Let $T$ be a tree with induced partial order $\preceq$. We investigate centered Gaussian processes $X=(X_t)_{t\in T}$ represented as $$
X_t=σ(t)\sum_{v \preceq t}α(v)ξ_v $$ for given weight functions $α$ and $σ$ on $T$ and with $(ξ_v)_{v\in T}$ i.i.d. standard normal. In a first part we treat general trees and weights and derive necessary and sufficient conditions for the a.s. boundedness of $X$ in terms of compactness properties of $(T,d)$. Here $d$ is a special metric defined via $α$ and $σ$, which, in general, is not comparable with the Dudley metric generated by $X$. In a second part we investigate the boundedness of $X$ for the binary tree and for homogeneous weights. Assuming some mild regularity assumptions about $α$ we completely characterize weights $α$ and $σ$ with $X$ being a.s. bounded.
We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.
We continue the research of Latała on improving estimates of $p$-th moments of sums of independent random variables. We generalize some of his results in the case when $2 \leq p \leq 4$ and present a combinatorial approach for even moments.
We study the persistence probability for some two-sided discrete-time Gaussian sequences that are discrete-time analogs of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous-time in [Molchan, Commun. Math. Phys., 1999] and [Molchen, J. Stat. Phys., 2017] to a wide class of discrete-time processes.
We consider the small deviation probabilities (SDP) for sums of stationary Gaussian sequences. For the cases of constant boundaries and boundaries tending to zero, we obtain quite general results. For the case of the boundaries tending to infinity, we focus our attention on the discrete analogs of the fractional Brownian motion (FBM). It turns out that the lower bounds for the SDP can be transferred from the well studied FBM caseto the discrete time setting under the usual assumptions that imply weak convergence while the transfer of the corresponding upper bounds necessarily requires a deeper…
We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums.
Let $\mathcal{T}$ be a rooted tree endowed with the natural partial order $\preceq$. Let $(Z(v))_{v\in \mathcal{T}}$ be a sequence of independent standard Gaussian random variables and let $α= (α_k)_{k=1}^\infty$ be a sequence of real numbers with $\sum_{k=1}^\infty α_k^2<\infty$. Set $α_0 =0$ and define a Gaussian process on $\mathcal{T}$ in the following way: \[ G(\mathcal{T}, α; v): = \sum_{u\preceq v} α_{|u|} Z(u), \quad v \in \mathcal{T}, \] where $|u|$ denotes the graph distance between the vertex $u$ and the root vertex. Under mild assumptions on $\mathcal{T}$, we obtain a necessary and…
The long-standing Gaussian product inequality (GPI) conjecture states that $E [\prod_{j=1}^{n}X_j^{2m_j}]\geq\prod_{j=1}^{n}E[X_j^{2m_j}]$ for any centered Gaussian random vector $(X_1,\dots,X_n)$ and $m_1,\dots,m_n\in\mathbb{N}$. In this paper, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of this novel method, we apply it to prove two new GPIs: $E[X_1^{2m_1}X_2^{6}X_3^{4}]\ge E[X_1^{2m_1}]E[X_2^{6}]E[X_3^{4}]$ and $E[X_1^{2m_1}X_2^{2}X_3^{2}X_4^{2}]\ge E[X_1^{2m_…
We consider the Gaussian approximation for functionals of a Poisson process that are expressible as sums of region-stabilizing (determined by the points of the process within some specified regions) score functions and provide a bound on the rate of convergence in the Wasserstein and the Kolmogorov distances. While such results have previously been shown in Lachièze-Rey, Schulte and Yukich (2019), we extend the applicability by relaxing some conditions assumed there and provide further insight into the results. This is achieved by working with stabilization regions that may differ from balls o…
Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain ``clumps'' around high-level crossings. We also discuss how these results are related t…
For a martingale $(X_n)$ converging almost surely to a random variable $X$, the sequence $(X_n - X)$ is called martingale tail sum. Recently, Neininger [Random Structures Algorithms, 46 (2015), 346-361] proved a central limit theorem for the martingale tail sum of R{é}gnier's martingale for the path length in random binary search trees. Gr{ü}bel and Kabluchko [to appear in Annals of Applied Probability, (2016), arXiv 1410.0469] gave an alternative proof also conjecturing a corresponding law of the iterated logarithm. We prove the central limit theorem with convergence of higher moments and the…
We give a necessary and sufficient condition for symmetric infinitely divisible distribution to have Gaussian component. The result can be applied to approximation the distribution of finite sums of random variables. Particularly, it shows that for a large class of distributions with finite variance stable approximation appears to be better than Gaussian. keywords: infinitely divisible distributions; Gaussian component; approximations of sums of random variables.
