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Suppose you and your friend both do $n$ tosses of an unfair coin with probability of heads equal to $alpha$. 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 $alpha$ vary. Our main tools are integral representations based on Fourier analysis.
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For a binomial random variable $xi$ with parameters $n$ and $b/n$, it is well known that the median equals $b$ when $b$ is an integer. In 1968, Jogdeo and Samuels studied the behaviour of the relative difference between ${sf P}(xi=b)$ and $1/2-{sf P}(xi<b)$. They proved its monotonicity in $n$ and posed a question about its monotonicity in $b$. This question is motivated by the solved problem proposed by Ramanujan in 1911 on the monotonicity of the same quantity but for a Poisson random variable with an integer parameter $b$. In the paper, we answer this question and introduce a simple way to analyse the monotonicity of similar functions.
The distribution of sum of independent non-identical binomial random variables is frequently encountered in areas such as genomics, healthcare, and operations research. Analytical solutions to the density and distribution are usually cumbersome to find and difficult to compute. Several methods have been developed to approximate the distribution, and among these is the saddlepoint approximation. However, implementation of the saddlepoint approximation is non-trivial and, to our knowledge, an R package is still lacking. In this paper, we implemented the saddlepoint approximation in the textbf{sinib} package. We provide two examples to illustrate its usage. One example uses simulated data while the other uses real-world healthcare data. The textbf{sinib} package addresses the gap between the theory and the implementation of approximating the sum of independent non-identical binomials.
Let $X$ be the constrained random walk on ${mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $tau_n$ be the first time the sum of the components of $X$ equals $n$. $p_n doteq P_x(tau_n < tau_0)$ is a key performance measure for the queueing system represented by $X$, stability implies $p_nrightarrow 0$ exponentially. Currently the only analytic method available to approximate $p_n$ is large deviations analysis, which gives the exponential decay rate of $p_n$. Finer results are available via rare event simulation. The present article develops a new method to approximate $p_n$ and related expectations. The method has two steps: 1) with an affine transformation, move the origin onto the exit boundary of $tau_n$, take limits to remove some of the constraints on the dynamics, this yields a limit unstable constrained walk $Y$ 2) Construct a basis of harmonic functions of $Y$ and use them to apply the classical superposition principle of linear analysis. The basis functions are linear combinations of $log$-linear functions and come from solutions of harmonic systems, which are graphs whose vertices represent points on the characteristic surface of $Y$, the edges between the vertices represent conjugacy relations between the points, the loops represent membership in the boundary characteristic surfaces. Using our method we derive explicit, simple and almost exact formulas for $P_x(tau_n < tau_0)$ for $d$-tandem queues, similar to the product form formulas for the stationary distribution of $X$. The same method allows us to approximate the Balayage operator mapping $f$ to $x rightarrow {mathbb E}_x left[ f(X_{tau_n}) 1_{{tau_n < tau_0}} right]$ for a range of stable constrained random walks in $2$ dimensions. We indicate how the ideas of the paper relate to more general processes and exit boundaries.
Let $X$ be the constrained random walk on $mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $lambda$, $mu_1$, $mu_2$,...,$mu_d$, where ${e_1,e_2,..,e_d}$ are the standard basis vectors. The process $X$ is assumed stable, i.e., $lambda < mu_i$ for all $i=1,2,3,...,d.$ Let $tau_n$ be the first time the sum of the components of $X$ equals $n$. We derive approximation formulas for the probability ${mathbb P}_x(tau_n < tau_0)$. For $x in bigcup_{i=1}^d Big{x in {mathbb R}^d_+: sum_{j=1}^{i} x(j)$ $> left(1 - frac{log lambda/min mu_i}{log lambda/mu_i}right) Big}$ and a sequence of initial points $x_n/n rightarrow x$ we show that the relative error of the approximation decays exponentially in $n$. The approximation formula is of the form ${mathbb P}_y(tau < infty)$ where $tau$ is the first time the sum of the components of a limit process $Y$ is $0$; $Y$ is the process $X$ as observed from a point on the exit boundary except that it is unconstrained in its first component (in particular $Y$ is an unstable process); $Y$ and ${mathbb P}_y(tau< infty)$ arise naturally as the limit of an affine transformation of $X$ and the probability ${mathbb P}_x(tau_n < tau_0).$ The analysis of the relative error is based on a new construction of supermartingales. We derive an explicit formula for ${mathbb P}_y(tau < infty)$ in terms of the ratios $lambda/mu_i$ which is based on the concepts of harmonic systems and their solutions and conjugate points on a characteristic surface associated with the process $Y$; the derivation of the formula assumes $mu_i eq mu_j$ for $i eq j.$
It is shown that functions defined on ${0,1,...,r-1}^n$ satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger ``local sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrands [Ann. Probab. 22 (1994) 1576--1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on ${0,1,...,r-1}^n$ for $rge2$.
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