Hyungsu Kim, Chul Ki Ko, Sungchul Lee
When we use the entropy method to get the tail bounds, typically the left tail bounds are not good comparing with the right ones. Up to now this asymmetry has been observed many times. Surprisingly we find an entropy method for the left tail that works in the exactly same way that it works for the right tail.
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In the paper, we find exact asymptotics of the left tail of renewal measure for a broad class of two-sided random walks. We only require that an exponential moment of the left tail is finite. Through a simple change of measure approach, our result turns out to be almost equivalent to Blackwell's Theorem.
Left Tail of the Derivative Martingale in a Gaussian BRW in the Entire Subcritical Regime
研究高斯分支随机游走在亚临界区域中导数鞅极限的左尾概率，通过建立双边估计，证实了特定猜想。
Following results of Kemperman and Pinelis, we show that if $X$ and $Y$ are real valued random variables such that $\mathbb{E}\left\vert Y\right\vert<\infty$ and for all non-decreasing convex $φ:\mathbb{R}\rightarrow [0,\infty)$, $\mathbb{E}φ(X)\leq\mathbb{E}φ(Y)$, then for all $s\in\mathbb{R}$ with $\mathbb{P}\left\{Y>s\right\}\neq 0$, $\mathbb{P}\left\{X\geq\mathbb{E}\left(Y:Y>s\right)\right\}\leq\mathbb{P}\left\{Y>s\right\}$. This bound is sharp in essentially the strictest possible sense: for any such $Y$ and $s$ there exists such an $X$ with $\mathbb{P}\left\{X\geq \mathbb{E}\left(Y:Y>s\r…
Complete left tail asymptotic for the density of branching processes in the Schröder case
本文在Schröder情形下为Galton-Watson过程的密度导出了由幂项与周期因子相乘构成的完全左尾渐近级数。
A formula for the periodic multiplier in left tail asymptotics for supercritical branching processes in the Schröder case
针对Schröder情形下超临界分支过程左尾渐近的幂律乘周期函数特性，研究给出了其中乘法周期函数的显式表达式。
Complete left-tail asymptotic for branching processes with immigration
本文针对带移民的Galton-Watson过程，首次推导出鞅极限密度处处收敛的完整左尾渐近级数，并给出可快速计算的密度近似。
Complete left tail asymptotic for supercritical multitype branching processes
本文推导了超临界多类型Galton-Watson过程鞅极限密度的完全左尾渐近级数，证明其处处收敛，并给出快速计算的近似，为首个多类型分支过程左尾完整结果。
Let $X$ be lognormal$(μ,σ^2)$ with density $f(x)$, let $θ>0$ and define ${L}(θ)=E e^{-θX}$. We study properties of the exponentially tilted density (Esscher transform) $f_θ(x) =e^{-θx}f(x)/{L}(θ)$, in particular its moments, its asymptotic form as $θ\to\infty$ and asymptotics for the Cramér function; the asymptotic formulas involve the Lambert W function. This is used to provide two different numerical methods for evaluating the left tail probability of lognormal sum $S_n=X_1+\cdots+X_n$: a saddlepoint approximation and an exponential twisting importance sampling estimator. For the latter we d…
Examples of measures with trivial left and non-trivial right random walk tail boundary
本文证明具有平凡左尾边界与非平凡右尾边界的非退化测度示例恰好存在于具有无限共轭类性质的非平凡因子的顺从群上。
We present in this paper a new sufficient condition for the so-called Prokhorov-Skorokhod continuity of random processes. Our conditions will be formulated in the terms of metric entropy generated by three-dimensional distribution of the considered random process (r.p.) in the parametric set, have a convenient and closed form, and generalize some previous results. We study also the conditions for weak compactness of the sequence of random processes in this space and as a consequence the Central Limit Theorem. Our consideration based on the theory of Prokhorov-Skorokhod spaces of random process…
The left tail of the subcritical derivative martingale in a branching random walk
本文在高斯分支随机游走框架下，精确估计了$L^4$相中导数鞅分布的左尾，回答了Lacoin等人提出的猜想。
Sharp Tail Bounds Beyond Twice the Mean
本文针对n个独立非负均值至多为1的随机变量，推导其总和超过阈值t（t≥2n+1）的尖锐尾界，通过松弛优化与动态规划证明，该界在特定二元独立同分布分布下取等。
In this short paper we find that the Sobolev inequality $$\frac 1{p-2}\left[\left(\int f^{p} dμ\right)^{2/p} - \int f^2 dμ\right] \le C \int |\nabla f|^2 dμ$$ ($p\ge 0$) is equivalent to the exponential convergence of the Markov diffusion semigroup $(P_t)$ to the invariant measure $μ$, in some $Φ$-entropy. We provide the estimate of the exponential convergence in total variation and a bounded perturbation result under the Sobolev inequality. Finally in the one-dimensional case we get some two-sided estimates of the Sobolev constant by means of the generalized Hardy inequality.
