PROCESO DE REPARACION Y/O MANTENIMIENTO DE

processes

Large deviation theory has been developed in various fields, ranging from queuing the- ory to statistics and from interacting particle systems to superexponential estimates, see

Varadhan (2008) for a survey of large deviation. In risk models, for example, large devi- ation principles are widely used to estimate the asymptotic behavior and the exponential rate of the total claims probability (Kl¨uppelberg and Mikosch, 1997; Tang et al., 2001). I am interested in the application of large deviation principles to the maximum of the stochastic processes, which is naturally related to the problems about the random loca- tions discussed in previous chapters, including the location of the path supremum, the first hitting time to high levels, etc. For example, for a stochastic process, one can consider how the distributions of first hitting times to a level over an interval evolve when both the length of the interval and the level grow to infinity in an appropriate way. I expect that large deviation principles to be a powerful tool to investigate the asymptotic behaviors of the set of first hitting times. This can be regarded as a compatibility problem of the distributions of this random location with extreme values and larger and larger intervals.

Index

ϕ-stationary intrinsic random location,4,

89, 100 f -divergence, 34, 36 almost compatible, 2,20 compatible,2, 9, 24,29, 30 conditionally atomless, 21, 24,30 control measure, 4,96 convex order, 2,12 ergodic process, 55 exchangeable process,116

first-time intrinsic location functional, 3,

78, 80

heterogeneity order, 8, 14

intrinsic location functional, 2, 3, 53, 58

intrinsic random location, 4,88

invariant intrinsic location functional, 3,

69, 72

joint mix,81

large deviation principle,5, 117

max-stable process,5, 116

partially ordered random set representation,60

relative stationary process, 52

stationary intrinsic random location,89,

96

stochastic order, 36

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