Let $(S_0,S_1,...)$ be a supermartingale relative to a nondecreasing sequence of $σ$-algebras $H_{\le0},H_{\le1},...$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that $X_i\le d$ and $\mathsf {Var}(X_i|H_{\le i-1})\le σ_i^2$ a.s. for every $i=1,2,...$, where $d>0$ and $σ_i>0$ are non-random constants. Let $T_n:=Z_1+...+Z_n$, where $Z_1,...,Z_n$ are i.i.d. r.v.'s each taking on only two values, one of which is $d$, and satisfying the conditions $\mathsf {E}Z_i=0$ and $\mathsf {Var}Z_i=σ^2:=\frac{1}{n}(σ_1^2+...+σ_n^2)$. Then, based on a comparison inequality between generalized moments of $S_n$ and $T_n$ for a rich class of generalized moment functions, the tail comparison inequality $$ \mathsf P(S_n\ge y) \le c \mathsf P^{\mathsf Lin,\mathsf L C}(T_n\ge y+\tfrach2)\quad\forall y\in \mathbb R$$ is obtained, where $c:=e^2/2=3.694...$, $h:=d+σ^2/d$, and the function $y\mapsto \mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y)$ is the least log-concave majorant of the linear interpolation of the tail function $y\mapsto \mathsf {P}(T_n\ge y)$ over the lattice of all points of the form $nd+kh$ ($k\in \mathbb {Z}$). An explicit formula for $\mathsf {P}^{\mathsf {Lin},\mathsf {LC}}(T_n\ge y+\tfrac{h}{2})$ is given. Another, similar bound is given under somewhat different conditions. It is shown that these bounds improve significantly upon known bounds.
Let $S$ and $X$ be independent random variables, assuming values in the set of non-negative integers, and suppose further that both $\mathbb{E}(S)$ and $\mathbb{E}(X)$ are integers satisfying $\mathbb{E}(S)\ge \mathbb{E}(X)$. We establish a sufficient condition for the tail probability $\mathbb{P}(S\ge \mathbb{E}(S))$ to be larger than $\mathbb{P}(S+X\ge \mathbb{E}(S+X))$. We also apply this result to sums of independent binomial and Poisson random variables.
Suppose you and your friend both do $n$ tosses of an unfair coin with probability of heads equal to $α$. What is the behavior of the probability that you obtain at least $d$ more heads than your friend if you make $r$ additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of $n$, and demonstrate surprising phase transition phenomenons as parameters $ d, r$ and $α$ vary. Our main tools are integral representations based on Fourier analysis.
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 …
The variation of a martingale $p_0^k=p_0,...,p_k$ of probabilities on a finite (or countable) set $X$ is denoted $V(p_0^k)$ and defined by $V(p_0^k)=E(\sum_{t=1}^k|p_t-p_{t-1}|_1)$. It is shown that $V(p_0^k)\leq \sqrt{2kH(p_0)}$, where $H(p)$ is the entropy function $H(p)=-\sum_xp(x)\log p(x)$ and $\log$ stands for the natural logarithm. Therefore, if $d$ is the number of elements of $X$, then $V(p_0^k)\leq \sqrt{2k\log d}$. It is shown that the order of magnitude of the bound $\sqrt{2k\log d}$ is tight for $d\leq 2^k$: there is $C>0$ such that for every $k$ and $d\leq 2^k$ there is a marting…
We consider an incomplete multi-asset binomial market model. We prove that for a wide class of contingent claims the extremal multi-step martingale measure is a power of the corresponding single-step extremal martingale measure. This allows for closed form formulas for the bounds of a no-arbitrage contingent claim price interval. We construct a feasible algorithm for computing those boundaries as well as for the corresponding hedging strategies. Our results apply, for example, to European basket call and put options and Asian arithmetic average options.
We study the joint laws of a continuous, uniformly integrable martingale, its maximum, and its minimum. In particular, we give explicit martingale inequalities which provide upper and lower bounds on the joint exit probabilities of a martingale, given its terminal law. Moreover, by constructing explicit and novel solutions to the Skorokhod embedding problem, we show that these bounds are tight. Together with previous results of Azéma & Yor, Perkins, Jacka and Cox & Obłój, this allows us to completely characterise the upper and lower bounds on all possible exit/no-exit probabilities, subject to…
The Bernstein inequality is a tight upper bound on tail probabilities for independent random variables. Freedman extended the Bernstein inequality to martingales with differences bounded from above, and then Dzhaparidze and van Zanten generalized Freedman's result to non-bounded locally square integrable martingales. In this paper, we derive some Dzhaparidze-van Zanten type inequalities for self-normalized martingales with square integrable and non-square integrable differences.
