A Gaussian bound for convolutions of functions on locally compact groups

Nick Dungey

Displaying similar documents to “A Gaussian bound for convolutions of functions on locally compact groups”

Two-parameter non-commutative Central Limit Theorem

Natasha Blitvić (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In 1992, Speicher showed the fundamental fact that the probability measures playing the role of the classical Gaussian in the various non-commutative probability theories (viz. fermionic probability, Voiculescu’s free probability, and $q$ -deformed probability of Bożejko and Speicher) all arise as the limits in a generalized Central Limit Theorem. The latter concerns sequences of non-commutative random variables (elements of a $*$ -algebra equipped with a state) drawn from an ensemble of pair-wise...

Note on the multidimensional Gebelein inequality

Marek Beśka, Agnieszka Wałachowska (2013)

Applicationes Mathematicae

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We generalize the Gebelein inequality for Gaussian random vectors in $ℝ^{d}$ .

Small ball probability estimates in terms of width

Rafał Latała, Krzysztof Oleszkiewicz (2005)

Studia Mathematica

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A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body K ⊆ ℝⁿ with inradius w and γₙ(K) ≤ 1/2 we have $γ ₙ (s K) \leq {(2 s)}^{w ² / 4} γ ₙ (K)$ for any s ∈ [0,1], where γₙ is the standard Gaussian probability measure. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.

Upper bounds for the density of solutions to stochastic differential equations driven by fractional brownian motions

Fabrice Baudoin, Cheng Ouyang, Samy Tindel (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H g t; 1 / 3$ . We show that under some geometric conditions, in the regular case $H g t; 1 / 2$ , the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $H g t; 1 / 3$ and under the same geometric conditions, we show that the density of the solution is smooth and...

Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails

Rafał M. Łochowski (2006)

Banach Center Publications

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Two kinds of estimates are presented for tails and moments of random multidimensional chaoses $S = \sum a_{i ₁, . . ., i_{d}} X_{i ₁}^{(1)} \dots X_{i_{d}}^{(d)}$ generated by symmetric random variables $X_{i ₁}^{(1)}, . . ., X_{i_{d}}^{(d)}$ with logarithmically concave tails. The estimates of the first kind are generalizations of bounds obtained by Arcones and Giné for Gaussian chaoses. They are exact up to constants depending only on the order d. Unfortunately, suprema of empirical processes are involved. The second kind estimates are based on comparison between moments of S and moments...

On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables

Alexander R. Pruss (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let Ω be a countable infinite product $Ω ₁^{ℕ}$ of copies of the same probability space Ω₁, and let Ξₙ be the sequence of the coordinate projection functions from Ω to Ω₁. Let Ψ be a possibly nonmeasurable function from Ω₁ to ℝ, and let Xₙ(ω) = Ψ(Ξₙ(ω)). Then we can think of Xₙ as a sequence of independent but possibly nonmeasurable random variables on Ω. Let Sₙ = X₁ + ⋯ + Xₙ. By the ordinary Strong Law of Large Numbers, we almost surely have $E_{*} [X ₁] \leq l i m i n f S ₙ / n \leq l i m s u p S ₙ / n \leq E * [X ₁]$ , where $E_{*}$ and E* are the lower and upper expectations....

Zeros of random functions in Bergman spaces

Joel H. Shapiro (1979)

Annales de l'institut Fourier

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Suppose $μ$ is a finite positive rotation invariant Borel measure on the open unit disc $Δ$ , and that the unit circle lies in the closed support of $μ$ . For $0 < p < \infty$ the $A_{μ}^{p}$ is the collection of functions in $L^{p} (μ)$ holomorphic on $Δ$ . We show that whenever a Gaussian power series $f (z) = Σ ζ_{n} a_{n} z^{n}$ almost surely lies in $A_{μ}^{p}$ but not in $⋃_{q > p} A_{μ}^{p}$ , then almost surely: a) the zero set $Z (f)$ of $f$ is not contained in any $A_{μ}^{q}$ zero set ( $q > p$ , and b) $Z (f + 1) \cup Z (f - 1)$ is not contained in any $A_{μ}^{q}$ zero set.

Some results on $A_{p} (G)$ submodules of $P M_{p} (G)$

Edmond E. Granirer (1987)

Colloquium Mathematicae

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On the powers of Voiculescu's circular element

Ferenc Oravecz (2001)

Studia Mathematica

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The main result of the paper is that for a circular element c in a C*-probability space, $(c ⁿ, c^{n *})$ is an R-diagonal pair in the sense of Nica and Speicher for every n = 1,2,... The coefficients of the R-series are found to be the generalized Catalan numbers of parameter n-1.

