Small Ball Probabilities Research Articles

Small ball probability estimates for log-concave measures

We establish a small ball probability inequality for isotropic log \log -concave probability measures: there exist absolute constants c 1 , c 2 > 0 c_{1}, c_{2}>0 such that if X X is an isotropic log \log -concave random vector in R n {\mathbb R}^n with ψ 2 \psi _{2} constant and bounded by b b and if A A is a non-zero n × n n\times n matrix, then for every ε ∈ ( 0 , c 1 ) \varepsilon \in (0,c_{1}) and y ∈ R n y \in \mathbb R^n , \[ P ( ‖ A x − y ‖ 2 ≤ ε ‖ A ‖ HS ) ≤ ε ( c 2 b ‖ A ‖ HS ‖ A ‖ op ) 2 , \mathbb P \left ( \|Ax-y\|_{2} \leq \varepsilon \|A\|_{\textrm {HS}} \right ) \leq \varepsilon ^{\big ( \frac {c_{2}}{b}\frac {\|A\|_{\textrm {HS}}}{\|A\|_{\textrm {op}}} \big )^{2} }, \] where c 1 , c 2 > 0 c_{1}, c_{2}>0 are absolute constants.

Kernel regression with functional response

We consider kernel regression estimate when both the response variable and the explanatory one are functional. The rates of uniform almost complete convergence are stated as function of the small ball probability of the predictor and as function of the entropy of the set on which uniformity is obtained.

Strong uniform consistency of a nonparametric estimator of a conditional quantile for censored dependent data and functional regressors

Let (T, C, X) be a vector of random variables (rvs) where T, C and X are the interest variable, a right censoring rv and a covariate, respectively. In this paper, we study the kernel conditional quantile estimation in the dependent case and when the covariable takes values in an infinite-dimension space. An estimator of the conditional quantile is given and, under some regularity conditions, among which the small-ball probability for the covariate, its uniform strong convergence with rates is established.

Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

For any 0 n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.

Functional Chung laws for small increments of the empirical process and a lower bound in the strong invariance principle

Consider the set Θ n of all a n -sized increment processes of the uniform empirical process α n on [0, 1]. We assume that a n ↓ 0, na n ↑ ∞, d n = na n (log n)−1 → ∞ and na n (log n)−7/3 = O(1). In Berthet (1996, 2005) the fourth assumption was shown to be critical with respect to the pointwise rates of convergence in the functional law of Deheuvels and Mason (1992) for Θ n because strong approximation methods become ineffective at such a small scale a n . We are now able to study directly these small empirical increments and compute the exact rate of clustering of Θ n to any Strassen function having Lebesgue derivative of bounded variation by making use of a sharp small deviation estimate for a Poisson process of high intensity due to Shmileva (2003a). It turns out that the best rates are of order d 1/4 (log n)−1 and are faster than in the Brownian case whereas the slowest rates are of order d −1/2 and correspond to the apparently crude ones obtained in Berthet (2005) by means of Gaussian small ball probabilities. These different sharp properties of the empirical and Brownian paths imply an almost sure lower bound in the strong invariance principle and provide a new insight into the famous KMT approximation of α n .

Local linear regression for functional data

We study a non-linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method and derive its asymptotic mean square error. Computations involve a linear inverse problem as well as a representation of the small ball probability of the data and are based on recent advances in this area.

A functional LIL for integrated α stable process

Let {X(t), t ∈ ℝ+} be an integrated α stable process. In this paper, a functional law of the iterated logarithm (LIL) is derived via estimating the small ball probability of X. As a corollary, the classical Chung LIL of X is obtained. Furthermore, some results about the weighted occupation measure of X(t) are established.

On A Set Of Transformations of Gaussian Random Functions

Let $B(t)$, $0\leq t\leq 1$, be a standard Brownian bridge. P. Deheuvels [Statist. Probab. Lett., 77 (2007), pp. 1190–1200] introduced a set of Gaussian processes ${\cal Y}_K(t)=B(t)-6Kt(1-t)\int_0^1B(s)\,ds$, $t\in [0,1]$, $K\in{\bf R},$ and showed that the distributional equality ${\cal Y}_K(t)\stackrel {d}{=}{\cal Y}_{2-K}(t)$ holds true. Moreover, he obtained the explicit Karhunen–Loéve (KL) expansion for the process ${\cal Y}_1(t)$. We introduce one-parameter sets of transformations for zero mean-value Gaussian random functions. These transformations satisfy a similar distributional equality. If the $L_2$-norm of original function is finite a.s., we derive the explicit relation between exact asymptotics of $L_2$-small ball probabilities for the transformed function and for the original one. For one-dimensional processes generating boundary value problems to ordinary differential equations we also obtain the KL-expansion for a transformed process provided the KL-expansion for original process is known.

