Reproducing Kernel Hilbert Spaces and Covariance Functions

Let $K : T \times T \to \mathbb{C}$ be a covariance function such that the associated RKHS $\mathcal{H}_K$ is separable where $T \subset \mathbb{R}$. Then there exists a family of vector functions

$\displaystyle \Psi (t, x) = (\psi_n (t, x), n \geq 1) \forall t \in T$

and a Borel measure $\mu$ on $T$ such that $\psi_n (t, x) \in L^2 (T, \mu)$ in terms of which $K$ is representable as:

$\displaystyle K (s, t) = \int_T \sum_{n = 1}^{\infty} \psi_n (s, x) \overline{\psi_n (t, x)} d \mu (x)$

The vector functions $\Psi (s, .), s \in T$ and the measure $\mu$ may not be unique, but all such $(\Psi, .), .)$ determine $K$ and its reproducing kernel Hilbert space (RKHS) $H_K$ uniquely and the cardinality of the components determining $K$ remains the same. [1, ]

Remark 2. 1. If $\Psi (t, .)$ is a scalar, then we have

$\displaystyle K (s, t) = \int_T \Psi (s, x) \overline{\Psi (t, x)} d \mu (x)$

which includes the tri-diagonal triangular covariance with $\mu$ absolutely continuous relative to the Lebesgue measure.

2. The following notational simplification of (25) can be made. Let $n = R \times Z_+ = S \otimes P$, where $P$ is the power set of integers Z, and let P = u @ o where o is the counting measure. Then

$\displaystyle \Psi (t, n) = (\psi_n (t, x), n \in Z)$

Hence

$\displaystyle | \Psi^{\ast} (t) |^2_{L^2} = \int_T | \psi_n (t, x) |^2 d \mu (x)$

Bibliography

[1]

Malempati M. Rao. Stochastic Processes: Inference Theory. Springer Monographs in Mathematics. Springer, 2nd edition, 2014.

Aimeds, Tenghistor Gratifier

Interpretation of "Aimeds, Tenghistor Gratifier"

In a speculative context, "aimeds, tenghistor gratifier" can be interpreted as follows:

Aimeds could suggest the concept of focus or intention. It might refer to the state of being directed or purpose-driven, implying the act of setting intentions or aiming toward a specific outcome.

Tenghistor could evoke ideas of history or chronology. It might refer to the interconnectedness of past events and their influence on the present, symbolizing the weight of historical experiences in shaping current realities or a collective memory among people.

Gratifier might indicate something that provides fulfillment or satisfaction. In this context, it could represent the ultimate goal of the intentions set in "aimeds" and the historical context of "tenghistor." It implies that pursuing knowledge, understanding, or connection leads to a gratifying experience.

Putting it all together, "The focused pursuit of understanding, informed by the lessons of history, leads to a fulfilling and rewarding experience."

Stationary Dilations

1Stationary Dilations

Definition 1. Let $(\Omega, \mathcal{F}, P)$ and $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ be probability spaces. We say that $(\Omega, \mathcal{F}, P)$ is a factor of $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ if there exists a measurable surjective map $\phi : \tilde{\Omega} \to \Omega$ such that:

1. For all $A \in \mathcal{F}$, $\phi^{- 1} (A) \in \tilde{\mathcal{F}}$

2. For all $A \in \mathcal{F}$, $P (A) = \tilde{P} (\phi^{- 1} (A))$

In other words, $(\Omega, \mathcal{F}, P)$ can be obtained from $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ by projecting the larger space onto the smaller one while preserving the probability measure structure.

Remark 2. In the context of stationary dilations, this means that the original nonstationary process $\{X_t \}$ can be recovered from the stationary dilation $\{Y_t \}$ through a measurable projection that preserves the probabilistic structure of the original process.

Definition 3. (Stationary Dilation) Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $\{X_t \}_{t \in \mathbb{R}_+}$ be a nonstationary stochastic process. A stationary dilation of $\{X_t \}$ is a stationary process $\{Y_t \}_{t \in \mathbb{R}_+}$ defined on a larger probability space $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$ such that:

1. $(\Omega, \mathcal{F}, P)$ is a factor of $(\tilde{\Omega}, \tilde{\mathcal{F}}, \tilde{P})$

2. There exists a measurable projection operator $\Pi$ such that:

   $\displaystyle X_t = \Pi Y_t \quad \forall t \in \mathbb{R}_+$

Theorem 4. (Representation of Nonstationary Processes) For a continuous-time nonstationary process $\{X_t \}_{t \in \mathbb{R}_+}$, its stationary dilation exists which has sample paths $t \mapsto X_t (\omega)$ which are continuous with probability one when $X_t$:

• is uniformly continuous in probability over compact intervals:

  $\displaystyle \lim_{s \to t} P (|X_s - X_t | > \epsilon) = 0 \quad \forall \epsilon > 0, t \in [0, T], T > 0$

• has finite second moments:

  $\displaystyle \mathbb{E} [|X_t |^2] < \infty \quad \forall t \in \mathbb{R}_+$

• has an integral representation of the form:

  $\displaystyle X_t = \int_0^t \eta (s) ds$

  where $\eta (t)$ is a measurable random function that is stationary in the wide sense (with $\int_0^t \mathbb{E} [| \eta (s) |^2] \hspace{0.17em} ds < \infty$ for all $t$)

• and has a covariance operator

  $\displaystyle R (t, s) =\mathbb{E} [X_t X_s]$

  which is symmetric $(R (t, s) = R (s, t))$, positive definite and continuous

Under these conditions, there exists a representation:

$\displaystyle X_t = M (t) \cdot S_t$

where:

• $M (t)$ is a continuous deterministic modulation function

• $\{S_t \}_{t \in \mathbb{R}_+}$ is a stationary process

This representation can be obtained through the stationary dilation by choosing:

$\displaystyle Y_t = \left( \begin{array}{c} M (t)\\ S_t \end{array} \right)$

with the projection operator $\Pi$ defined as:

$\displaystyle \Pi Y_t = M (t) \cdot S_t$

Proposition 5. (Properties of Dilation) The stationary dilation satisfies:

1. Preservation of moments:

   $\displaystyle \mathbb{E} [|X_t |^p] \leq \mathbb{E} [|Y_t |^p] \quad \forall p \geq 1$

2. Minimal extension: Among all stationary processes that dilate $X_t$, there exists a minimal one (unique up to isomorphism) in terms of the probability space dimension

Corollary 6. For any nonstationary process satisfying the above conditions, the stationary dilation provides a canonical factorization into deterministic time-varying components and stationary stochastic components.