Sum Of Bernoulli Random Variables Research Articles

Poisson approximation for the expectation of call function with application in collateralized debt obligation

Let W be a sum of Bernoulli random variables which satisfies local dependence and hz be a call function. This function has many applications in finance. In this article, we give bounds of Poisson approximation for E [ h z ( W ) ] by using Stein-Chen’s method. Our results improve the previous results. Finally, we apply these results to approximate the mean of percentage loss for each tranche in the collateralized debt obligation (CDO) tranche pricing.

The local limit theorem for general weighted sums of Bernoulli random variables

The local limit theorem (LLT) is one of the well-known limit theorems which can be used to estimate the probability at a particular point of a random variable. In this paper, we generalize weighted sums which are introduced by Giuliano and Weber and give the LLT with explicit error bounds.

A Limit Theorem for Bernoulli Convolutions and the $$\Phi $$-Variation of Functions in the Takagi Class

We consider a probabilistic approach to compute the Wiener–Young \(\Phi \)-variation of fractal functions in the Takagi class. Here, the \(\Phi \)-variation is understood as a generalization of the quadratic variation or, more generally, the pth variation of a trajectory computed along the sequence of dyadic partitions of the unit interval. The functions \(\Phi \) we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function \(\Phi \), our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and linear \(\Phi \)-variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear \(\Phi \)-variation for each function \(\Phi \) from our class. The proof of our main result relies on a limit theorem for certain sums of Bernoulli random variables that converge to an infinite Bernoulli convolution.

Using a Chen-Stein identity to obtain low variance simulation estimators

AbstractThis paper is concerned with developing low variance simulation estimators of probabilities related to the sum of Bernoulli random variables. It shows how to utilize an identity used in the Chen-Stein approach to bounding Poisson approximations to obtain low variance estimators. Applications and numerical examples in such areas as pattern occurrences, generalized coupon collecting, system reliability, and multivariate normals are presented. We also consider the problem of estimating the probability that a positive linear combination of Bernoulli random variables is greater than some specified value, and present a simulation estimator that is always less than the Markov inequality bound on that probability.

Tree Bounds for Sums of Bernoulli Random Variables: A Linear Optimization Approach

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of n dependent Bernoulli random variables exceeds an integer k. Under knowledge of all pairs of bivariate distributions denoted by a complete graph, the bounds are NP-hard to compute. When the bivariate distributions are specified on a tree graph, we show that tight bounds are computable in polynomial time using a compact linear program. These bounds provide robust probability estimates when the assumption of conditional independence in a tree-structured graphical model is violated. We demonstrate, through numericals, the computational advantage of our compact linear program over alternate approaches. A comparison of bounds under various knowledge assumptions, such as univariate information and conditional independence, is provided. An application is illustrated in the context of Chow–Liu trees, wherein our bounds distinguish between various trees that encode the maximum possible mutual information.

Bivariate Bernoulli Weighted Sums and Distribution of Single-Period Tontine Benefits

This paper studies the distribution of particular weighted sums of Bernoulli random variables. The computing methods are applied to derive the probability distribution of the random amount of survivor credits to be shared among surviving participants in single-period tontine schemes. The effectiveness of this new arrangement can then be evaluated beyond the classical analysis based on crude approximations for the two first moments, only.

Poisson Approximations for Sum of Bernoulli Random Variables and its Application to Ewens Sampling Formula

Poisson Approximations for Sum of Bernoulli Random Variables and its Application to Ewens Sampling Formula

Concavity of entropy along binomial convolutions

Motivated by a generalization of Sturm-Lott-Villani theory to discrete spaces and by a conjecture stated by Shepp and Olkin about the entropy of sums of Bernoulli random variables, we prove the concavity in $t$ of the entropy of the convolution of a probability measure $a$, which has the law of a sum of independent Bernoulli variables, by the binomial measure of parameters $n\geq 1$ and $t$.

Lower estimations of the deviation probabilities for sums of Bernoulli random variables

We obtain lower bilateral and unilateral estimations of the deviation probability of the arithmetic mean of independent Bernoulli random variables from mathematical expectation.

Law of the Sum of Bernoulli Random Variables

Let $\Delta_n$ be the set of all possible joint distributions of n Bernoulli random variables $X_{1},\ldots,X_n$. Suppose that $\Delta_n$, which is a simplex in the $2^n$-dimensional space, is endowed with the normalized Lebesgue measure $\mu_n$. Suppose also that the integer n is large. Then we show that there is a subset $\Delta $ of $\Delta_n$, whose measure $\mu_n(\Delta )$ is very close to 1, such that if the joint distribution of $(X_{1},\ldots,X_n)$ is in $\Delta $, then the law of the sum $X_{1}+\cdots+X_n$ is close to the binomial law ${\cal B}(n,\frac12)$. This result does not need any independence assumption. Next, we show a result of the same kind when $\Delta_n$ is endowed with another probability measure $\nu_n$.

Law of the sum of Bernoulli random variables

Law of the sum of Bernoulli random variables

Short Communications On Inequalities for Large Deviations in the Bernoulli Scheme

We discuss known inequalities for probabilities of large deviations for sums of Bernoulli random variables and improve some of them.

The stochastic precedence ordering with applications in sampling and testing

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.

The stochastic precedence ordering with applications in sampling and testing

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.

Random sums of random vectors and multitype families ofproductive individuals

We prove limit theorems for a family of random vectors whose coordinates are a special form of random sums of Bernoulli random variables. Applying these limit theorems, we study the number of productive individuals in n‐type indecomposable critical branching stochastic processes with types of individuals T1, …, Tn.

Comparing sums of independent bounded random variables and sums of Bernoulli random variables

This paper deals with the comparison of tail probabilities for sums of independent bounded random variables and those for sums of Bernoulli random variables. As a consequence, we obtain a new sufficient criterion for the strong law of large numbers for a certain class of sequences of independent random variables satisfying boundedness and second moment conditions, which - in a sense - cannot be improved upon.

Some applications of the Stein-Chen method for proving Poisson convergence

Let W be a sum of Bernoulli random variables and U λ a Poisson random variable having the same mean λ = EW. Using the Stein-Chen method and suitable couplings, general upper bounds for the variational distance between W and U λ are given. These bounds are applied to problems of occupancy, using sampling with and without replacement and Pólya sampling, of capture-recapture, of spacings and of matching and ménage.

Some applications of the Stein-Chen method for proving Poisson convergence

Let W be a sum of Bernoulli random variables and Uλ a Poisson random variable having the same mean λ = EW. Using the Stein-Chen method and suitable couplings, general upper bounds for the variational distance between W and Uλ are given. These bounds are applied to problems of occupancy, using sampling with and without replacement and Pólya sampling, of capture-recapture, of spacings and of matching and ménage.

Some density functions of a counting variable in the simple random walk

Abstract Extract Let 〈Xi 〉 be a sequence of completely independent and identically distributed random variables such that Xi = + 1, −1 with probabilities p (0 (µ + λ)n holds for only finitely many n-values, where λ is any positive number less than 2 to avoid triviality. Define the indicator variables Yn ≡ Yn (p, λ) by Yn = 1 if Sn > (µ + λ)n and Yn = 0 otherwise. The counting variable of interest here is N∞ ≡ ∞ (p, λ) and is defined by . Hence, P{N∞ < ∞} = 1, that is, N∞ is an honest random variable or N∞ has a proper distribution. The following theorem provides some exact density functions of N∞ for selected combinations of p and λ and thus elucidates the nature of chance fluctuations of sums of Bernoulli random variables with respect to the “finitely man...