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We give a new sufficient condition for a continuous distribution to be completely mixable, and we use this condition to show that the worst-possible value-at-risk for the sum of d inhomogeneous risks is equivalent to the worst-possible expected shortfall under the same marginal assumptions, in...
Persistent link: https://www.econbiz.de/10011046639
Recent crises in the financial industry have shown weaknesses in the modeling of Risk-Weighted Assets (RWAs). Relatively minor model changes may lead to substantial changes in the RWA numbers. Similar problems are encountered in the Value-at-Risk (VaR)-aggregation of risks. In this article, we...
Persistent link: https://www.econbiz.de/10011030553
We introduce the concept of an extremely negatively dependent (END) sequence of random variables with a given common marginal distribution. An END sequence has a partial sum which, subtracted by its mean, does not diverge as the number of random variables goes to infinity. We show that an END...
Persistent link: https://www.econbiz.de/10011208475
Following the results of Rüschendorf and Uckelmann (2002) [20], we introduce the completely mixable distributions on and prove that the distributions with monotone density and moderate mean are completely mixable. Using this method, we solve the minimization problem for convex functions f...
Persistent link: https://www.econbiz.de/10009194649
In various frameworks, to assess the joint distribution of a k-dimensional random vector X=(X1,…,Xk), one selects some putative conditional distributions Q1,…,Qk. Each Qi is regarded as a possible (or putative) conditional distribution for Xi given (X1,…,Xi−1,Xi+1,…,Xk). The Qi are...
Persistent link: https://www.econbiz.de/10011041946
Let be the space of real cadlag functions on with finite limits at ±[infinity], equipped with uniform distance, and let Xn be the empirical process for an exchangeable sequence of random variables. If regarded as a random element of , Xn can fail to converge in distribution. However, in this...
Persistent link: https://www.econbiz.de/10008874727
Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>L</mi> </math> </EquationSource> </InlineEquation> be a linear space of real random variables on the measurable space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$(\varOmega ,\mathcal {A})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="italic">Ω</mi> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>. Conditions for the existence of a probability <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$P$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>P</mi> </math> </EquationSource> </InlineEquation> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\mathcal {A}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="script">A</mi> </math> </EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$E_P|X|\infty $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>E</mi> <mi>P</mi> </msub> <mrow> <mo stretchy="false">|</mo> <mi>X</mi> <mo stretchy="false">|</mo> </mrow> <mo></mo> <mi>∞</mi> </mrow> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$E_P(X)=0$$</EquationSource> <EquationSource Format="MATHML">...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation>
Persistent link: https://www.econbiz.de/10011151903
Persistent link: https://www.econbiz.de/10005613357
For an exchangeable sequence of random variables, almost surely, the difference between the empirical and the predictive distribution functions converges to zero uniformly.
Persistent link: https://www.econbiz.de/10005313944
Let $(S,\mathcal{B},\Gamma)$ and $(T,\mathcal{C},Q)$ be probability spaces, with $Q$ nonatomic, and $\mathcal{H}=\{H\in\mathcal{C}:Q(H)0\}$. In some economic models, the following conditional law of large numbers (LLN) is requested. There are a probability space $(\Omega,\mathcal{A},P)$ and a...
Persistent link: https://www.econbiz.de/10008502024