Laws of the iterated logarithm for a class of iterated processes

Let X = { X ( t ) , t ≥ 0 } be a Brownian motion or a spectrally negative stable process of index 1 < α < 2 . Let E = { E ( t ) , t ≥ 0 } be the hitting time of a stable subordinator of index 0 < β < 1 independent of X . We use a connection between X ( E ( t ) ) and the stable subordinator of index β / α to derive information on the path behavior of X ( E ( t ) ) . This is an extension of the connection of iterated Brownian motion and ( 1 4 )-stable subordinator due to Bertoin [Bertoin, J., 1996a. Iterated Brownian motion and stable ( 1 4 ) subordinator. Statist. Probab. Lett., 27, 111–114; Bertoin, J., 1996b, Lévy Processes. Cambridge University Press]. Using this connection, we obtain various laws of the iterated logarithm for X ( E ( t ) ) . In particular, we establish the law of the iterated logarithm for local time Brownian motion, X ( L ( t ) ) , where X is a Brownian motion (the case α = 2 ) and L ( t ) is the local time at zero of a stable process Y of index 1 < γ ≤ 2 independent of X . In this case E ( ρ t ) = L ( t ) with β = 1 − 1 / γ for some constant ρ > 0 . This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper [Meerschaert, M.M., Nane, E., Xiao, Y.. 2008. Large deviations for local time fractional Brownian motion and applications. J. Math. Anal. Appl. 346, 432–445]. We also obtain exact small ball probability for X ( E ( t ) ) using ideas from Aurzada and Lifshits [Aurzada, F., Lifshits, M., On the small deviation problem for some iterated processes. preprint: arXiv:0806.2559].

CHUNG-TYPE LAW OF THE ITERATED LOGARITHM OF l∞-VALUED GAUSSIAN PROCESSES

In this paper, by estimating small ball probabilities of <TEX>$l^{\infty}$</TEX>-valued Gaussian processes, we investigate Chung-type law of the iterated logarithm of <TEX>$l^{\infty}$</TEX>-valued Gaussian processes. As an application, the Chung-type law of the iterated logarithm of <TEX>$l^{\infty}$</TEX>-valued fractional Brownian motion is established.

Log-level comparison principle for small ball probabilities

We prove a new variant of comparison principle for logarithmic L 2 -small ball probabilities of Gaussian processes. As an application, we obtain logarithmic small ball asymptotics for some well-known processes with smooth covariances.

Rates of contraction of posterior distributions based on Gaussian process priors

We derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on Gaussian processes. The rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process and the small ball probabilities of the Gaussian process. We determine these quantities for a range of examples of Gaussian priors and in several statistical settings. For instance, we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a (fractionally integrated) Brownian motion. We also consider regression with Gaussian errors and smooth classification under a logistic or probit link function combined with various priors.

Entropy Estimate for $k$-Monotone Functions via Small Ball Probability of Integrated Brownian Motions

Metric entropy of the class of probability distribution functions on $[0,1]$ with a $k$-monotone density is studied through its connection with the small ball probability of $k$-times integrated Brownian motions.

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

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.

Small ball probabilities for Gaussian random fields and tensor products of compact operators

We find the logarithmic L 2 -small ball asymptotics for a large class of zero mean Gaussian fields with covariances having the structure of product. The main condition imposed on marginal covariances is the regular behavior of their eigenvalues at infinity that is valid for a multitude of Gaussian random functions including the fractional Brownian sheet, Ornstein - Uhlenbeck sheet, etc. So we get the far-reaching generalizations of well-known results by Csaki (1982) and by Li (1992). Another class of Gaussian fields considered is the class of additive fields studied under the supremum-norm by Chen and Li (2003). Our theorems are based on new results on spectral asymptotics for the tensor products of compact self-adjoint operators in Hilbert space which are of independent interest.

Small ball probabilities for stable convolutions

We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function f: ]0, +∞[→ R with a real SαS Levy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of Lifshits and Simon, Ann. Inst. H. Poincare Probab. Statist. 41 (2005) 725-752 where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as Lifshits and Simon, Ann. Inst. H. Poincare Probab. Statist. 41 (2005) 725-752 is obtained with respect to the norms, thanks to a weak decorrelation inequality due to Li, Elec. Comm. Probab. 4 (1999) 111-118. In the more difficult non-Gaussian case, we use a different method relying on comparison of entropy numbers and restrict ourselves to Holder and L p -norms.

Small ball probability and Dvoretzky’s Theorem

Large deviation estimates are by now a standard tool in Asymptotic Convex Geometry, contrary to small deviation results. In this note we present a novel application of a small deviations inequality to a problem that is related to the diameters of random sections of high dimensional convex bodies. Our results imply an unexpected distinction between the lower and upper inclusions in Dvoretzky’s Theorem.

Posterior consistency of Gaussian process prior for nonparametric binary regression

Consider binary observations whose response probability is an unknown smooth function of a set of covariates. Suppose that a prior on the response probability function is induced by a Gaussian process mapped to the unit interval through a link function. In this paper we study consistency of the resulting posterior distribution. If the covariance kernel has derivatives up to a desired order and the bandwidth parameter of the kernel is allowed to take arbitrarily small values, we show that the posterior distribution is consistent in the L1-distance. As an auxiliary result to our proofs, we show that, under certain conditions, a Gaussian process assigns positive probabilities to the uniform neighborhoods of a continuous function. This result may be of independent interest in the literature for small ball probabilities of Gaussian processes.

Chung LIL for integrated [formula omitted] stable process

Let { X ( t ) , t ∈ R + } be an integrated α stable process. With the help of the distributional relationship between a stable process and a Gaussian process, we study the small deviations of X ( t ) . Using a small ball probability estimate, we obtain a Chung law of the iterated logarithm (LIL) for this process.

A Functional LIL for m-Fold Integrated Brownian Motion*

Let {X m (t); t ∈ R +} be an m-Fold integrated Brownian motion. In this paper, with the help of small ball probability estimate, a functional law of the iterated logarithm (LIL) for X m (t) is established. This extends the classic Chung type liminf result for this process. Furthermore, a result about the weighted occupation measure for X m (t) is also obtained.