Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue

Abstract

Let Sn = X1 + · · · + Xn be a random walk with negative drift μ < 0, let F(x) = P(Xk ≦ x), v(u) =inf{n : Sn > u} and assume that for some γ > 0 is a proper distribution with finite mean Various limit theorems for functionals of X1,· · ·, Xv(u) are derived subject to conditioning upon {v(u)< ∞} with u large, showing similar behaviour as if the Xi were i.i.d. with distribution For example, the deviation of the empirical distribution function from properly normalised, is shown to have a limit in D, and an approximation for by means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and the GI/G/1 queue considered either within a busy cycle or in the steady state. The methods produce an alternate approach to known asymptotic formulae for ruin probabilities as well as related waiting-time approximations for the GI/G/1 queue. For example uniformly in N, with WN the waiting time of the Nth customer.

Busy period analysis, rare events and transient behavior in fluid flow models

“…For (f), let similarly t(u) I(Ir + (u)/u-1/g'(7) > e), and appeal to Lemma 7.2 below. El Theorem 6.1 may be seen as an analog of GI/G/1 results of Asmussen [4] (see in particular Theorem 5.1 of that paper). See also Anantharam [1].…”

Diffusions conditioned on occupation measures

“…Knowing that k → −∞ is equivalent to total deconfinement in S, we find however that N (−∞) = 0 is the only stable fixed point reached in that limit. Similarly, since k → ∞ corresponds to total confinement in S, we must have N (+∞) = I. Equations (49), (50), and (53) define a set of coupled nonlinear differential equations that can be used to find the SCGF λ(k), which corresponds to the dominant eigenvalue λ 0 (k), and its associated eigenfunction Ψ, which corresponds to Ψ 0 (k). From the dominant eigenfunction, we then find the driven potential U k as in the previous section.…”

“…(55) Figure 8 shows the perturbation results for λ(k) and I(r) obtained by integrating Eqs. (49), (50), and (53) starting from the known eigenvalues λ n (0) and eigenstates Ψ n (0) of the quantum harmonic oscillator. The results are for the unit interval occupation, S = [0, 1], and are also obtained by truncating N (k) to a finite size M .…”

On the integral of the workload process of the single server queue

“…Define the new probability space (Ĩ P, {F n }) by the exponential change of measure with the likehood ratio up to time n: respectively. We show that for large x conditioning on event {h > x} the system up to the time of reaching the cycle maximum behaves like the new system with the service and the interarrival time distribution tails given in (4.4) and (4.5), respectively (see Asmussen [3] for the similar result when we are conditioning on the event {τ (x) < ∞}, where τ (x) := inf{t ≥ 0 : W (t) > x}). ByĨ E we mean the expectation with respect toĨ P. Let ρ :=Ĩ Eσ/Ĩ Eτ (4.6) be the traffic load in the new system.…”