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

1982 ◽  
Vol 14 (01) ◽  
pp. 143-170 ◽  
Author(s):  
Søren Asmussen
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Limit Theorems ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Brownian Bridge ◽  
Ruin Probabilities ◽  
Asymptotic Formulae ◽  
Negative Drift ◽  
Busy Cycle

LetSn=X1+ · · · +Xnbe a random walk with negative drift μ < 0, letF(x) =P(Xk≦x),v(u) =inf{n:Sn>u} and assume that for some γ > 0is a proper distribution with finite meanVarious limit theorems for functionals ofX1,· · ·,Xv(u)are derived subject to conditioning upon {v(u)< ∞} withularge, showing similar behaviour as if theXiwere i.i.d. with distributionFor example, the deviation of the empirical distribution function fromproperly normalised, is shown to have a limit inD, and an approximation forby means of Brownian bridge is derived. Similar results hold for risk reserve processes in the time up to ruin and theGI/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 theGI/G/1 queue. For exampleuniformly inN, withWNthe waiting time of the Nth customer.

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

1982 ◽  
Vol 14 (1) ◽  
pp. 143-170 ◽  
Author(s):  
Søren Asmussen
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Limit Theorems ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Brownian Bridge ◽  
Ruin Probabilities ◽  
Asymptotic Formulae ◽  
Negative Drift ◽  
Busy Cycle

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.

scholarly journals How linear reinforcement affects Donsker’s theorem for empirical processes

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1173-1192 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Long Range ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Empirical Processes ◽  
Constant Factor ◽  
Empirical Measure ◽  
Reinforced Random Walks ◽  
Bernoulli Processes

Abstract A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

On the maximum and absorption time of left-continuous random walk

1978 ◽  
Vol 15 (2) ◽  
pp. 292-299 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Initial Position ◽  
Absorption Time ◽  
Conditional Limit Theorems ◽  
Negative Drift ◽  
Continuous Random Walk ◽  
Conditional Limit ◽  
Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

Comment on ‘Corrected diffusion approximations in certain random walk problems'

1986 ◽  
Vol 23 (1) ◽  
pp. 89-96 ◽  
Author(s):  
Michael L. Hogan
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Exponential Family ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Negative Drift ◽  
Family Case ◽  
Correction Terms

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.

On Some Limit Theorems Involving the Empirical Distribution Function

1967 ◽  
Vol 10 (5) ◽  
pp. 739-741
Author(s):  
Miklós Csörgo
Keyword(s):  
Distribution Function ◽  
Limit Theorems ◽  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Continuous Distribution ◽  
Random Variables ◽  
Continuous Distribution Function ◽  
Independent Random Variables ◽  
Mutually Independent

Let X1 …, Xn be mutually independent random variables with a common continuous distribution function F (t). Let Fn(t) be the corresponding empirical distribution function, that isFn(t) = (number of Xi ≤ t, 1 ≤ i ≤ n)/n.Using a theorem of Manija [4], we proved among others the following statement in [1].

On the maximum and absorption time of left-continuous random walk

1978 ◽  
Vol 15 (02) ◽  
pp. 292-299 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Initial Position ◽  
Absorption Time ◽  
Conditional Limit Theorems ◽  
Negative Drift ◽  
Continuous Random Walk ◽  
Conditional Limit ◽  
Transformed Distribution

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift. Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

Comment on ‘Corrected diffusion approximations in certain random walk problems'

1986 ◽  
Vol 23 (01) ◽  
pp. 89-96
Author(s):  
Michael L. Hogan
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Exponential Family ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Negative Drift ◽  
Family Case ◽  
Correction Terms

Correction terms for the diffusion approximation to the maximum and ruin probabilities for a random walk with small negative drift, obtained by Siegmund (1979) in the exponential family case, are extended by different methods to some non-exponential family cases.

scholarly journals Exponential Behavior in the Presence of Dependence in Risk Theory

2006 ◽  
Vol 43 (1) ◽  
pp. 257-273 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Jef L. Teugels
Keyword(s):  
Random Walk ◽  
Waiting Time ◽  
Finite Time ◽  
Risk Theory ◽  
Ruin Probabilities ◽  
Exponential Behavior ◽  
Exponential Estimates ◽  
Insurance Portfolio ◽  
Actual Size ◽  
Dependence Structures

We consider an insurance portfolio situation in which there is possible dependence between the waiting time for a claim and its actual size. By employing the underlying random walk structure we obtain explicit exponential estimates for infinite- and finite-time ruin probabilities in the case of light-tailed claim sizes. The results are illustrated in several examples, worked out for specific dependence structures.

Sign in / Sign up

Export Citation Format

Share Document