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- Erhardsson, Torkel
(author)
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Strong memoryless times and rare events in Markov renewal point processes
- 2004
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In: Annals of Probability. - : Institute of Mathematical Statistics. - 0091-1798 .- 2168-894X. ; 32:3B, s. 2446-2462
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Journal article (peer-reviewed)abstract
- Let W be the number of points in (0, t] of a stationary finite-state Markov renewal point process. We derive a bound for the total variation distance between the distribution of W and a compound Poisson distribution. For any nonnegative random variable (, we construct a "strong memoryless time" zeta such that zeta - t is exponentially distributed conditional on {zeta less than or equal to t, zeta > t}, for each t. This is used to embed the Markov renewal point process into another such process whose state space contains a frequently observed state which represents loss of memory in the original process. We then write W as the accumulated reward of an embedded renewal reward process, and use a compound Poisson approximation error bound for this quantity by Erhardsson. For a renewal process, the bound depends in a simple way on the first two moments of the interrenewal time distribution, and on two constants obtained from the Radon-Nikodym derivative of the interrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.
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| 4. |
- Erhardsson, Torkel, 1966-
(author)
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Strong memoryless times and rare events in Markov renewal point processes
- 2004
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In: Annals of Probability. - 0091-1798 .- 2168-894X. ; 32:3B, s. 2446-2462
-
Journal article (peer-reviewed)abstract
- Let $W$ be the number of points in $(0,t]$ of a stationary finite-state Markov ren ewal point process. We derive a bound for the total variation distance between the distribution of $W$ and a compound Poisson distribution. For any nonnegative rand om variable $\zeta$ we construct a ``strong memoryless time'' $\hat\zeta$ such tha t $\zeta-t$ is exponentially distributed conditional on $\{\hat\zeta\leq t,\zeta>t \}$, for each $t$. This is used to embed the Markov renewal point process into ano ther such process whose state space contains a frequently observed state which rep resents loss of memory in the original process. We then write $W$ as the accumulat ed reward of an embedded renewal reward process, and use a compound Poisson approx imation error bound for this quantity by Erhardsson. For a renewal process, the bo und depends in a simple way on the first two moments of the interrenewal time dist ribution, and on two constants obtained from the Radon-Nikodym derivative of the i nterrenewal time distribution with respect to an exponential distribution. For a Poisson process, the bound is 0.
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- Gut, A, et al.
(author)
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Precise asymptotics in the law of the iterated logarithm
- 2000
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In: ANNALS OF PROBABILITY. - : INST MATHEMATICAL STATISTICS. - 0091-1798. ; 28:4, s. 1870-1883
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Journal article (peer-reviewed)abstract
- Let X, X-1, X-2,... be i.i random variables with mean 0 and positive, finite variance sigma (2), and set S-n = X-1 + ... + X-n, n greater than or equal to 1. Continuing earlier work related to strong laws, we prove the following analogs for the law of the
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- Klass, Michael J., et al.
(author)
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An improvement of Hoffmann-Jorgensen's inequality
- 2000
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In: Annals of Probability. - 0091-1798. ; 28:2, s. 851-862
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Journal article (peer-reviewed)abstract
- Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞.
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