scholarly journals Limit distributions for the ratio of the random sum of squares to the square of the random sum with applications to risk measures

2006 ◽  
Vol 80 (94) ◽  
pp. 219-240 ◽  
Author(s):  
Sophie Ladoucette ◽  
Jef Teugels
Keyword(s):  
Coefficient Of Variation ◽  
Counting Process ◽  
Risk Measures ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Random Sum ◽  
Convergence Conditions ◽  
Sample Coefficient Of Variation ◽  
Sample Dispersion ◽  
Erratic Behavior

Let {X1,X2, . . .} be a sequence of independent and identically distributed positive random variables of Pareto-type and let {N(t); t >_ 0} be a counting process independent of the Xi?s. For any fixed t> _ 0, define: TN(t) := X2 1 + X2 2 + ? ? ? + X2N (t) (X1 + X2 + ? ? ? + XN(t))2 if N(t) >_ 1 and TN(t) := 0 otherwise. We derive limits in distribution for TN(t) under some convergence conditions on the counting process. This is even achieved when both the numerator and the denominator defining TN(t) exhibit an erratic behavior (EX1 = ?) or when only the numerator has an erratic behavior (EX1 < ? and EX2 1 = ?). Armed with these results, we obtain asymptotic properties of two popular risk measures, namely the sample coefficient of variation and the sample dispersion. References.

scholarly journals Domains of attraction of the real random vector (X,X2) and applications

2009 ◽  
Vol 86 (100) ◽  
pp. 41-53
Author(s):  
Edward Omey ◽  
Gulck van
Keyword(s):  
Asymptotic Behaviour ◽  
Random Vector ◽  
Coefficient Of Variation ◽  
Domain Of Attraction ◽  
Stable Law ◽  
Domains Of Attraction ◽  
Full Discussion ◽  
Definition Of ◽  
Sample Coefficient Of Variation ◽  
Sample Dispersion

Many statistics are based on functions of sample moments. Important examples are the sample variance s2(n), the sample coefficient of variation SV (n), the sample dispersion SD(n) and the non-central t-statistic t(n). The definition of these quantities makes clear that the vector defined by (?ni=1Xi, ?ni=1Xi2)plays an important role. In the paper we obtain conditions under which the vector (X,X2) belongs to a bivariate domain of attraction of a stable law. Applying simple transformations then leads to a full discussion of the asymptotic behaviour of SV(n) and t(n).

The tail behaviour of a random sum of subexponential random variables and vectors

Extremes ◽  
2007 ◽  
Vol 10 (1-2) ◽  
pp. 21-39 ◽  
Author(s):  
D. J. Daley ◽  
Edward Omey ◽  
Rein Vesilo
Keyword(s):  
Random Variables ◽  
Random Sum ◽  
Tail Behaviour

On the percentage points of the sample coefficient of variation

Biometrika ◽  
1968 ◽  
Vol 55 (3) ◽  
pp. 580-581 ◽  
Author(s):  
BORIS IGLEWICZ ◽  
RAYMOND H. MYERS ◽  
RICHARD B. HOWE
Keyword(s):  
Coefficient Of Variation ◽  
Percentage Points ◽  
Sample Coefficient Of Variation

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
Author(s):  
Sunder Sethuraman
Keyword(s):  
Distribution Function ◽  
Positive Integer ◽  
Order Statistics ◽  
Law Of Large Numbers ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Large Numbers ◽  
The Law ◽  
Sample Maximum ◽  
Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

A point-process approach to almost-sure behaviour of record values and order statistics

1996 ◽  
Vol 28 (02) ◽  
pp. 426-462 ◽  
Author(s):  
Charles M. Goldie ◽  
Ross A. Maller
Keyword(s):  
Order Statistics ◽  
Point Process ◽  
Relative Stability ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Record Values ◽  
Process Approach ◽  
Integral Conditions ◽  
Comprehensive Investigation ◽  
Trimmed Sums

Point-process and other techniques are used to make a comprehensive investigation of the almost-sure behaviour of partial maxima(the rth largest among a sample ofni.i.d. random variables), partial record valuesand differences and quotients involving them. In particular, we obtain characterizations of such asymptotic properties asa.s. for some finite constantc, ora.s. for some constantcin [0,∞], which tell us, in various ways, how quickly the sequences increase. These characterizations take the form of integral conditions on the tail ofF,which furthermore characterize such properties as stability and relative stability of the sequence of maxima. We also develop their relation to the large-sample behaviour of trimmed sums, and discuss some statistical applications.

On risk measuring in the variance-gamma model

2018 ◽  
Vol 35 (1-2) ◽  
pp. 23-33 ◽  
Author(s):  
Roman V. Ivanov
Keyword(s):  
Value At Risk ◽  
Hypergeometric Functions ◽  
Risk Measures ◽  
Random Variables ◽  
Gamma Model ◽  
Variance Gamma ◽  
Stochastic Volatilities ◽  
Investment Returns ◽  
Variance Gamma Model ◽  
Analytical Expressions

AbstractIn this paper, we discuss the problem of calculating the primary risk measures in the variance-gamma model. A portfolio of investments in a one-period setting is considered. It is supposed that the investment returns are dependent on each other. In terms of the variance-gamma model, we assume that there are relations in both groups of the normal random variables and the gamma stochastic volatilities. The value at risk, the expected shortfall and the entropic monetary risk measures are discussed. The obtained analytical expressions are based on values of hypergeometric functions.

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