary Gaussian process with covariance function $r(t)$, we can express the equality in terms of the standard normal distribution function $\Phi$:

$$P(\sup_{t \in [0,T]} X(t) > u) \approx T \cdot (1 - \Phi(u)) \quad \text{as } u \to \infty$$

This approximation becomes exact in the limit, forming a connection between the Bonferroni equality and the asymptotic behavior of Gaussian processes.