Regularity of Gaussian white noise on the d-dimensional torus

Mark C. Veraar (2011)

Banach Center Publications

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In this paper we prove that a Gaussian white noise on the d-dimensional torus has paths in the Besov spaces $B_{p, \infty}^{- d / 2} (^{d})$ with p ∈ [1,∞). This result is shown to be optimal in several ways. We also show that Gaussian white noise on the d-dimensional torus has paths in the Fourier-Besov space $b ̂_{p, \infty}^{- d / p} (^{d})$ . This is shown to be optimal as well.

Universality for random tensors

Razvan Gurau (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We prove two universality results for random tensors of arbitrary rank $D$ . We first prove that a random tensor whose entries are $N^{D}$ independent, identically distributed, complex random variables converges in distribution in the large $N$ limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability...

Note on the variance of the sum of Gaussian functionals

Marek Beśka (2010)

Applicationes Mathematicae

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Let $(X_{i}, i = 1, 2, . . .)$ be a Gaussian sequence with $X_{i} \in N (0, 1)$ for each i and suppose its correlation matrix $R = {(ρ_{i j})}_{i, j \geq 1}$ is the matrix of some linear operator R:l₂→ l₂. Then for $f_{i} \in L ² (μ)$ , i=1,2,..., where μ is the standard normal distribution, we estimate the variation of the sum of the Gaussian functionals $f_{i} (X_{i})$ , i=1,2,... .

Universality of the asymptotics of the one-sided exit problem for integrated processes

Frank Aurzada, Steffen Dereich (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We consider the one-sided exit problem – also called one-sided barrier problem – for ( $α$ -fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $α \mapsto θ (α)$ such that the probability that any $α$ -fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{- θ (α) + o (1)}$ for large $T$ . We also investigate when the fixed level can be replaced by a different...

Sub-Laplacian with drift in nilpotent Lie groups

Camillo Melzi (2003)

Colloquium Mathematicae

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We consider the heat kernel $ϕ_{t}$ corresponding to the left invariant sub-Laplacian with drift term in the first commutator of the Lie algebra, on a nilpotent Lie group. We improve the results obtained by G. Alexopoulos in [1], [2] proving the “exact Gaussian factor” exp(-|g|²/4(1+ε)t) in the large time upper Gaussian estimate for $ϕ_{t}$ . We also obtain a large time lower Gaussian estimate for $ϕ_{t}$ .

Fermat test with Gaussian base and Gaussian pseudoprimes

José María Grau, Antonio M. Oller-Marcén, Manuel Rodríguez, Daniel Sadornil (2015)

Czechoslovak Mathematical Journal

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The structure of the group ${(ℤ / n ℤ)}^{☆}$ and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Euler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group $𝒢_{n} : = {a + b i \in ℤ [i] / n ℤ [i] : a^{2} + b^{2} \equiv 1 (mod n)}$ . In particular, we characterize Gaussian Carmichael...

Asymptotic rate of convergence in the degenerate U-statistics of second order

Olga Yanushkevichiene (2010)

Banach Center Publications

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Let X,X₁,...,Xₙ be independent identically distributed random variables taking values in a measurable space (Θ,ℜ ). Let h(x,y) and g(x) be real valued measurable functions of the arguments x,y ∈ Θ and let h(x,y) be symmetric. We consider U-statistics of the type $T (X ₁, . . ., X ₙ) = n^{- 1} \sum_{1 \leq i L e t q_{i} (i \geq 1) b e e i g e n v a l u e s o f t h e H i l b e r t - S c h m i d t o p e r a t o r a s s o c i a t e d w i t h t h e k e r n e l h (x, y), a n d q ₁ b e t h e l a r g e s t i n a b s o l u t e v a l u e o n e . W e p r o v e t h a t}$ Δn = ρ(T(X₁,...,Xₙ),T(G₁,..., Gₙ)) ≤ (cβ’1/6)/(√(|q₁|) n1/12) $,$ where $G_{i}$ , 1 ≤ i ≤ n, are i.i.d. Gaussian random vectors, ρ is the Kolmogorov (or uniform) distance and $β^{'} : = E | h (X, X ₁) | ³ + {E | h (X, X ₁) |}^{18 / 5} + E | g (X) | ³ + {E | g (X) |}^{18 / 5} + 1 < \infty$ .

