Higher moments of Banach space valued random variables

Janson, Svante; Kaijser, Sten

doi:10.1090/memo/1127

Cited by 16 publications

(58 citation statements)

References 66 publications

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“…Proof Using the independence of the

ξ_{i}, i \in N

, we obtain

\begin{matrix} true \\ left E ([], {(‖‖ i \circ (\sum_{i = 1}^{\infty} λ_{i}^{\frac{1}{2}} ξ_{i} S_{i}) \otimes (\sum_{i^{'} = 1}^{\infty} λ_{i^{'}}^{\frac{1}{2}} ξ_{i^{'}} S_{i^{'}}))}_{L^{1} ({[0, 1]}^{2}; R^{d^{2}})}) \\ left \leq {(\sum_{i = 1}^{\infty} λ_{i}^{\frac{1}{2}} {(E [ξ_{i}^{2}])}^{\frac{1}{2}} {∥ S_{i} ∥}_{L^{1} ([0, 1]; {double-struckR}^{d})})}^{2} \\ left \leq {(\sum_{i = 1}^{d} λ_{i}^{\frac{1}{2}} + \sum_{m = 0}^{\infty} 2^{- \frac{m + 4}{2}} λ_{d 2^{m + 1}}^{\frac{1}{2}})}^{2} < \infty, \end{matrix}

where the second line follows from . Thus the existence and representation of Θ defined in follows from [, Theorem 10.2]. Without further reference, we will use the inequality

P (\sum_{i} (||, ζ_{i}) \geq a) \leq \sum_{i} P ((||, ζ_{i}) \geq b_{i} a)

several times in the proof.…”

Section: Tensor Quadratic Variationmentioning

confidence: 97%

“…Let

ξ_{i}, i \in N

, be independent standard normal random variables and define

Θ : = 2 E [((), \sum_{i = 1}^{\infty}, λ_{i}^{\frac{1}{2}}, ξ_{i}, S_{i}) \otimes ((), \sum_{i^{'} = 1}^{\infty}, λ_{i^{'}}^{\frac{1}{2}}, ξ_{i^{'}}, S_{i^{'}})]

provided that this expression exists in

L^{1} ([0, 1]; R^{d}) {\overset{\otimes}{true}}_{π} L^{1} ([0, 1]; R^{d})

. To ensure compatibility with , , and all integrals with Banach space valued integrands, as for example in the definitions of

{[X]}^{\otimes}

and Θ, are Bochner integrals. Proposition Suppose and , i.e.…”

Section: Tensor Quadratic Variationmentioning

confidence: 99%

“…In other words,

η_{t}, t \geq δ

, is a martingale. For well‐definiteness see again [, Theorem 10.2]. Using , , , and Doob's inequality we obtain for

T \geq δ

and

ε > 0

\begin{matrix} true \\ left P (() \underset{0 \leq t \leq T}{trueprefixsup} {(‖‖ \int_{0}^{t} {(() Y_{s + δ} - Y_{s})}^{\otimes^{2}} d s - δ t Θ)}_{π} \geq δ ε) \\ left \leq P (() \underset{0 \leq t \leq δ}{trueprefixsup} {(‖‖ \int_{0}^{t} {(() Y_{s + δ} - Y_{s})}^{\otimes^{2}} d s - δ t Θ)}_{π} \geq \frac{1}{2} δ ε) \\ left + P (() \underset{δ \leq t \leq T}{trueprefixsup} {(‖‖ \int_{0}^{t} {(() Y_{s + δ} - Y_{s})}^{\otimes^{2}} d s - δ t Θ)}_{π} \geq \frac{1}{2} δ ε) \\ left \leq h (δ; ε) + P (() \underset{δ \leq t \leq T}{trueprefixsup} {(‖‖, δ, η_{t})}_{π} \geq \frac{1}{4} δ ε) + g (δ; ε) \\ left \leq h (δ; ε) + \frac{16}{ε^{2} δ^{2}} E ([], {(‖‖, δ, η_{T})}_{π}^{2}) + g (δ; ε) \\ left \leq \frac{16}{ε^{2} δ^{2}} E ([], (‖‖, ())) \end{matrix}

…”

Section: Tensor Quadratic Variationmentioning

confidence: 99%

“…From Lemma it follows that the right‐hand side converges in the sense of to

\sum_{i = 1}^{k} \int_{0}^{t} \frac{\partial}{\partial x_{i}} f (s; ⟨ S_{1}, X_{s} ⟩, ..., ⟨ S_{k}, X_{s} ⟩) \cdot d ⟨ S_{i}, X_{s} ⟩ as α \to 0 .