Within a contest there is some probability M_i(t) that contestant i will be the winner, given information available at time t, and M_i(t) must be a martingale in t. Assume continuous paths, to capture the idea that relevant information is acquired slowly. Provided each contestant's initial winning probability is at most b, one can easily calculate, without needing further model specification, the expectations of the random variables N_b = number of contestants whose winning probability ever exceeds b, and D_{ab} = total number of downcrossings of the martingales over an interval [a,b]. The dis…
We investigate the properties of a discrete-time martingale $\{X_m\}_{m\in \mathbb Z_{\geq 0}}$, where all differences between adjacent random variables are limited to be not more than a constant as a promise. In this situation, it is known that the Azuma-Hoeffding inequality holds, which gives an upper bound of a probability for exceptional events. The inequality gives a simple form of the upper bound, and it has been utilized for many investigations. However, the inequality is not tight. We give an explicit expression of a tight upper bound, and we show that it and the bound obtained from th…
Let $M_n= \fsu X1n$ be a sum of independent random variables such that $ X_k\leq 1$, $\E X_k =0$ and $\E X_k^2=\s_k^2$ for all $k$. Hoeffding 1963, Theorem 3, proved that $$¶{M_n \geq nt}\leq H^n(t,p),\quad H(t,p)= \bgl(1+qt/p\bgr)^{p +qt} \bgl({1-t}\bgr)^{q -qt}$$ with $$q=\ffrac 1{1+\s^2},\quad p=1-q, \quad \s^2 =\ffrac {\s_1^2+...+\s_n^2}n,\quad 0<t<1.$$ Bentkus 2004 improved Hoeffding's inequalities using binomial tails as upper bounds. Let $\ga_k =\E X_k^3/\s_k^3$ and $ \vk_k= \E X_k^4/\s_k^4$ stand for the skewness and kurtosis of $X_k$. In this paper we prove (improved) counterparts of …
We give a necessary and sufficient condition on a sequence of functions on a set $Ω$ under which there is a measure on $Ω$ which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities.
We derive simple but nearly tight upper and lower bounds for the binomial lower tail probability (with straightforward generalization to the upper tail probability) that apply to the whole parameter regime. These bounds are easy to compute and are tight within a constant factor of $89/44$. Moreover, they are asymptotically tight in the regimes of large deviation and moderate deviation. By virtue of a surprising connection with Ramanujan's equation, we also provide strong evidences suggesting that the lower bound is tight within a factor of $1.26434$. It may even be regarded as the natural lowe…
Lundberg-type inequalities for ruin probabilities of non-homogeneous risk models are presented in this paper. By employing martingale method, the upper bounds of ruin probabilities are obtained for the general risk models under weak assumptions. In addition, several risk models, including the newly defined united risk model and quasi-periodic risk model with interest rate, are studied.
In this paper we present various new inequalities for tail proabilities for distributions that are elements of the most improtant exponential families. These families include the Poisson distributions, the Gamma distributions, the binomial distributions, the negative binomial distributions and the inverse Gaussian distributions. All these exponential families have simple variance functions and the variance functions play an important role in the exposition. All the inequalities presented in this paper are formulated in terms of the signed log-likelihood. The inequalities are of a qualitative n…
Hoeffding has shown that tail bounds on the distribution for sampling from a finite population with replacement also apply to the corresponding cases of sampling without replacement. (A special case of this result is that binomial tail bounds apply to the corresponding hypergeometric tails.) We give a new proof of Hoeffding's result by constructing a martingale coupling between the sampling distributions. This construction is given by an explicit combinatorial procedure involving balls and urns. We then apply this construction to create martingale couplings between other pairs of sampling dist…
Sequential Monte Carlo methods which involve sequential importance sampling and resampling are shown to provide a versatile approach to computing probabilities of rare events. By making use of martingale representations of the sequential Monte Carlo estimators, we show how resampling weights can be chosen to yield logarithmically efficient Monte Carlo estimates of large deviation probabilities for multidimensional Markov random walks.
We study the tail asymptotic of the stationary joint queue length distribution for a generalized Jackson network (GJN for short), assuming its stability. For the two station case, this problem has been recently solved in the logarithmic sense for the marginal stationary distributions under the setting that arrival processes and service times are of phase-type. In this paper, we study similar tail asymptotic problems on the stationary distribution, but problems and assumptions are different. First, the asymptotics are studied not only for the marginal distribution but also the stationary probab…
Freedman's inequality is a supermartingale counterpart to Bennett's inequality. This result shows that the tail probabilities of a supermartingale is controlled by the quadratic characteristic and a uniform upper bound for the supermartingale difference sequence. Replacing the quadratic characteristic by $\textrm{H}_k^y:= \sum_{i=1}^k\left(\mathbf{E}(ξ_i^2 |\mathcal{F}_{i-1}) +ξ_i^2\textbf{1}_{\{|ξ_i|> y\}}\right),$ Dzhaparidze and van Zanten (\emph{Stochastic Process. Appl.}, 2001) have extended Freedman's inequality to martingales with unbounded differences. In this paper, we prove that $\te…
We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so. We demonstrate that this is the case when the norm on the space can be approximated, within an absolute constant factor, by a norm which is differentiable on the unit sphere with a Lipschitz continuous gradient. We also present various examples of spaces possessing the latter property.