We apply Lindeberg's method, invented to prove a central limit theorem, to analyze the moderate deviations around such a central limit theorem. In particular, we will show moderate deviation principles for martingales as well as for random sums, in the latter situation both, in the case when the limit distribution is Gaussian or non-Gaussian. Moreover in the Gaussian case we show moderate deviations for random sums using bounds on cumulants, alternatively. Finally, we also prove a large deviation principle for certain random sums.
With $\{ξ_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with non-negative correlation function $ρ(i):=\mathbb{E}[ ξ_0ξ_i]$ and $\{σ(i)\}_{i\ge 1}$ a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum $\sum_{i=1}^\ell σ(i) ξ_i$, $\ell\ge 1$. For summable correlations $ρ$, we show that the persistence exponent is universal. On the contrary, for non-summable $ρ$, even for polynomial weight functions $σ(i)\sim i^p$ the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter $H$) and on the poly…
Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent identically distributed Rademacher random variables, that is $\mathbb{P}\{\varepsilon_i=\pm1\}=1/2$. Let $S_n=a_1\varepsilon_1+\cdots+a_n\varepsilon_n$, where $\mathbf{a}=(a_1,\ldots,a_n)\in\mathbb{R}^n$ is a vector such that ${a_1^2+\cdots+a_n^2\leq1}$. We find the smallest possible constant $c$ in the inequality \[\mathbb{P}\{S_n\geq x\}\leq c\mathbb{P}\{η\geq x\}\qquad for all x\in \mathbb{R},\] where $η\sim N(0,1)$ is a standard normal random variable. This optimal value is equal to \[c_*=\bigl(4\mathbb{P}\{η\geq\sqrt{2}\}\bigr)^ {-1}…
For fixed $t\ge 2$, we consider the class of representations of $1$ as sum of unit fractions whose denominators are powers of $t$ or equivalently the class of canonical compact $t$-ary Huffman codes or equivalently rooted $t$-ary plane "canonical" trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution).
We study the asymptotic behavior of the $ν$-symmetric Riemman sums for functionals of a self-similar centered Gaussian process $X$ with increment exponent $0<α<1$. We prove that, under mild assumptions on the covariance of $X$, the law of the weak $ν$-symmetric Riemman sums converge in the Skorohod topology when $α=(2\ell+1)^{-1}$, where $\ell$ denotes the smallest positive integer satisfying $\int_{0}^{1}x^{2j}ν(dx)=(2j+1)^{-1}$ for all $j=0,\dots, \ell-1$. In the case $α>(2\ell+1)^{-1}$, we prove that the convergence holds in probability.
This note investigates invariance principles for sums of N(nt) iid radom variables, where n is an integer, t is a positive real number and N(u) is a stochastic process with nonnegative integer values. We show that the sequence of sums of these random variables denoted S(n,t), when appropriately centered and normalized, weakly converges to a Gaussian process. We give sufficient conditions depending on the expectation of N(nt) which allows to rescale S(n,t) into a stochastic S(n,a(t)) weakly converging to a Brownian motion.
We prove a dimension-free tail comparison between the Euclidean norms of sums of independent random vectors uniformly distributed in centred Euclidean spheres and properly rescaled standard Gaussian random vectors.
It is well-known that the central limit theorem holds for partial sums of a stationary sequence $(X_i)$ of $m$-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if $\mathrm{Var}(X_i)\neq0$. We show that this happens only in the case when $X_i-\mathbb E X_i=Y_i-Y_{i-1}$ for an $(m-1)$-dependent stationary sequence $(Y_i)$ with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to degenerate. Two applications to subtre…
We consider a Bernoulli bond percolation on a random recursive tree of size $n\gg 1$, with supercritical parameter $p_n=1-c/\ln n$ for some $c>0$ fixed. It is known that with high probability, there exists then a unique giant cluster of size $G_n\sim \e^{-c}$, and it follows from a recent result of Schweinsberg \cite{Sch} that $G_n$ has non-gaussian fluctuations. We provide an explanation of this by analyzing the effect of percolation on different phases of the growth of recursive trees. This alternative approach may be useful for studying percolation on other classes of trees, such as for ins…
We construct a tree-based dependence structure for the representation of binomial, Poisson and Gaussian random vectors having a given covariance matrix, using sums of independent random variables. This construction allows us to characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. Our method relies on the representation of dependent components using binary trees on the discrete $d$-dimensional hypercube $C_d$, and on Möbius inversion techniques. In the case of Poisson random vectors this approach involves Lévy measures on $C_…
For the Gaussian free field on a $(d + 1)$-regular tree with $d \geq 2$, we study the percolative properties of its level sets in the critical and the near-critical regime. In particular, we show the continuity of the percolation probability, derive an exact symptotic tail estimate for the cardinality of the connected component of the critical level set, and describe the asymptotic behaviour of the percolation probability in the near-critical regime.
