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.

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.

Corrected diffusion approximations in certain random walk problems

1979 ◽  
Vol 11 (4) ◽  
pp. 701-719 ◽  
Author(s):  
D. Siegmund
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Diffusion Approximations ◽  
Infinite Time ◽  
Ladder Height ◽  
Correction Terms

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk

1997 ◽  
Vol 29 (03) ◽  
pp. 695-712 ◽  
Author(s):  
C. D. Fuh
Keyword(s):  
Random Walk ◽  
State Space ◽  
Stopping Times ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Markov Random ◽  
Markov Random Walk ◽  
Finite State ◽  
Correction Terms ◽  
Finite State Space

Let (X, S) = {(Xn , Sn ); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn ∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | X o = i}, and a two-barrier probability P{ST ≧b | X o = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.

Corrected diffusion approximations in certain random walk problems

1979 ◽  
Vol 11 (04) ◽  
pp. 701-719 ◽  
Author(s):  
D. Siegmund
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Diffusion Approximations ◽  
Infinite Time ◽  
Ladder Height ◽  
Correction Terms

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk

1997 ◽  
Vol 29 (3) ◽  
pp. 695-712 ◽  
Author(s):  
C. D. Fuh
Keyword(s):  
Random Walk ◽  
State Space ◽  
Stopping Times ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Markov Random ◽  
Markov Random Walk ◽  
Finite State ◽  
Correction Terms ◽  
Finite State Space

Let (X, S) = {(Xn, Sn); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | Xo= i}, and a two-barrier probability P{ST ≧b | Xo = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.

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 &lt; 0, letF(x) =P(Xk≦x),v(u) =inf{n:Sn&gt;u} and assume that for some γ &gt; 0is a proper distribution with finite meanVarious limit theorems for functionals ofX1,· · ·,Xv(u)are derived subject to conditioning upon {v(u)&lt; ∞} 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 On the Maximum of a Random Walk with Small Negative Drift

1983 ◽  
Vol 11 (3) ◽  
pp. 491-505 ◽  
Author(s):  
Michael J. Klass
Keyword(s):  
Random Walk ◽  
Negative Drift

scholarly journals Uniform renewal theory with applications to expansions of random geometric sums

2007 ◽  
Vol 39 (4) ◽  
pp. 1070-1097 ◽  
Author(s):  
J. Blanchet ◽  
P. Glynn
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Renewal Theory ◽  
Random Variable ◽  
Higher Order ◽  
Order Correction ◽  
Diffusion Approximations ◽  
Geometric Sums ◽  
Correction Terms

Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

scholarly journals Exponential Family Graph Embeddings

2020 ◽  
Vol 34 (04) ◽  
pp. 3357-3364
Author(s):  
Abdulkadir Celikkanat ◽  
Fragkiskos D. Malliaros
Keyword(s):  
Random Walk ◽  
Exponential Family ◽  
Representation Learning ◽  
Learning Problems ◽  
Interaction Patterns ◽  
Network Representation ◽  
Learning Tasks ◽  
Learning Techniques ◽  
Real World Datasets ◽  
Low Dimensional

Representing networks in a low dimensional latent space is a crucial task with many interesting applications in graph learning problems, such as link prediction and node classification. A widely applied network representation learning paradigm is based on the combination of random walks for sampling context nodes and the traditional Skip-Gram model to capture center-context node relationships. In this paper, we emphasize on exponential family distributions to capture rich interaction patterns between nodes in random walk sequences. We introduce the generic exponential family graph embedding model, that generalizes random walk-based network representation learning techniques to exponential family conditional distributions. We study three particular instances of this model, analyzing their properties and showing their relationship to existing unsupervised learning models. Our experimental evaluation on real-world datasets demonstrates that the proposed techniques outperform well-known baseline methods in two downstream machine learning tasks.

Sign in / Sign up

Export Citation Format

Share Document