The paper investigates a discrete time Binomial risk model with different types of polices and shock events may influence some of the claim sizes. It is shown that this model can be considered as a particular case of the classical compound Binomial model. As far as we work with parallel Binomial counting processes in infinite time, if we consider them as independent, the probability of the event they to have at least once simultaneous jumps would be equal to one. We overcome this problem by using thinning instead of convolution operation. The bivariate claim counting processes are expressed in…
A clique colouring of a graph is a colouring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colours in such a colouring is the clique chromatic number. In this paper, we study the asymptotic behaviour of the clique chromatic number of the random graph G(n,p) for a wide range of edge-probabilities p=p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.
Research in psychology and neuroscience has successfully modeled decision making as a process of noisy evidence accumulation to a decision bound. While there are several variants and implementations of this idea, the majority of these models make use of a noisy accumulation between two absorbing boundaries. A common assumption of these models is that decision parameters, e.g., the rate of accumulation (drift rate), remain fixed over the course of a decision, allowing the derivation of analytic formulas for the probabilities of hitting the upper or lower decision threshold, and the mean decisio…
We obtain Azuma bounds for the probabilities of being away from the limit for a class of urn models. The method consists of relating the variables to certain linear combinations using eigenvectors of the replacement matrix, thus bringing in appropriate martingales. Some cases of repeated eigenvalues are also considered using Jordan vectors.
Let $d\in \mathbb{N}$ and let $γ_i\in [0,\infty)$, $x_i\in (0,1)$ be such that $\sum_{i=1}^{d+1} γ_i = M\in (0,\infty)$ and $\sum_{i=1}^{d+1} x_i = 1$. We prove that \begin{equation*} a \mapsto \frac{Γ(aM + 1)}{\prod_{i=1}^{d+1} Γ(a γ_i + 1)} \prod_{i=1}^{d+1} x_i^{aγ_i} \end{equation*} is completely monotonic on $(0,\infty)$. This result generalizes the one found by Alzer (2018) for binomial probabilities ($d=1$). As a consequence of the log-convexity, we obtain some combinatorial inequalities for multinomial coefficients. We also show how the main result can be used to derive asymptotic form…
We find a simple expression for the probability density of $\int \exp (B_s - s/2) ds$ in terms of its distribution function and the distribution function for the time integral of $\exp (B_s + s/2)$. The relation is obtained with a change of measure argument where expectations over events determined by the time integral are replaced by expectations over the entire probability space. We develop precise information concerning the lower tail probabilities for these random variables as well as for time integrals of geometric Brownian motion with arbitrary constant drift. In particular, $E[ \exp\big…
A novel multinomial theorem for commutative idempotents is shown to lead to new results about the moments, central moments, factorial moments, and their generating functions for any random variable $X = \sum_{i} Y_i $ expressible as a sum of Bernoulli indicator random variables $Y_i$. The resulting expressions are functions of the expectation of products of the Bernoulli indicator random variables. These results are used to derive novel expressions for the various moments in a large number of familiar examples and classic problems including: the binomial, hypergeometric, Poisson limit of the b…
In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the us…
For every given real value of the ratio $μ:=A_X/G_X>1$ of the arithmetic and geometric means of a positive random variable $X$ and every real $v>0$, exact upper bounds on the right- and left-tail probabilities $\mathsf{P}(X/G_X\ge v)$ and $\mathsf{P}(X/G_X\le v)$ are obtained, in terms of $μ$ and $v$. In particular, these bounds imply that $X/G_X\to1$ in probability as $A_X/G_X\downarrow1$. Such a result may be viewed as a converse to a reverse Jensen inequality for the strictly concave function $f=\ln$, whereas the well-known Cantelli and Chebyshev inequalities may be viewed as converses to a…
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…
Kolmogorov-Type Maximal Inequalities for Independent and Dependent Negative Binomial Random Variables: Sharp Bounds, Sub-Exponential Refinements, and Applications to Overdispersed Count Data
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…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Duke University（杜克大学）
The non-asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probability in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user-friendly schemes for developing non-asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub-Gaussian and sub-exponential random variables, which match the classic Hoeffding-type and Bernstein-type concentration inequal…