For the renormalised sums of the random $\pm 1$-colouring of the connected components of $\mathbb Z$ generated by the coalescing renewal processes in the "power law Pólya's urn" of Hammond and Sheffield we prove functional convergence towards fractional Brownian motion, closing a gap in the tightness argument of their paper. In addition, in the regime of the strong renewal theorem we gain insights into the coalescing renewal processes in the Hammond-Sheffield urn (such as the asymptotic depth of most recent common ancestors) and are able to control the coalescence probabilities of two, three a…
We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n.
We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian approximation of this process of partial sums with the aid of a certain class of Gaussian processes, and obtain sufficient conditions for the $C$-convergence in the invariance principle in the Donsker form
For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension. The first family are the Gaussian polytopes defined as convex hulls of i.i.d. samples from a non-degenerate Gaussian distribution in $\mathbb R^d$. The second family are convex hulls of random walks with exchangeable increments satisfying certain mild general position assumptio…
We take an $L_1$-dense class of functions $\Cal F$ on a measurable space $(X,\Cal X)$ and a sequence of i.i.d. $X$-valued random variables $ξ_1,\dots,ξ_n$, and give a good estimate on the tail behaviour of $\sup\limits_{f\in\Cal F}\sum\limits_{j=1}^nf(ξ_j)$ if the conditions $\sup\limits_{x\in X}|f(x)|\le1$, $Ef(ξ_1)=0$ and $Ef(ξ_1)^2<σ^2$ with some $0\leσ\le1$ hold for all $f\in\Cal F$. Roughly speaking this estimate states that under some natural conditions the above considered supremum is not much larger than the worst element taking part in it. The proof heavily depends on the main result …
In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the c…
We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand (1995). We also complete Talagrand's inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cramér (1938), Bahadur-Rao (1960) and Sakhanenko (1991). We also show that…
Sums of independent random variables form the basis of many fundamental theorems in probability theory and statistics, and therefore, are well understood. The related problem of characterizing products of independent random variables seems to be much more challenging. In this work, we investigate such products of normal random variables, products of their absolute values, and products of their squares. We compute power-log series expansions of their cumulative distribution function (CDF) based on the theory of Fox H-functions. Numerically we show that for small arguments the CDFs are well appr…
The exact expression for the probability density $p_{_N}(x)$ for sums of a finite number $N$ of random independent terms is obtained. It is shown that the very tail of $p_{_N}(x)$ has a Gaussian form if and only if all the random terms are distributed according to the Gauss Law. In all other cases the tail for $p_{_N}(x)$ differs from the Gaussian. If the variances of random terms diverge the non-Gaussian tail is related to a Levy distribution for $p_{_N}(x)$. However, the tail is not Gaussian even if the variances are finite. In the latter case $p_{_N}(x)$ has two different asymptotics. At sm…
An explicit upper bound on the tail probabilities for the normalized Rademacher sums is given. This bound, which is best possible in a certain sense, is asymptotically equivalent to the corresponding tail probability of the standard normal distribution, thus affirming a longstanding conjecture by Efron. Applications to sums of general centered uniformly bounded independent random variables and to the Student test are presented.
We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. This result applies when the dimension of random vectors ($p$) is large compared to the sample size ($n$); in fact, $p$ can be much larger than $n$, without restricting correlations of the coordinates of these vectors. We also show that the distribution of the maximum of a sum…
We consider the generalized differential entropy of normalized sums of independent and identically distributed (IID) continuous random variables. We prove that the Rényi entropy and Tsallis entropy of order $α (α>0)$ of the normalized sum of IID continuous random variables with bounded moments are convergent to the corresponding Rényi entropy and Tsallis entropy of the Gaussian limit, and obtain sharp rates of convergence.
