Large Deviations for Processes with Independent Increments

Lynch, James; Sethuraman, Jayaram

doi:10.1214/aop/1176992161

Cited by 145 publications

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“…Indeed, for this rate function, Mogulskii refers to the paper by Lynch and Sethuraman ( [2]), and according to the latter, the value for  …”

Section: Notation Previous Results and The Rate Functionmentioning

confidence: 99%

“…random variables with generating functions finite on a neighborhood of the origin. In [2], Lynch and Sethuraman gave large deviations results for stochastic processes with independent and stationary increments. The analysis was done on the space of functions of bounded variation on   0,1 endowed with the weak * -topology.…”

Section: Introductionmentioning

confidence: 99%

“…That is the general approach we follow for Theorem 1: we examine the LDP for a sequence of risk processes with respect to the Skorohod topology under suitable time and scale modifications. We follow Mogulskii's approach [1], whose results are based on Lynch and Sethuraman [2] to obtain an LDP for risk processes on rized counterpart, proving the LDP, and then dealing with the random time via exponential equivalence of the times. In this direction there is work by Feng and Kurtz, [9] and our Theorem 2 for aggregate claims processes.…”

Section: Introductionmentioning

confidence: 99%

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Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims

García¹,

Meda²

2012

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In this paper we examine the large deviations principle (LDP) for sequences of classic Cramér-Lundberg risk processes under suitable time and scale modifications, and also for a wide class of claim distributions including (the non-superexponential) exponential claims. We prove two large deviations principles: first, we obtain the LDP for risk processes onwith the Skorohod topology. In this case, we provide an explicit form for the rate function, in which the safety loading condition appears naturally. The second theorem allows us to obtain the LDP for Aggregate Claims processes on with a different time-scale modification. As an application of the first result we estimate the ruin probability, and for the second result we work explicit calculations for the case of exponential claims.

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“…Indeed, for this rate function, Mogulskii refers to the paper by Lynch and Sethuraman ( [2]), and according to the latter, the value for  …”

Section: Notation Previous Results and The Rate Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims

García¹,

Meda²

2012

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“…The case of processes with independent increments was considered in [13][14][15]. The paper is organized as follows: in the rest of Section 1 we outline our approach to obtaining the large deviation principle.…”

Section: Main Notationsmentioning

confidence: 99%

“…A sequence X n = (X n t ) t≥0 , n ≥ 1 of processes with paths in D is said to be exponential tight if for any C there exists a compact K C in D such that lim n n −1 log P (X n ∈ D \ K C ) ≤ −C. Other names for exponential tightness are "large deviation tightness" [15] and "strong tightness" [19].…”

Section: Main Notationsmentioning

confidence: 99%

Limit theorems on large deviations for semimartingales

Liptser

Pukhalskii

1992

Stochastics and Stochastic Reports

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Abstract. We consider a sequence X n = (X n t ) t≥0 , n ≥ 1 of semimartingales. Each X n is a weak solution to an Itô equation with respect to a Wiener process and a Poissonian martingale measure and is in general non-Markovian process. For this sequence, we prove the large deviation principle in the Skorokhod space D = D [0,∞) . We use a new approach based on of exponential tightness. This allows us to establish the large deviation principle under weaker assumptions than before.Main notationsthe set of stopping times (relative to a filtration F not exceeding L;the Skorokhod space of all right continuous, having left hand limits real valued functionsthe space of all right continuous functions fromthe Lindvall-Skorokhod metric on D;" P − → ′′ , convergence in probability; x ∧ y = min(x, y), x ∨ y = max(x, y);1991 Mathematics Subject Classification. 60F10.

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Colliding stacks: A large deviations analysis

Maier

1991

Random Struct Algorithms

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We analyze the performance of a prototypical scheme for shared storage allocation: two initially empty stacks evolving in a contiguous block of memory of size m. We treat the case in which the stacks are more likely to shrink than grow, but with the probabilities of insertion and deletion allowed to depend arbitrarily on stack height as a fraction of m. New results are obtained on the m+ m asymptotics of the stack collision time, and of the final stack heights. The results of Wentzell and Freidlin on the large deviations of Markov chains are used, and the relation of their formalism to the Hamiltonian formulation of classical mechanics is emphasized. Certain results on higher-order asymptotics follow from WKB expansions.

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Large Deviations for Processes with Independent Increments

Cited by 145 publications

References 0 publications

Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims

Applications of Mogulskii, and Kurtz-Feng Large Deviation Results to Risk Reserve Processes with Aggregate Claims

Limit theorems on large deviations for semimartingales

Colliding stacks: A large deviations analysis

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