A note on small ball probability of a Gaussian process with stationary increments

1993 ◽  
Vol 6 (3) ◽  
pp. 595-602 ◽  
Author(s):  
Qi-Man Shao
Keyword(s):  
Gaussian Process ◽  
Small Ball Probability ◽  
Small Ball ◽  
Stationary Increments

scholarly journals Small ball probability estimates in terms of width

2005 ◽  
Vol 169 (3) ◽  
pp. 305-314 ◽  
Author(s):  
Rafał Latała ◽  
Krzysztof Oleszkiewicz
Keyword(s):  
Small Ball Probability ◽  
Small Ball ◽  
Probability Estimates

An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments

1985 ◽  
Vol 22 (02) ◽  
pp. 454-460 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Asymptotic Formula ◽  
Gaussian Process ◽  
Variance Function ◽  
Stationary Increments ◽  
Distribution Of The Maximum

Let X(t), t≧0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function σ 2(t) is convex and σ 2(t) = o(t) for t → 0, then P(max[o,t] X(s) > u) ~ P(X(t) > u) for u → ∞ and each t > 0.

scholarly journals On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors

2008 ◽  
Vol 2008 ◽  
pp. 1-16 ◽  
Author(s):  
Charles El-Nouty
Keyword(s):  
General Result ◽  
Gaussian Processes ◽  
Brownian Sheet ◽  
Random Vectors ◽  
Fractional Brownian Sheet ◽  
Integral Test ◽  
Small Ball Probability ◽  
Small Ball ◽  
Sufficiency Part ◽  
Necessity Part

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

scholarly journals Zero-free neighborhoods around the unit circle for Kac polynomials

2021 ◽  
Author(s):  
Gerardo Barrera ◽  
Paulo Manrique
Keyword(s):  
Unit Circle ◽  
Random Variables ◽  
Common Denominator ◽  
Small Ball Probability ◽  
Small Ball ◽  
Probability Inequalities ◽  
Independent And Identically Distributed

AbstractIn this paper, we study how the roots of the Kac polynomials $$W_n(z) = \sum _{k=0}^{n-1} \xi _k z^k$$ W n ( z ) = ∑ k = 0 n - 1 ξ k z k concentrate around the unit circle when the coefficients of $$W_n$$ W n are independent and identically distributed nondegenerate real random variables. It is well known that the roots of a Kac polynomial concentrate around the unit circle as $$n\rightarrow \infty $$ n → ∞ if and only if $${\mathbb {E}}[\log ( 1+ |\xi _0|)]<\infty $$ E [ log ( 1 + | ξ 0 | ) ] < ∞ . Under the condition $${\mathbb {E}}[\xi ^2_0]<\infty $$ E [ ξ 0 2 ] < ∞ , we show that there exists an annulus of width $${\text {O}}(n^{-2}(\log n)^{-3})$$ O ( n - 2 ( log n ) - 3 ) around the unit circle which is free of roots with probability $$1-{\text {O}}({(\log n)^{-{1}/{2}}})$$ 1 - O ( ( log n ) - 1 / 2 ) . The proof relies on small ball probability inequalities and the least common denominator used in [17].

scholarly journals Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments

2017 ◽  
Vol 46 (3-4) ◽  
pp. 67-78 ◽  
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Sergiy Shklyar
Keyword(s):  
Brownian Motion ◽  
Maximum Likelihood ◽  
Gaussian Process ◽  
Unknown Parameter ◽  
Strong Consistency ◽  
Likelihood Function ◽  
Likelihood Estimator ◽  
Stationary Increments ◽  
Drift Estimation ◽  
Mixed Fractional Brownian Motion

The paper deals with the regression model X_t = \theta t + B_t , t\in[0, T ],where B=\{B_t, t\geq 0\} is a centered Gaussian process with stationary increments.We study the estimation of the unknown parameter $\theta$ and establish the formula for the likelihood function in terms of a solution to an integral equation.Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.

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