The Infinite Random Order of Dimension $k$

Peter Winkler (1985)

Publications du Département de mathématiques (Lyon)

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Geometrically strictly semistable laws as the limit laws

Marek T. Malinowski (2007)

Discussiones Mathematicae Probability and Statistics

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A random variable X is geometrically infinitely divisible iff for every p ∈ (0,1) there exists random variable $X_{p}$ such that $X \overset{d}{=} \sum_{k = 1}^{T (p)} X_{p, k}$ , where $X_{p, k}$ ’s are i.i.d. copies of $X_{p}$ , and random variable T(p) independent of $X_{p, 1}, X_{p, 2}, . . .$ has geometric distribution with the parameter p. In the paper we give some new characterization of geometrically infinitely divisible distribution. The main results concern geometrically strictly semistable distributions which form a subset of geometrically infinitely divisible distributions....

The absolute continuity of the invariant measure of random iterated function systems with overlaps

Balázs Bárány, Tomas Persson (2010)

Fundamenta Mathematicae

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We consider iterated function systems on the interval with random perturbation. Let $Y_{ε}$ be uniformly distributed in [1-ε,1+ ε] and let $f_{i} \in C^{1 + α}$ be contractions with fixpoints $a_{i}$ . We consider the iterated function system $Y_{ε} f_{i} + a_{i} (1 - Y_{ε}) ⁿ_{i = 1}$ , where each of the maps is chosen with probability $p_{i}$ . It is shown that the invariant density is in L² and its L² norm does not grow faster than 1/√ε as ε vanishes. The proof relies on defining a piecewise hyperbolic dynamical system on the cube with an SRB-measure whose projection...

Tail and moment estimates for sums of independent random variables with logarithmically concave tails

E. Gluskin, S. Kwapień (1995)

Studia Mathematica

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For random variables $S = \sum_{i = 1}^{\infty} α_{i} ξ_{i}$ , where $(ξ_{i})$ is a sequence of symmetric, independent, identically distributed random variables such that $l n P (| ξ_{i} | \geq t)$ is a concave function we give estimates from above and from below for the tail and moments of S. The estimates are exact up to a constant depending only on the distribution of ξ. They extend results of S. J. Montgomery-Smith [MS], M. Ledoux and M. Talagrand [LT, Chapter 4.1] and P. Hitczenko [H] for the Rademacher sequence.

Volume thresholds for Gaussian and spherical random polytopes and their duals

Peter Pivovarov (2007)

Studia Mathematica

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Let g be a Gaussian random vector in ℝⁿ. Let N = N(n) be a positive integer and let $K_{N}$ be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes $V_{N} : = v o l (K_{N} \cap R B ₂ ⁿ) / v o l (R B ₂ ⁿ)$ . For a large range of R = R(n), we establish a sharp threshold for N, above which $V_{N} \to 1$ as n → ∞, and below which $V_{N} \to 0$ as n → ∞. We also consider the case when $K_{N}$ is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both...

Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

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We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy $E$ in the localized phase. Assume the density of states function is not too flat near $E$ . Restrict it to some large cube $Λ$ . Consider now $I_{Λ}$ , a small energy interval centered at $E$ that asymptotically contains infintely many eigenvalues when the volume of the cube $Λ$ grows to infinity. We prove that, with probability one in the...

Gebelein's inequality and its consequences

M. Beśka, Z. Ciesielski (2006)

Banach Center Publications

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Let $(X_{i}, i = 1, 2, . . .)$ be the normalized gaussian system such that $X_{i} \in N (0, 1)$ , i = 1,2,... and let the correlation matrix $ρ_{i j} = E (X_{i} X_{j})$ satisfy the following hypothesis: $C = s u p_{i \geq 1} \sum_{j = 1}^{\infty} | ρ_{i, j} | < \infty$ . We present Gebelein’s inequality and some of its consequences: Borel-Cantelli type lemma, iterated log law, Levy’s norm for the gaussian sequence etc. The main result is that (f(X₁) + ⋯ + f(Xₙ))/n → 0 a.s. for f ∈ L¹(ν) with (f,1)ν = 0.

Random ε-nets and embeddings in $ℓ_{\infty}^{N}$

Y. Gordon, A. E. Litvak, A. Pajor, N. Tomczak-Jaegermann (2007)

Studia Mathematica

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We show that, given an n-dimensional normed space X, a sequence of $N = {(8 / ε)}^{2 n}$ independent random vectors ${(X_{i})}_{i = 1}^{N}$ , uniformly distributed in the unit ball of X*, with high probability forms an ε-net for this unit ball. Thus the random linear map $Γ : ℝ \to ℝ^{N}$ defined by $Γ x = {(⟨ x, X_{i} ⟩)}_{i = 1}^{N}$ embeds X in $ℓ_{\infty}^{N}$ with at most 1 + ε norm distortion. In the case X = ℓ₂ⁿ we obtain a random 1+ε-embedding into $ℓ_{\infty}^{N}$ with asymptotically best possible relation between N, n, and ε.