Let $χ$ be a non-real Dirichlet character modulo a prime $q$. In this paper we prove that the distribution of the short character sum $S_{χ,H}(x)=\sum_{x< n\leq x+H} χ(n)$, as $x$ runs over the positive integers below $q$, converges to a two-dimensional Gaussian distribution on the complex plane, provided that $\log H=o(\log q)$ and $H\to\infty$ as $q\to\infty$. Furthermore, we use a method of Selberg to give an upper bound on the rate of convergence.
We study the number of random records in an arbitrary split tree (or equivalently, the number of random cuttings required to eliminate the tree). We show that a classical limit theorem for convergence of sums of triangular arrays to infinitely divisible distributions can be used to determine the distribution of this number. After normalization the distributions are shown to be asymptotically weakly 1-stable. This work is a generalization of our earlier results for the random binary search tree, which is one specific case of split trees. Other important examples of split trees include $m$-ary s…
In this paper we determine two asymptotic results for Jack measures on partitions, a model defined by two specializations of Jack polynomials proposed by Borodin-Olshanski in [European J. Combin. 26.6 (2005): 795-834]. Assuming these two specializations are the same, we derive limit shapes and Gaussian fluctuations for the anisotropic profiles of these random partitions in three asymptotic regimes associated to diverging, fixed, and vanishing values of the Jack parameter. To do so, we introduce a generalization of Motzkin paths we call "ribbon paths", show for general Jack measures that certai…
We extend, in the free probability framework, an invariance principle for multilinear homogeneous sums with low influences recently established in [E. Mossel, R. O'Donnell and K. Oleszkiewicz (2010). Noise stability of functions with low influences: invariance and optimality. {\it Ann. Math.} {\bf 171}, no. 1, 295-341]. To do so, a hypercontractivity property for those homogeneous sums is necessary, and to prove it has turned out to be our main task. Finally, we deduce from our extension several universality phenomenons, in the spirit of the paper [I. Nourdin, G. Peccati and G. Reinert (2010).…
Consider the classical Gaussian unitary ensemble of size $N$ and the real Wishart ensemble $W_N(n,I)$. In the limits as $N \to \infty$ and $N/n \to γ> 0$, the expected number of eigenvalues that exit the upper bulk edge is less than one, 0.031 and 0.170 respectively, the latter number being independent of $γ$. These statements are consequences of quantitative bounds on tail sums of eigenvalues outside the bulk which are established here for applications in high dimensional covariance matrix estimation.
Let $\{η_{j}\}_{j = 0}^{N}$ be a sequence of independent, identically distributed random complex Gaussian variables, and let $\{f_{j} (z)\}_{j = 0}^{N}$ be a sequence of given analytic functions that are real-valued on the real number line. We prove an exact formula for the expected density of the distribution of complex zeros of the random equation $\sum_{j = 0}^{N} η_{j} f_{j} (z) = \mathbf{K}$, where $\mathbf{K} \in \mathds{C}$. The method of proof employs a formula for the expected absolute value of quadratic forms of Gaussian random variables. We then obtain the limiting behaviour of the …
We investigate level-set percolation of the Gaussian free field on transient trees, for instance on super-critical Galton-Watson trees conditioned on non-extinction. Recently developed Dynkin-type isomorphism theorems provide a comparison with percolation of the vacant set of random interlacements, which is more tractable in the case of trees. If $h_*$ and $u_*$ denote the respective (non-negative) critical values of level-set percolation of the Gaussian free field and of the vacant set of random interlacements, we show here that $h_* < \sqrt{2u}_*$ in a broad enough set-up, but provide an exa…
The maximum correlation of functions of a pair of random variables is an important measure of stochastic dependence. It is known that this maximum nonlinear correlation is identical to the absolute value of the Pearson correlation for a pair of Gaussian random variables or a pair of finite sums of iid random variables. This paper extends these results to pairwise Gaussian vectors and processes, nested sums of iid random variables, and permutation symmetric functions of sub-groups of iid random variables. It also discusses applications to additive regression models.
We consider the spectrum of random Laplacian matrices of the form $L_n=A_n-D_n$ where $A_n$ is a real symmetric random matrix and $D_n$ is a diagonal matrix whose entries are equal to the corresponding row sums of $A_n$. If $A_n$ is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution the empirical spectral measure of $L_n$ is known to converge to the free convolution of a semicircle distribution and a standard real Gaussian distribution. We consider real symmetric random matrices $A_n$ with independent entries (up to symmetry) whose row sums converge to a purely…
