Large deviations of heavy-tailed random sums with applications in insurance and finance

1997 ◽  
Vol 34 (2) ◽  
pp. 293-308 ◽  
Author(s):  
C. Klüppelberg ◽  
T. Mikosch
Keyword(s):  
Distribution Function ◽  
Large Deviation ◽  
Random Variables ◽  
Compound Poisson Process ◽  
Random Sum ◽  
Random Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Heavy Tailed ◽  
The Right

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

Large deviations of heavy-tailed random sums with applications in insurance and finance

1997 ◽  
Vol 34 (02) ◽  
pp. 293-308 ◽  
Author(s):  
C. Klüppelberg ◽  
T. Mikosch
Keyword(s):  
Distribution Function ◽  
Large Deviation ◽  
Random Variables ◽  
Compound Poisson Process ◽  
Random Sum ◽  
Random Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Heavy Tailed ◽  
The Right

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

The central limit problem for trimmed sums

1987 ◽  
Vol 102 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Limit Problem ◽  
Absolute Value ◽  
Trimmed Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Central Limit Problem ◽  
Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

1984 ◽  
Vol 21 (1) ◽  
pp. 98-107 ◽  
Author(s):  
Minoru Yoshida
Keyword(s):  
Distribution Function ◽  
Stopping Time ◽  
Random Variables ◽  
Secretary Problem ◽  
Independent Random Variables ◽  
Distribution Law ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Random Moment ◽  
The Moment

Before some random moment θ, independent identically distributed random variables x1, · ··, xθ–1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ+1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's.In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.

A Note on Normal Attraction to a Stable Law

1971 ◽  
Vol 14 (3) ◽  
pp. 451-452
Author(s):  
M. V. Menon ◽  
V. Seshadri
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Stable Law ◽  
Common Distribution ◽  
Normal Attraction ◽  
Common Distribution Function ◽  
The Common ◽  
Necessary And Sufficient

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

Asymptotic distributions of extremes of extremal Markov sequences

1994 ◽  
Vol 31 (01) ◽  
pp. 256-261
Author(s):  
S. R. Adke ◽  
C. Chandran
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Arbitrary Distribution ◽  
Extreme Statistics ◽  
Common Distribution ◽  
Common Distribution Function ◽  
The Common

Let {ξ n , n ≧1} be a sequence of independent real random variables, F denote the common distribution function of identically distributed random variables ξ n , n ≧1 and let ξ 1 have an arbitrary distribution. Define Xn+ 1 = k max(Xn, ξ n +1), Yn + 1 = max(Yn, ξ n +1) – c, Un +1 = l min(Un, ξ n +1), Vn+ 1 = min(Vn, ξ n +1) + c, n ≧ 1, 0 < k < 1, l > 1, 0 < c < ∞, and X 1 = Υ 1 = U 1 = V 1 = ξ 1. We establish conditions under which the limit law of max(X 1, · ··, Xn ) coincides with that of max(ξ 2, · ··, ξ n+ 1) when both are appropriately normed. A similar exercise is carried out for the extreme statistics max(Y 1, · ··, Yn ), min(U 1,· ··, Un ) and min(V 1, · ··, Vn ).

A martingale approach to strong convergence of the number of records

2001 ◽  
Vol 33 (4) ◽  
pp. 864-873 ◽  
Author(s):  
Raúl Gouet ◽  
F. Javier López ◽  
Miguel San Miguel
Keyword(s):  
Distribution Function ◽  
Strong Convergence ◽  
Wide Class ◽  
Random Variables ◽  
Strong Law ◽  
Martingale Approach ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Independent Identically Distributed

Let (Xn) be a sequence of independent, identically distributed random variables, with common distribution function F, possibly discontinuous. We use martingale arguments to connect the number of upper records from (Xn) with sums of minima of related random variables. From this relationship we derive a general strong law for the number of records for a wide class of distributions F, including geometric and Poisson.

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