Dimension-free Gaussian tail estimates for linear functionals on convex bodies
论文研究了在单位体积的凸体上随机向量的线性泛函的高斯尾估计，证明了存在绝对常数和一组正交向量，使得对于所有θ∈Θ，随机向量的泛函的p阶矩与二阶矩之间存在上下界。
We present explicit estimates of right and left tails and exact (up to universal, multiplicative constants) estimates of tails and moments of hitting times of Bessel processes. The latter estimates are obtained from more general estimates of moments and tails established for convolutions of elementary mixtures of exponential distributions.
Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\mathbb{E}\log m(ξ_{0})=\infty$. We show that (1) there exists no proper $c_{n}$ such that $\{Z_{n}/c_{n}\}$ has a proper, non-degenerate limit, (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\left(\barξ,Z_{n}(\barξ)\right)$ converges almost surely to a random variable $Y(\barξ)$, where $Y\in(0,1)~η$-a.s., (3) finally, we give a necessary and sufficient conditions for the almost sure convergence of $\left\{\frac{U(\barξ,Z_{n}(\bar…
A number of recent works have sought to generalize the Kolmogorov-Sinai entropy of probability-preserving transformations to the setting of Markov operators acting on the integrable functions on a probability space $(X,μ)$. These have culminated in a proof by Downarovicz and Frej that these definitions all coincide, and that the resulting quantity is uniquely characterized by certain properties. On the other hand, Makarov has shown that this `operator entropy' is always dominated by the Kolmogorov-Sinai entropy of a classical system that may be constructed from a Markov operator, and that thes…
Left-tail expansions for Schröder branching processes with explicit convergence rates
该研究移除了Schröder分支过程鞅极限密度左尾展开证明中的两个假设，推导了带显式收敛速度的展开系数界，分析了临界角行为及其对振荡修正项的影响。
Complete left tail asymptotic for branching processes in random environments
针对随机环境中分枝过程，提出利用稀有事件概率之比的极限分布生成函数满足广义Schröder型方程，提取完全左尾渐近级数，并讨论与奇异分形测度的联系。
Let $N_{\triangle}(G)$ be the number of triangles in a graph $G$. In [14] and [25] (respectively) the following bounds were proved on the lower tail behaviour of triangle counts in the dense Erdős-Rényi random graphs $G_m\sim G(n,m)$: \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ\left(δ^2n^3\right)\right) \qquad \text{if $n^{-3/2}\ll δ\ll n^{-1}$} \] and \[ \mathbb{P}\big(N_{\triangle}(G_m) \, < \, (1-δ)\mathbb{E}[N_{\triangle}(G_m)]\big) \,=\, \exp\left(-Θ(δ^{2/3}n^2) \right) \qquad \text{if $n^{-3/4} \ll δ\ll 1$.} \] Neeman, Radin an…
We show that the naive mean-field approximation correctly predicts the leading term of the logarithmic lower tail probabilities for the number of copies of a given subgraph in $G(n,p)$ and of arithmetic progressions of a given length in random subsets of the integers in the entire range of densities where the mean-field approximation is viable. Our main technical result provides sufficient conditions on the maximum degrees of a uniform hypergraph $\mathcal{H}$ that guarantee that the logarithmic lower tail probabilities for the number of edges induced by a binomial random subset of the vertice…
