Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Abstract

We use Stein's method to obtain explicit bounds on the rate of convergence for the Laplace approximation of two different sums of independent random variables; one being a random sum of mean zero random variables and the other being a deterministic sum of mean zero random variables in which the normalisation sequence is random. We make technical advances to the framework of Pike and Ren [37] for Stein's method for Laplace approximation, which allows us to give bounds in the Kolmogorov and Wasserstein metrics. Under the additional assumption of vanishing third moments, we obtain faster convergence rates in smooth test function metrics. As part of the derivation of our bounds for the Laplace approximation for the deterministic sum, we obtain new bounds for the solution, and it first two derivatives, of the Rayleigh Stein equation.where the test function h is real-valued. The second step is to solve (1.2) for f h and obtain suitable bounds for the solution. Finally, to approximate the distribution of a random variable of interest W by the target distribution q, one may evaluate both sides of (1.2) at W , take expectations and finally take the supremum of both sides over a class of functions H to obtain d H (W, Y ) := sup h∈H |E[h(W )] − E[h(Y )]| = sup h∈H E[Af h (W )].

Stein’s method for the single server queue in heavy traffic

“…We prove that the rate of convergence of our bound is optimal. This follows a recent argument used in the proof of Theorem 5.10 of Gaunt [16]. As such this result was not found in [23] and [27]; however, one can readily adapt our argument to show that the rates of convergence in the analogous results of [23] and [27] are optimal.…”

“…Here we note that both the M/G/1 and G/G/1 queue have a stationary distribution that is the convolution of a geometrically distributed number of IID random variables, and we prove a variant of results in Peköz and Röllin [23] and Ross [27]. In addition, we provide a new result following Gaunt [16] which proves that the rate of convergence considered in optimal. From our results we can deduce the following bound for the G/G/1 queue.…”

Inequalities for Integrals of the Modified Struve Function of the First Kind

“…In the recent papers [9] and [11], simple lower and upper bounds, involving the modified Bessel function of the first kind I ν (x), were obtained for the integrals where x > 0, 0 ≤ γ < 1 and ν > − 1 2 . For γ = 0 there does not exist simple closed form expressions for the integrals in (1.1) The inequalities of [9,11] were needed in the development of Stein's method [18,6,17] for variance-gamma approximation [7,8,10], although as they are simple and surprisingly accurate the inequalities may also prove useful in other problems involving modified Bessel functions; see for example, [5] in which inequalities for modified Bessel functions of the first kind were used to obtain lower and upper bounds for integrals involving modified Bessel functions of the first kind.…”

Inequalities for Some Integrals Involving Modified Lommel Functions of the First Kind

“…Finally, we observe that the bounds are most accurate in the case µ−ν = −0.5. 0.0001 0.0052 0.0129 0.0147 0.0125 0.0080 0.0045 (7,5) 0.0000 0.0011 0.0034 0.0048 0.0050 0.0037 0.0023 (12,10) 0.0000 0.0002 0.0006 0.0010 0.0015 0.0014 0.0010 (4.75 − 0.25) 0.0014 0.1217 0.2999 0.3120 0.2137 0.1134 0.0584 (5,0) 0.0006 0.0534 0.1361 0.1468 0.1034 0.0558 0.0290 (7.5, 2.5) 0.0000 0.0030 0.0095 0.0136 0.0125 0.0080 0.0045 (10,5) 0.0000 0.0007 0.0026 0.0043 0.0050 0.0037 0.0023 (15,10) 0.0000 0.0001 0.0005 0.0009 0.0014 0.0014 0.0010…”