We recall that

Θ = 2 E [((), \sum_{i = 1}^{\infty}, λ_{i}^{\frac{1}{2}}, ξ_{i}, S_{i}) \otimes ((), \sum_{i^{'} = 1}^{\infty}, λ_{i^{'}}^{\frac{1}{2}}, ξ_{i^{'}}, S_{i^{'}})]

where

ξ_{i}, i \in N

, are independent standard normal random variables. Using [, Theorem 10.2], together with we obtain from

\begin{matrix} true \\ left \frac{1}{2} \int_{0}^{t}_{{(B {\overset{\otimes}{true}}_{π} B)}^{*}} {〈D^{2} F^{(α)} (s, X_{s}), normalΘ〉}_{{(B {\overset{\otimes}{true}}_{π} B)}^{* *}} d s \\ left = \frac{1}{2} \sum_{l, l^{'} = 1}^{k} \int_{0}^{t} \frac{\partial^{2}}{\partial x_{l} ...} \end{matrix}

…”

Section: Itô's Formulamentioning

confidence: 99%

“…To ensure compatibility with [9], [10], and [21] all integrals with Banach space valued integrands, as for example in the definitions of [X ] ⊗ and , are Bochner integrals.…”

Section: Tensor Quadratic Variationmentioning

confidence: 99%

See 4 more Smart Citations

Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation, quadratic variation, and Itô formula

Karlsson

Löbus

2016

Mathematische Nachrichten

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The paper studies a class of Ornstein–Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron–Martin space. It is shown that the distributions of certain finite dimensional Ornstein–Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein–Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

show abstract

“…Proof Using the independence of the

ξ_{i}, i \in N

, we obtain

\begin{matrix} true \\ left E ([], {(‖‖ i \circ (\sum_{i = 1}^{\infty} λ_{i}^{\frac{1}{2}} ξ_{i} S_{i}) \otimes (\sum_{i^{'} = 1}^{\infty} λ_{i^{'}}^{\frac{1}{2}} ξ_{i^{'}} S_{i^{'}}))}_{L^{1} ({[0, 1]}^{2}; R^{d^{2}})}) \\ left \leq {(\sum_{i = 1}^{\infty} λ_{i}^{\frac{1}{2}} {(E [ξ_{i}^{2}])}^{\frac{1}{2}} {∥ S_{i} ∥}_{L^{1} ([0, 1]; {double-struckR}^{d})})}^{2} \\ left \leq {(\sum_{i = 1}^{d} λ_{i}^{\frac{1}{2}} + \sum_{m = 0}^{\infty} 2^{- \frac{m + 4}{2}} λ_{d 2^{m + 1}}^{\frac{1}{2}})}^{2} < \infty, \end{matrix}

where the second line follows from . Thus the existence and representation of Θ defined in follows from [, Theorem 10.2]. Without further reference, we will use the inequality

P (\sum_{i} (||, ζ_{i}) \geq a) \leq \sum_{i} P ((||, ζ_{i}) \geq b_{i} a)

several times in the proof.…”

Section: Tensor Quadratic Variationmentioning

confidence: 97%

“…Let

ξ_{i}, i \in N

, be independent standard normal random variables and define

Θ : = 2 E [((), \sum_{i = 1}^{\infty}, λ_{i}^{\frac{1}{2}}, ξ_{i}, S_{i}) \otimes ((), \sum_{i^{'} = 1}^{\infty}, λ_{i^{'}}^{\frac{1}{2}}, ξ_{i^{'}}, S_{i^{'}})]

provided that this expression exists in

L^{1} ([0, 1]; R^{d}) {\overset{\otimes}{true}}_{π} L^{1} ([0, 1]; R^{d})

. To ensure compatibility with , , and all integrals with Banach space valued integrands, as for example in the definitions of

{[X]}^{\otimes}

and Θ, are Bochner integrals. Proposition Suppose and , i.e.…”

Section: Tensor Quadratic Variationmentioning

confidence: 99%

“…In other words,

η_{t}, t \geq δ

, is a martingale. For well‐definiteness see again [, Theorem 10.2]. Using , , , and Doob's inequality we obtain for