Let $(W,H,μ)$ be the classical Wiener space where $H$ is the Cameron-Martin space which consists of the primitives of the elements of $L^2([0,1],\,dt)\otimes \R^d$, we denote by $L^2_a(μ,H)$ the equivalence classes w.r.t. $dt\times dμ$ whose Lebesgue densities $s\to\dot{u}(s,w)$ are almost surely adapted to the canonical Brownian filtration. If $f$ is a Wiener functional s.t. $\frac{1}{E[e^{-f}]}e^{-f}dμ$ is of finite relative entropy w.r.t. $μ$, we prove that \beaa J_\star&=& \inf\left(E_μ\left[f\circ U+\half |u|_H^2\right]: u\in L_a^2(μ,H)\right)\\ &\geq&-\log E_μ[e^{-f}]=\inf\left(\int_W fd…
The Zeta Tail Distribution: A Novel Event-Count Model
提出Zeta Tail(a)分布作为风险分析中随机损害事件计数的新模型，推导其基本性质，并评估其作为几何分布替代的理论与应用价值。
Let $X_{1},..,X_{n}$ denote an i.i.d. sample with light tail distribution and $S_{1}^{n}$ denote the sum of its terms; let $a_{n}$ be a real sequence\ going to infinity with $n.$\ In a previous paper (\cite{BoniaCao}) it is proved that as $n\rightarrow\infty$, given $\left(S_{1}^{n}/n>a_{n}\right) $ all terms $X_{i_{\text{}}}$ concentrate around $a_{n}$ with probability going to 1. This paper explores the asymptotic distribution of $X_{1}$ under the conditioning events $\left(S_{1}^{n}/n=a_{n}\right) $ and $\left(S_{1}^{n}/n\geq a_{n}\right)$ . It is proved that under some regulatity property,…
We present novel and sharp lower bounds for higher load moments in the classical problem of mapping $M$ balls into $N$ bins by $q$-universal hashing, specialized to the case when $M=N$. As a corollary we prove a tight counterpart for the result about min-entropy condensers due to Dodis, Pietrzak and Wichs (CRYPTO'14), which has found important applications in key derivation. It states that condensing $k$ bits of min-entropy into a $k$-bit string $ε$-close to almost full min-entropy (precisely $ k-\log\log(1/ε)$ bits of entropy) can be achieved by the use of $q$-independent hashing with $q= \lo…
Exact Signature Tail Asymptotics for Pure Rough Paths
针对纯粗糙路径的签名尾渐近，证明并推广了BGS20猜想，通过有限维展开和范数循环构造给出匹配下界，将签名尾与局部m-变差关联。
We give asymptotics for the left and right tails of the limiting Quicksort distribution. The results agree with, but are less precise than, earlier non-rigorous results by Knessl and Spankowski.
Suppose that $(Z_n)_{n\geq0}$ is a supercritical branching process in independent and identically distributed random environment. The right tail function of the scaled growth rate for $(Z_n)_{n\geq0}$ is studied. The upper bounds for $\displaystyle\mathbb{P}\left[\frac{\log Z_n}{Mn}-μ\geq x\right]$ for any $x\geq3$ are obtained, by applying an extension of the Hoeffding type inequalities.
We give upper and lower asymptotic bounds for the left tail and for the right tail of the continuous limiting QuickSort density f that are nearly matching in each tail. The bounds strengthen results from a paper of Svante Janson (2015) concerning the corresponding distribution function F. Furthermore, we obtain similar bounds on absolute values of derivatives of f of each order.
