Simulation Output Analysis Based on Excursions

On Exact Simulation Algorithms for Some Distributions Related to Brownian Motion and Brownian Meanders

“…For x ≤ π/ √ 2 (which is ≥ Δ), using | sin x| ≤ |x|, and, for α, β, θ ≥ 0, |α sin θ − β cos θ | ≤ αθ + β, We conclude that Further references relevant to the material in this paper include Asmussen et al (1995), Bertoin et al (1999), Bonaccorsi and Zambotti (2004), Calvin (2004), Ciesielski and Taylor (1962), Fujita and Yor (2007), Lévy (1948), Pitman and Yor (2003), Revuz andYor (1991), andZambotti (2003).…”

Confidence intervals for quantiles with standardized time series

“…The limiting random variable K(y, 1)/ζ (K(y, · )) then has the distribution of the ratio of a standard normal over a standard Brownian excursion height, so H(x) = 1 − πx 2 ∑ ∞ n=1 nK 1 (πnx), where K 1 is the modified Bessel function (Calvin 2004). The 0.95-critical point of H is γ ≈ 1.39739 (i.e., H(γ) = 0.95), which we can use in (12) to obtain an asymptotic 90% confidence interval for Q(y), where, for n ≥ max{(1 − y)/y, y/(1 − y)},…”

Using Excursions to Analyze Simulation Output

“…A variation on the estimator is to fix the memory allocation m and keep only the m longest excursions. This estimator was presented, without proofs, in Calvin [3].…”

“…In this section we illustrate the approach with one example. A preliminary version of this approach appeared in Calvin [3].…”

“…The bias correction terms described in Section 3.1 were incorporated into the estimator (with twice the term defined in (6) added to the range component). To avoid numerical instability, in cases where the last zero crossing takes place prior to time 0.05n, we consider only the meander part of the estimator defined in Theorem 4.3, and, similarly, if the last zero occurs after time 0.95n, we use only the excursion part of the estimator (as described in [3]). The quantiles for the estimator based on only the meander are computed from the distribution of N/ √ γ , which has the distribution P N √ γ > z = 1 2 1 − 1 1 + 2/z 2 for z > 0, which can be established by a calculation similar to the one used to derive (8).…”