1,721,008 research outputs found
Finitely Additive Equivalent Martingale Measures
Full text linkLet L be a linear space of real bounded random variables on the probability space (omega,A, P0). There is a finitely additive probability P on A, such that P tilde P0 and EP (X) = 0 for all X in L, if and only if cEQ(X) = ess sup(-X), X in L, for some constant c > 0 and (countably additive) probability Q on A such that Q tilde P0. A necessary condition for such a P to exist is L - L+(inf) n L+(inf) = {0}, where the closure is in the norm-topology. If P0 is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability P on A, such that PArbitrage, de Finetti’s coherence principle, equivalent martingale measure, finitely additive probability, fundamental theorem of asset pricing.
Some duality results for equivalence couplings and total variation
Full text linkLet be a measurable space and . Suppose that and the relation on defined as is reflexive, symmetric and transitive. Following \cite{JAFFE}, say that is strongly dualizable if there is a sub--field such that
\min_{P\in\Gamma(\mu,\nu)}(1-P(E))=\max_{A\in\mathcal{G}}\,\abs{\mu(A)-\nu(A)}
for all probabilities and on . This paper investigates strong duality. Essentially, it is shown that is strongly dualizable provided some mild modifications are admitted. Let be the -invariant sub--field of . One result is that, for all probabilities and on , there is a probability on such that
\begin{gather*}
\nu_0=\nu\text{ on }\mathcal{G}_0\quad\text{and}\quad\min_{P\in\Gamma(\mu,\nu_0)}(1-P(E))=\max_{A\in\mathcal{G}_0}\,\abs{\mu(A)-\nu(A)}.
\end{gather*}
In the other results, is a standard Borel space and the min over is replaced by the inf over in the definition of strong duality. Then, is strongly dualizable provided is allowed to depend on or it is taken to be the universally measurable version of the -invariant -field
Exchangeable Sequences Driven by an Absolutely Continuous Random Measure
Full text linkLet S be a Polish space and (Xn : n = 1) an exchangeable sequence of S-valued random variables. Let an(·) = P( Xn+1 in · | X1, . . . ,Xn) be the predictive measure and a a random probability measure on S such that an (weak) --> a a.s.. Two (related) problems are addressed. One is to give conditions for a 0, where ||·|| is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that (Xn) is conditionally identically distributed, in the sense of [2].Conditional identity in distribution, Exchangeability, Predictive measure, Random probability measure.
Skorohod Representation Theorem Via Disintegrations
Full text linkLet (µn : n >= 0) be Borel probabilities on a metric space S such that µn -> µ0 weakly. Say that Skorohod representation holds if, on some probability space, there are S-valued random variables Xn satisfying Xn - µn for all n and Xn -> X0 in probability. By Skorohod’s theorem, Skorohod representation holds (with Xn -> X0 almost uniformly) if µ0 is separable. Two results are proved in this paper. First, Skorohod representation may fail if µ0 is not separable (provided, of course, non separable probabilities exist). Second, independently of µ0 separable or not, Skorohod representation holds if W(µn, µ0) -> 0 where W is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a W is a version of Wasserstein distance which can be defined for any metric space S satisfying a mild condition. To prove the quoted results (and to define W), disintegrable probability measures are fundamental.Disintegration, Separable probability measure, Skorohod representation theorem, Wasserstein distance, Weak convergence of probability measures.
Atomic Intersection of s-Fields and Some of Its Consequences
Full text linkLet (omega,F,P) be a probability space. For each G in F, define G as the s-field generated by G and those sets f in F satisfying P(f) in {0, 1}. Conditions for P to be atomic on the intersection of the complements of Ai for i=1,..,k, with A1, . . . ,Ak in F sub-s-fields, are given. Conditions for P to be 0-1-valued on the intersection of the complements of Ai for i=1,..,k are given as well. These conditions are useful in various fields, including Gibbs sampling, iterated conditional expectations and the intersection property.Atomic probability measure, Gibbs sampling, Graphical models, Intersection property, Iterated conditional expectations.
On the existence of continuous processes with given one-dimensional distributions
No full textLet be the collection of Borel probability measures on , equipped with the weak topology, and let be a continuous map. Say that is presentable if , , for some real process with continuous paths. It may be that fails to be presentable. Hence, firstly, conditions for presentability are given. For instance, is presentable if is supported by an interval (possibly, by a singleton) for all but countably many . Secondly, assuming presentable, we investigate whether the quantile process induced by has continuous paths. The latter is defined, on the probability space Lebesgue measure, by egin{gather*} Q_t(alpha)=inf,igl{xinmathbb{R}:mu_t(-infty,x]gealphaigl}quadquad ext{for all }tin [0,1] ext{ and }alphain (0,1). end{gather*} A few open problems are discussed as well
Convergence in Total Variation to a Mixture of Gaussian Laws
No full textIt is not unusual that Xn⟶distVZ where Xn, V, Z are real random variables, V is independent of Z and Z∼N(0,1). An intriguing feature is that PVZ∈A=EN(0,V2)(A) for each Borel set A⊂R, namely, the probability distribution of the limit VZ is a mixture of centered Gaussian laws with (random) variance V2. In this paper, conditions for dTV(Xn,VZ)→0 are given, where dTV(Xn,VZ) is the total variation distance between the probability distributions of Xn and VZ. To estimate the rate of convergence, a few upper bounds for dTV(Xn,VZ) are given as well. Special attention is paid to the following two cases: (i) Xn is a linear combination of the squares of Gaussian random variables; and (ii) Xn is related to the weighted quadratic variations of two independent Brownian motions
Central Limit Theorems For Multicolor Urns With Dominated Colors
Full text linkAn urn contains balls of d >= 2 colors. At each time n >= 1, a ball is drawn and then replaced together with a random number of balls of the same color. Let An =diag (An,1, . . . ,An,d) be the n-th reinforce matrix. Assuming EAn,j = EAn,1 for all n and j, a few CLT’s are available for such urns. In real problems, however, it is more reasonable to assume EAn,j = EAn,1 whenever n >= 1 and 1 limsup EAn,j whenever j > d0, for some integer 1 = 1) is independent but need not be identically distributed. Some statistical applications are given as well.Central limit theorem, Clinical trials, Random probability measure, Stable convergence, Urn model.
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