Let $S=\sum_{i=1}^{+\infty}λ_{i}Z_{i}$ where the $Z_{i}$'s are i.d.d. positive with $\mathbb{E}\| Z\| ^{3}<+\infty$ and $(λ_{i})_{i\in\mathbb{N}}$ a positive nonincreasing sequence such that $\sumλ_{i}<+\infty$. We study the small ball probability $\mathbb{P}(S<ε) $ when $ε\downarrow0$. We start from a result by Lifshits (1997) who computed this probability by means of the Laplace transform of $S$. We prove that $\mathbb{P}(S<\cdot) $ belongs to a class of functions introduced by de Haan, well-known in extreme value theory, the class of Gamma-varying functions, for which an exponential-integral representation is available. This approach allows to derive bounds for the rate in nonparametric regression for functional data at a fixed point $x_{0}$ : $\mathbb{E}(y|X=x_{0}%) $ where $(y_{i},X_{i})_{1\leq i\leq n}$ is a sample in $(\mathbb{R},\mathcal{F}) $ and $\mathcal{F}$ is some space of functions. It turns out that, in a general framework, the minimax lower bound for the risk is of order $(\log n)^{-τ}$ for some $τ>0$ depending on the regularity of the data and polynomial rates cannot be achieved.
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.
We prove comparison theorems for small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms. We find the sharp small ball asymptotics for many classical processes under quite general assumptions on the weight.
We prove that the probability that a sum of independent random variables in $\mathbb{R}^d$ with bounded densities lies in a ball is maximized by taking uniform distributions on balls. This in turn generalizes a result by Rogozin on the maximum density of such sums on the line.
This paper establishes small ball probabilities for a class of time-changed processes $X\circ E$, where $X$ is a self-similar process and $E$ is an independent continuous process, each with a certain small ball probability. In particular, examples of the outer process $X$ and the time change $E$ include an iterated fractional Brownian motion and the inverse of a general subordinator with infinite Lévy measure, respectively. The small ball probabilities of such time-changed processes show power law decay, and the rate of decay does not depend on the small deviation order of the outer process $X…
We investigate the Gaussian small ball probabilities with random centers, find their deterministic a.s.-equivalents and establish a relation to infinite-dimensional high-resolution quantization.
While small ball, or lower tail, asymptotic for Gaussian measures generated by solutions of stochastic ordinary differential equations is relatively well understood, a lot less is known in the case of stochastic partial differential equations. The paper presents exact logarithmic asymptotics of the small ball probabilities in a scale of Sobolev spaces when the Gaussian measure is generated by the solution of a diagonalizable stochastic parabolic equation. Compared to the finite-dimensional case, new effects appear in a certain range of the Sobolev exponents.
We study the behavior of the tail probabilities $P(V^2<r)$ as $r\to 0$, where $V^2$ is defined by the following double sum $$ V^2 =π^{-4}\,\sum\limits_{i,j\ge 1} \Big((i+b)\,(j + δ)\Big)^{-2}\,ξ_{ij}^2,$$ where $ \{ξ_{ij}\} $ are independent standard normal random variables, and $b$ and $δ$ are constants: $b>-1,\ δ>-1$.
In the paper the old results on probabilities of small balls for stable measures in a Hilbert space, obtained in 1977 and remaining unpublished, are presented. Apart of historical value these results are interesting even now, since they are comparable with recently obtained ones.
We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.
We study the Small Ball Probabilities (SBPs) of Gaussian rough paths. While many works on rough paths study the Large Deviations Principles (LDPs) for stochastic processes driven by Gaussian rough paths, it is a noticeable gap in the literature that SBPs have not been extended to the rough path framework. LDPs provide macroscopic information about a measure for establishing Integrability type properties. SBPs provide microscopic information and are used to establish a locally accurate approximation for a measure. Given the compactness of a Reproducing Kernel Hilbert space (RKHS) ball, its metr…
We obtain new upper tail probabilities of $m$-times integrated Brownian motions under the uniform norm and the $L^p$ norm. For the uniform norm, Talagrand's approach is used, while for the $L^p$ norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities (large ball probabilities) for general Gaussian random variable in Banach spaces. As applications, explicit bounds are given for the largest eigenvalue of t…
We study concentration properties of random vectors of the form $AX$, where $X = (X_1, ..., X_n)$ has independent coordinates and $A$ is a given matrix. We show that the distribution of $AX$ is well spread in space whenever the distributions of $X_i$ are well spread on the line. Specifically, assume that the probability that $X_i$ falls in any given interval of length $T$ is at most $p$. Then the probability that $AX$ falls in any given ball of radius $T \|A\|_{HS}$ is at most $(Cp)^{0.9 r(A)}$, where $r(A)$ denotes the stable rank of $A$ and $C$ is an absolute constant.
