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

Let S, = X1 + - - - + X, be a random walk with negative drift p u} and assume that for some y>O dF(x)= e"vx dF(x) is a proper distribution with finite mean Ft. Various limit theorems for functionals of X1, - - - , X,,) are derived subject to conditioning upon {v(u)< oo} with u large, showing similar behaviour as if the XN were i.i.d. with distribution F. For example, the deviation of the empirical distribution function from F, properly normalised, is shown to have a limit in D, and an approximation for (u-l[Sv(U)t]- tu])o0at 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

[1]  F. J. Anscombe,et al.  Large-sample theory of sequential estimation , 1949, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Charles Stone.  Weak convergence of stochastic processes defined on semi-infinite time intervals , 1963 .

[3]  J. Kingman.  A martingale inequality in the theory of queues , 1964 .

[4]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

[5]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[6]  R. Pyke.  The weak convergence of the empirical process with random sample size , 1968, Mathematical Proceedings of the Cambridge Philosophical Society.

[7]  L. Donald Iglehart.  Diffusion approximations in collective risk theory , 1969 .

[8]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[9]  J. Kingman.  Inequalities in the Theory of Queues , 1970 .

[10]  D. Iglehart,et al.  The Equivalence of Functional Central Limit Theorems for Counting Processes and Associated Partial Sums , 1971 .

[11]  W. Whitt.  Weak convergence of first passage time processes , 1971, Journal of Applied Probability.

[12]  Olof Thorin.  Further remarks on the ruin problem in case the epochs of claims form a renewal process , 1971 .

[13]  Douglas R. Miller.  Existence of Limits in Regenerative Processes , 1972 .

[14]  D. Iglehart.  Extreme Values in the GI/G/1 Queue , 1972 .

[15]  Vincent Hodgson,et al.  The Single Server Queue. , 1972 .

[16]  W. Vervaat.  Functional central limit theorems for processes with positive drift and their inverses , 1972 .

[17]  T. Lindvall.  Weak convergence of probability measures and random functions in the function space D [0,∞) , 1973 .

[18]  Miklós Csörgő,et al.  Some examples and results in the theory of mixing and random-sum central limit theorems , 1973 .

[19]  D. Iglehart.  Functional Central Limit Theorems for Random Walks Conditioned to Stay Positive , 1974 .

[20]  S. Ross.  Bounds on the delay distribution in GI/G/1 queues , 1974, Journal of Applied Probability.

[21]  B. V. Bahr.  Ruin probabilities expressed in terms of ladder height distributions , 1974 .

[22]  D. Iglehart.  Conditioned limit theorems for random walks , 1975, Advances in Applied Probability.

[23]  Erwin Bolthausen,et al.  On a Functional Central Limit Theorem for Random Walks Conditioned to Stay Positive , 1976 .

[24]  Greg Taylor,et al.  Use of differential and integral inequalities to bound ruin and queuing probabilities , 1976 .

[25]  W. D. Kaigh.  An Invariance Principle for Random Walk Conditioned by a Late Return to Zero , 1976 .

[26]  Jacob Cohen,et al.  On regenerative processes in queueing theory , 1976 .

[27]  Jan Grandell,et al.  A class of approximations of ruin probabilities , 1977 .

[28]  G. Eagleson.  Some Simple Conditions for Limit Theorems to Be Mixing , 1977 .

[29]  D. Aldous.  Weak convergence of randomly indexed sequences of random variables , 1978, Mathematical Proceedings of the Cambridge Philosophical Society.

[30]  R. Dudley.  Central Limit Theorems for Empirical Measures , 1978 .

[31]  A. Pakes.  On the maximum and absorption time of left-continuous random walk , 1978, Journal of Applied Probability.

[32]  Limiting diffusion for random walks with drift conditioned to stay positive , 1978 .

[33]  Jan Grandell,et al.  A remark on ‘A class of approximations of ruin probabilities’ , 1978 .

[34]  D. Siegmund.  Corrected diffusion approximations in certain random walk problems , 1979, Advances in Applied Probability.

[35]  Michio Shimura.  A limit theorem for conditional random walk , 1979 .

[36]  R. Durrett.  Conditioned limit theorems for random walks with negative drift , 1980 .

[37]  The Limit Distribution for a Random Walk with Absorption , 1981 .

[38]  S. Asmussen.  Equilibrium properties of the M/G/1 queue , 1981 .