Structure of lower tails in sparse random graphs
本文研究稀疏 Erdős–Rényi 随机图在低尾子图计数事件下的典型结构，通过改造熵增量方案并证明熵变分解的稳定性，表明其在子图计数与割范数上接近平均场熵极小元。
We consider quadratic Weyl sums $S_N(x;α,β)=\sum_{n=1}^N \exp\!\left[2πi\left( \left(\tfrac{1}{2}n^2+βn\right)\!x+αn\right)\right]$ for $(α,β)\in\mathbb{Q}^2$, where $x\in\mathbb{R}$ is randomly distributed according to a probability measure absolutely continuous with respect to the Lebesgue measure. We prove that the limiting distribution in the complex plane of $\frac{1}{\sqrt{N}}S_N(x;α,β)$ as $N\to\infty$ is either heavy tailed or compactly supported, depending solely on $α,β$. In the heavy tailed case, the probability (according to the limiting distribution) of landing outside a ball of r…
Let $X_1,\dots,X_n$ be independent nonnegative random variables (r.v.'s), with $S_n:=X_1+\dots+X_n$ and finite values of $s_i:=E X_i^2$ and $m_i:=E X_i>0$. Exact upper bounds on $E f(S_n)$ for all functions $f$ in a certain class $\mathcal{F}$ of nonincreasing functions are obtained, in each of the following settings: (i) $n,m_1,\dots,m_n,s_1,\dots,s_n$ are fixed; (ii) $n$, $m:=m_1+\dots+m_n$, and $s:=s_1+\dots+s_n$ are fixed; (iii)~only $m$ and $s$ are fixed. These upper bounds are of the form $E f(η)$ for a certain r.v. $η$. The r.v. $η$ and the class $\mathcal{F}$ depend on the choice of on…
We prove stability estimates for the Shannon-Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors $X,Y \in \mathbb{R}^d$, the deficit in the Shannon-Stam inequality is bounded from below by the expression $$ C \left(\mathrm{D}\left(X||G\right) + \mathrm{D}\left(Y||G\right)\right), $$ where $\mathrm{D}\left( \cdot ~ ||G\right)$ denotes the relative entropy with resp…
Full distribution of the number of distinct sites visited by a random walker in dimension $d \ge 2$
该研究针对 $d\ge2$ 维格点随机游走者访问不同位点数的分布，推导了其左右尾的大偏差标度形式与标度函数的界、解析结果，并通过数值模拟验证了理论预测。
The upper tail problem for the largest eigenvalue of the Erdős--Rényi random graph $\mathcal{G}_{n,p}$ is to estimate the probability that the largest eigenvalue of the adjacency matrix of $\mathcal{G}_{n,p}$ exceeds its typical value by a factor of $1+δ$. In this note we show that for $δ>0$ fixed, and $p \rightarrow 0$ such that $n^{\frac{1}{2}} p \rightarrow \infty$, the upper tail probability for the largest eigenvalue of $\mathcal{G}_{n,p}$ is $$\exp\left[-(1+o(1)) \min\left\{\tfrac{(1+δ)^2}{2}, δ(1+δ) \right\} n^{2}p^{2}\log (1/p)\right].$$ In the same regime of $p$, we show that the seco…
Anderson Hamiltonians with singular potentials
本文构建了具有狄利克雷和诺伊曼边界条件的随机薛定谔算子，研究了奇异随机势场下的积分态密度及生命尾行为与主特征值尾行为的关系。
Complete right tail asymptotic for the density of branching processes with fractional generating functions
针对具有分数概率生成函数的高尔顿-沃森过程，推导其密度的右尾渐近级数，分析级数收敛条件并与标准积分表示及左尾渐近对比，揭示指数因子频率的复平面分形结构。
The purpose of this letter is to improve Hoeffding's lemma and consequently Hoeffding's tail bounds. The improvement pertains to left skewed zero mean random variables $X\in[a,b]$, where $a<0$ and $-a>b$. The proof of Hoeffding's improved lemma uses Taylor's expansion, the convexity of $\exp(sx), s\in {\bf R}$ and an unnoticed observation since Hoeffding's publication in 1963 that for $-a>b$ the maximum of the intermediate function $τ(1-τ)$ appearing in Hoeffding's proof is attained at an endpoint rather than at $τ=0.5$ as in the case $b>-a$. Using Hoeffding's improved lemma we obtain one side…
We establish precise asymptotics near zero of the integrated density of states for the random Schrödinger operators $(-Δ)^{α/2} + V^ω$ in $L^2(\mathbb R^d)$ for the full range of $α\in(0,2]$ and a fairly large class of random nonnegative alloy-type potentials $V^ω$. The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit $$\lim_{s\to 0} s^{d/α}\ln\ell([0,s]) = -C \left(λ_d^{(α)}\right)^{d/α},$$ with $C \in (0,\infty]$. The constant $C$ is is finite if and only if the common distribution of the lattice random variables charges $\left\{0\right\}$. In this case, the c…
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