25 Result(s)
within
**n V. Li
English
Probability Theory and Stochastic Processes
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Article
Comparison results for the lower tail of Gaussian seminorms
Let ξ=(ξ n ) be i.i.d.N(0, 1) random variables andq(x), q′(x):R ∞→[0, ∞) be seminorms. We investigate necessary and sufficient conditions that the ratio ofP(q(ξ)<ε) andP(q′(ξ)<ε)...
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Chapter
On the Lower Tail of Gaussian Measures on l p
Throughout this paper \( (\xi_k)_{k=1}^{\infty} \) denotes a sequence of independent, centered, Gaussian random varia...
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Article
Lim inf results for the Wiener process and its increments under theL 2-norm
LetW(t) be a Wiener process. The lim inf behavior of theL 2-norm ofW(t) on the interval [T-a(T), T] and of |W(t+θT)−W(t)| on the interval [αT, βT] is given under suitable conditions.
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Article
Small ball estimates for Brownian motion and the Brownian sheet
Small ball estimates are obtained for Brownian motion and the Brownian sheet when balls are given by certain Hölder norms. As an application of these results we include a functional form of Chung's LIL in this...
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Chapter and Conference Paper
Some Large Deviation Results for Gaussian Measures
Let B denote a real separable Banach space with norm || · || and topological dual B*, and assume X is a centered B-valued Gaussian random vector with μ = L(X). If B is a Hilbert space H, then in [5] we obtained t...
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Article
The Gaussian measure of shifted balls
Let μ be a centered Gaussian measure on a Hilbert spaceH and let \(B_R \subseteq H\) be the centered ball of radiusR>0. Fora∈
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Article
On the future infima of some transient processes
Let (X(t), t∈S) be a real-valued stochastic process with ℙ(X(0)=0)=1 and $$\mathbb{P} (\mathop {\lim }\limits_{t \to \infty } X(t) = \infty ) ...
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Article
Small ball probabilities for Gaussian processes with stationary increments under Hölder norms
Small ball probabilities are estimated for Gaussian processes with stationary increments when the small balls are given by various Hölder norms. As an application we establish results related to Chung's functi...
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Chapter and Conference Paper
Some Shift Inequalities for Gaussian Measures
Let μ be a centered Gaussian measure on a Banach space B and suppose h∈H μ, the generating Hilbert space of ∈. If E is a Borel subset of B, we establish some inequalities between ∈.(E) and ∈(E+h) wh...
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Chapter and Conference Paper
A Central Limit Theorem for the Sock-Sorting Problem
The problem of arranging 2n objects into n pairs in a prescribed way, when the objects are presented one at a time in random order, is considered. Using tools from the theory of empirical processes, we derive a f...
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Article
Small Ball Estimates for Gaussian Processes under Sobolev Type Norms
A sharp small ball estimate under Sobolev type norms is obtained for certain Gaussian processes in general and for fractional Brownian motions in particular. New method using the techniques in large deviation ...
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Article
Small Deviations for Gaussian Markov Processes Under the Sup-Norm
Let {X(t); 0≤t≤1} be a real-valued continuous Gaussian Markov process with mean zero and covariance σ(s, t) = EX(s) X(t) ≠ 0 for 0<s, t<1. It is known that we can write σ(s, t) = G(min(s, t)) H(max(s, t)) with G>...
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Chapter and Conference Paper
A Note on the Gaussian Correlation Conjecture
We show that a special setting of the well-known Gaussian correlation conjecture, namely for sets of equal measure, can be useful in proving the existence of small ball constants. We hope this sheds light on t...
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Article
Capture time of Brownian pursuits
Let B 0,B 1, ⋯ ,B n be independent standard Brownian motions, starting at 0. We investigate the tail of the capture time
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Article
A normal comparison inequality and its applications
Let $\xi=(\xi_i, 1 \leq i \leq n)$ and $\eta= (\eta_i, 1 \leq i \leq n)$ be standard normal random variables with covariance matrices $R^1=(r_{ij}^1)$ and $R^0=(r_{ij}^0)$, respectively. Slepian's lemma says ...
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Article
A Functional LIL and Some Weighted Occupation Measure Results for Fractional Brownian Motion
Weighted occupation measure results are obtained for fractional Brownian motion. Proofs depend on small ball probability estimates of the sup-norm for these processes, which are then used to obtain a functiona...
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Chapter and Conference Paper
Small Deviation Estimates for Some Additive Processes
We study the small deviation probabilities for real valued additive processes. This naturally leads to the small deviation for the corresponding range process. Our general results can be applied to a wide rang...
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Article
Small Deviations of Stable Processes via Metric Entropy
Let X=(X(t)) t∈T be a symmetric α-stable, 0<α<2, process with paths in the dual E * of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some L α ...
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Article
Large and moderate deviations for intersection local times
We study the large and moderate deviations for intersection local times generated by, respectively, independent Brownian local times and independent local times of symmetric random walks. Our result in the Bro...
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Article
Logarithmic Level Comparison for Small Deviation Probabilities
Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L 2 norm, multiple sums motivated by tensor product of ...