We consider the stochastic heat equation on $[0,\,1]$ with periodic boundary conditions and driven by space-time white noise. Under various natural conditions, we study small ball probabilities for the Hölder semi-norms of the solutions, and provide near optimal bounds on these probabilities. As an application, we prove a support theorem in these Hölder semi-norms.
Asymptotic factorizations for the small-ball probability (SmBP) of a Hilbert valued random element $X$ are rigorously established and discussed. In particular, given the first $d$ principal components (PCs) and as the radius $\varepsilon$ of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joint density of the first $d$ PCs, (b) the volume of the $d$-dimensional ball with radius $\varepsilon$, and (c) a correction factor weighting the use of a truncated version of the process expansion. Moreover, under suitable assumptions on the spectrum of the covariance operator of…
We consider the following stochastic partial differential equation on $t \geq 0, x\in[0,J], J \geq 1$ where we consider $[0,J]$ to be the circle with end points identified: \begin{equation*} \partial_t{\mathbf u}(t,x) =\frac{1}{2}\,\partial_x^2 {\mathbf u}(t,x) + {\mathbf g}(t,x,\mathbf u) + {\mathbf σ}(t,x, {\mathbf u})\dot {\mathbf W}(t,x) , \end{equation*} and $\dot {\mathbf W }(t,x)$ is 2-parameter $d$-dimensional vector valued white noise and ${\mathbf σ}$ is function from ${\mathbb R}_+\times {\mathbb R} \times {\mathbb R}^d \rightarrow {\mathbb R}^d$ to space of symmetric $d\times d$ ma…
Consider a discrete-time martingale $\{X_t\}$ taking values in a Hilbert space $\mathcal H$. We show that if for some $L \geq 1$, the bounds $\mathbb{E} \left[\|X_{t+1}-X_t\|_{\mathcal H}^2 \mid X_t\right]=1$ and $\|X_{t+1}-X_t\|_{\mathcal H} \leq L$ are satisfied for all times $t \geq 0$, then there is a constant $c = c(L)$ such that for $1 \leq R \leq \sqrt{t}$, \[\mathbb{P}(\|X_t\|_{\mathcal H} \leq R \mid X_0 = x_0) \leq c \frac{R}{\sqrt{t}} e^{-\|x_0\|_{\mathcal H}^2/(6 L^2 t)}\,.\] Following [Lee-Peres, Ann. Probab. 2013], this has applications to diffusive estimates for random walks on …
We find the logarithmic $L_2$-small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of "tensor product". The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein--Uhlenbeck process, etc., in the case of special self-similar measure of integration. Our results are based on new theorem on spectral asymptotics for …
In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the Lévy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with respect…
We compare the classical Kolmogorov and quantum probability models. We show that the gap between these model is not so huge as it was commonly believed. The main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the Kolmogorov model. In particular, we obtain ``interference of probabilities'' without to appeal to the Hilbert space formalism. We interpret ``interference of probabilities'' as a perturbation (by a $\cos$-term) of the conventiona…
We significantly advance the research program initiated in "Moment-Based Evidence for Simple Rational-Valued Hilbert-Schmidt Generic 2 x 2 Separability Probabilities" (J. Phys. A, 45, 095305 [2012]). A function P(alpha), incorporating a family of six hypergeometric functions, all with argument 27/64 =(3/4)^3, is obtained here. It reproduces a series, alpha = 1/2, 1, 3/2, 2,...,32$, of sixty-four Hilbert-Schmidt generic 2 x 2 generalized separability probabilities, advanced on the basis of systematic high-accuracy probability-distribution-reconstruction computations, employing 7,501 determinant…
Compelling evidence-though yet no formal proof-has been adduced that the probability that a generic (standard) two-qubit state ($ρ$) is separable/disentangled is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal $4F3$-hypergeometric moment formula (Appendix A), we reach, {\it via} density-approximation procedures, the conclusion that one-half ($\frac{4}{33}$) of this probability arises when the determinantal inequality $|ρ^{PT}|>|ρ|$, where $PT$ denotes the partial transpose, is satisfied, and, the oth…
For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, consider the geodesic $Γ_n$ joining $(0,0)$ and $(n,n)$ for large $n$. It is well known that the transversal fluctuation of $Γ_n$ around the line $x=y$ is $n^{2/3+o(1)}$ with high probability. We obtain the exponent governing the decay of the small ball probability for $Γ_{n}$ and establish that for small $δ$, the probability that $Γ_{n}$ is contained in a strip of width $δn^{2/3}$ around the diagonal is $\exp (-Θ(δ^{-3/2}))$ uniformly in high $n$. We also obtain optimal small deviation estimates for the …
