Chapter 7Extreme Values Statistics for Markov Chains with Applications to Finance and Insurance

Patrice Bertail1, Stéphan Clémençon2 and Charles Tillier1

1MODAL'X, Université Paris-Ouest, Nanterre, France

2TSI, TelecomParisTech, Paris, France

AMS 2000 Mathematics Subject Classification: 60G70, 60J10, 60K20.

7.1 Introduction

Extremal events for (strongly or weakly) dependent data have received an increasing attention in the statistical literature in the last past years (see (Newell, 1964); (Loynes, 1965); (O'Brien, 1974), (O'Brien, 1987); (Hsing, 1988), (Hsing, 1991), (Hsing, 1993); (Resnick and Stărică, 1995); (Rootzén, 2009), for instance). A major issue for evaluating risks and understanding extremes and their possible replications is to take into account some dependencies. Indeed, whereas extreme values naturally occur in an isolated fashion in the identically independent distributed (i.i.d.) setup, since extreme values may be highly correlated, they generally tend to take place in small clusters for weakly dependent sequences. Most methods for statistical analysis of extremal events in weakly dependent setting rely on (fixed length) blocking techniques, which consist, roughly speaking, in dividing an observed data series into (overlapping or nonoverlapping) blocks of fixed length. Examining how extreme values occur over these data segments allows to capture the tail and the dependency structure of extreme values.

As originally pointed out in Rootzén (1988), the extremal ...

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