T \geq δ

and

ε > 0

\begin{matrix} true \\ left P (() \underset{0 \leq t \leq T}{trueprefixsup} {(‖‖ \int_{0}^{t} {(() Y_{s + δ} - Y_{s})}^{\otimes^{2}} d s - δ t Θ)}_{π} \geq δ ε) \\ left \leq P (() \underset{0 \leq t \leq δ}{trueprefixsup} {(‖‖ \int_{0}^{t} {(() Y_{s + δ} - Y_{s})}^{\otimes^{2}} d s - δ t Θ)}_{π} \geq \frac{1}{2} δ ε) \\ left + P (() \underset{δ \leq t \leq T}{trueprefixsup} {(‖‖ \int_{0}^{t} {(() Y_{s + δ} - Y_{s})}^{\otimes^{2}} d s - δ t Θ)}_{π} \geq \frac{1}{2} δ ε) \\ left \leq h (δ; ε) + P (() \underset{δ \leq t \leq T}{trueprefixsup} {(‖‖, δ, η_{t})}_{π} \geq \frac{1}{4} δ ε) + g (δ; ε) \\ left \leq h (δ; ε) + \frac{16}{ε^{2} δ^{2}} E ([], {(‖‖, δ, η_{T})}_{π}^{2}) + g (δ; ε) \\ left \leq \frac{16}{ε^{2} δ^{2}} E ([], (‖‖, ())) \end{matrix}

…”

Section: Tensor Quadratic Variationmentioning

confidence: 99%

“…From Lemma it follows that the right‐hand side converges in the sense of to

\sum_{i = 1}^{k} \int_{0}^{t} \frac{\partial}{\partial x_{i}} f (s; ⟨ S_{1}, X_{s} ⟩, ..., ⟨ S_{k}, X_{s} ⟩) \cdot d ⟨ S_{i}, X_{s} ⟩ as α \to 0 .

We recall that

Θ = 2 E [((), \sum_{i = 1}^{\infty}, λ_{i}^{\frac{1}{2}}, ξ_{i}, S_{i}) \otimes ((), \sum_{i^{'} = 1}^{\infty}, λ_{i^{'}}^{\frac{1}{2}}, ξ_{i^{'}}, S_{i^{'}})]

where

ξ_{i}, i \in N

, are independent standard normal random variables. Using [, Theorem 10.2], together with we obtain from

\begin{matrix} true \\ left \frac{1}{2} \int_{0}^{t}_{{(B {\overset{\otimes}{true}}_{π} B)}^{*}} {〈D^{2} F^{(α)} (s, X_{s}), normalΘ〉}_{{(B {\overset{\otimes}{true}}_{π} B)}^{* *}} d s \\ left = \frac{1}{2} \sum_{l, l^{'} = 1}^{k} \int_{0}^{t} \frac{\partial^{2}}{\partial x_{l} ...} \end{matrix}

…”

Section: Itô's Formulamentioning

confidence: 99%

“…To ensure compatibility with [9], [10], and [21] all integrals with Banach space valued integrands, as for example in the definitions of [X ] ⊗ and , are Bochner integrals.…”

Section: Tensor Quadratic Variationmentioning

confidence: 99%

See 3 more Smart Citations

Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation, quadratic variation, and Itô formula

Karlsson

Löbus

2016

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

Detecting deviations from second-order stationarity in locally stationary functional time series

2019

View full text Add to dashboard Cite

A time-domain test for the assumption of second order stationarity of a functional time series is proposed. The test is based on combining individual cumulative sum tests which are designed to be sensitive to changes in the mean, variance and autocovariance operators, respectively. The combination of their dependent p-values relies on a joint dependent block multiplier bootstrap of the individual test statistics. Conditions under which the proposed combined testing procedure is asymptotically valid under stationarity are provided. A procedure is proposed to automatically choose the block length parameter needed for the construction of the bootstrap. The finitesample behavior of the proposed test is investigated in Monte Carlo experiments and an illustration on a real data set is provided.

show abstract

Detecting relevant differences in the covariance operators of functional time series: a sup-norm approach

Dette

Kokot

2021

Ann Inst Stat Math

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Higher moments of Banach space valued random variables

Cited by 16 publications

References 66 publications

Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation, quadratic variation, and Itô formula

Infinite dimensional Ornstein–Uhlenbeck processes with unbounded diffusion – Approximation, quadratic variation, and Itô formula

Detecting deviations from second-order stationarity in locally stationary functional time series

Detecting relevant differences in the covariance operators of functional time series: a sup-norm approach

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