We introduce the Hilbert-Galton board as a variant of the classical Galton board. Balls fall into a row of bins at a rate depending on the bin, and at random times, each bin gets shifted one unit to the right and an empty bin is added to the left. We compute the stationary distribution of this Markov chain and show the existence of an enriched Markov chain on triangular arrays of numbers which projects down to the Hilbert-Galton board. We also define finite-ball projections of the Hilbert-Galton board, for which we compute the stationary distribution, the full spectrum and the grand coupling t…
Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting--abelian or nonabelian groups, or vector spaces, or Banach spaces--we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing th…
Employing Hilbert-Schmidt measure, we explicitly compute and analyze a number of determinantal product (bivariate) moments |rho|^k |rho^{PT}|^n, k,n=0,1,2,3,..., PT denoting partial transpose, for both generic (9-dimensional) two-rebit (alpha = 1/2) and generic (15-dimensional) two-qubit (alpha=1) density matrices rho. The results are, then, incorporated by Dunkl into a general formula (Appendix D6), parameterized by k, n and alpha, with the case alpha=2, presumptively corresponding to generic (27-dimensional) quaternionic systems. Holding the Dyson-index-like parameter alpha fixed, the induce…
We consider diffusion processes in Hilbert spaces with constant non-degenerate diffusion operators and show that, under broad assumptions on the drift, the transition probabilities of the process are positive on ellipsoids associated with the diffusion operator. This is an infinite-dimensional analogue of positivity of densities of transition probabilities. Our results apply to diffusions corresponding to stochastic partial differential equations.
The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.
We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio, yielded, in the course of almost seventeen billion iterations (recorded at intervals of five million), two-qubit estimates repeatedly close to nine decimal places to $\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.073313783$. Howeer, despite the use of over twenty-three billion it…
We provide sharp lower and upper bounds for the Hausdorff dimension of the intersection of a typical random covering set with a fixed analytic set both in Ahlfors regular metric spaces and in the $d$-dimensional torus. In metric spaces, we consider covering sets generated by balls and, in the torus, we deal with general analytic generating sets.
We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker's inequality via the Kullback-Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space.…
We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, existence and uniqueness of a measure valued solution, for the corresponding Fokker--Planck equation. In particular, we verify the Chapman--Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.
Consider a symmetric $α$-stable Lévy process with $α\in (1,2)$. We study shifted small ball probabilities for these processes in the uniform topology, when the shift function is an arbitrary continuous function which starts at 0. We obtain the exact rate of decrease for these probabilities including constants. Using these small ball estimates, we obtain a functional LIL for $α$-stable Lévy process with attracting functions that are continuous. It occurs that the limit set for the family of renormalized $α$-stable Lévy processes is equal to the set of all continuous functions on $[0,1]$ which s…
The Bernoulli sieve is the infinite Karlin "balls-in-boxes" scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals $n$, we prove several functional limit theorems (FLTs) in the Skorohod space $D[0,1]$ endowed with the $J_{1}$- or $M_{1}$-topology for the number $K_{n}^{*}(t)$ of boxes containing at most $[n^{t}]$ balls, $t\in[0,1]$, and the random distribution function $K_{n}^{*}(t)/K_{n}^{*}(1)$, as $n\to\infty$. The limit processes for $K_{n}^{*}(t)$ are of the form $(X(1)-X((1-t)-))_{t\in[0,1]}$, where $X$ is either a Brownian motion, a spec…
Let $E=\limsup\limits_{n\to\infty}(g_n+ξ_n)$ be the random covering set on the torus $\mathbb{T}^d$, where $\{g_n\}$ is a sequence of ball-like sets and $ξ_n$ is a sequence of independent random variables uniformly distributed on $\T^d$. We prove that $E\cap F\neq\emptyset$ almost surely whenever $F\subset\mathbb{T}^d$ is an analytic set with Hausdorff dimension, $\dim_H(F)>d-α$, where $α$ is the almost sure Hausdorff dimension of $E$. Moreover, examples are given to show that the condition on $\dim_H(F)$ cannot be replaced by the packing dimension of $F$.
