1 results
Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue
-
- Journal:
- Advances in Applied Probability / Volume 14 / Issue 1 / March 1982
- Published online by Cambridge University Press:
- 01 July 2016, pp. 143-170
- Print publication:
- March 1982
-
- Article
- Export citation
-
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.
is a proper distribution with finite mean 
Various limit theorems for functionals of 
For example, the deviation of the empirical distribution function from 
properly normalised, is shown to have a limit in 
by means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and the 
uniformly in 