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Pietro Rigo

Publications and source records attributed to Pietro Rigo.

30 records · Page 2Linked to original sources
arXiv2014

Two versions of the fundamental theorem of asset pricing

Let $L$ be a convex cone of real random variables on the probability space $(Ω,\mathcal{A},P_0)$. The existence of a probability $P$ on $\mathcal{A}$ such that $$ P \sim P_0,\quad E_P \abs{X}< \infty\, \text{ and } \, E_P(X) \leq 0\, \text{ for all }X \in L $$ is investigated. Two results are provided. In the first, $P$ is a finitely additive probability, while $P$ is $σ$-additive in the second. If $L$ is a linear space then $-X\in L$ whenever $X\in L$, so that $E_P(X)\leq 0$ turns into $E_P(X)=0$. Hence, the results apply to various significant frameworks, including equivalent martingale measures and equivalent probability measures with given marginals.

math.PR↗
arXiv2013

Exchangeable sequences driven by an absolutely continuous random measure

Let $S$ be a Polish space and $(X_n:n\geq1)$ an exchangeable sequence of $S$-valued random variables. Let $α_n(\cdot)=P(X_{n+1}\in \cdot\mid X_1,\...,X_n)$ be the predictive measure and $α$ a random probability measure on $S$ such that $α_n\stackrel{\mathrm{weak}}{\longrightarrow}α$ a.s. Two (related) problems are addressed. One is to give conditions for $α\llλ$ a.s., where $λ$ is a (nonrandom) $σ$-finite Borel measure on $S$. Such conditions should concern the finite dimensional distributions $\mathcal {L}(X_1,\...,X_n)$, $n\geq1$, only. The other problem is to investigate whether $\Vert\alp ha_n-α\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$, where $\Vert\cdot\Vert$ is total variation norm. Various results are obtained. Some of them do not require exchangeability, but hold under the weaker assumption that $(X_n)$ is conditionally identically distributed, in the sense of [Ann. Probab. 32 (2004) 2029-2052].

math.PR↗
arXiv2012

An Anscombe-type theorem

Let (X_n) be a sequence of random variables (with values in a separable metric space) and (N_n) a sequence of random indices. Conditions for X_{N_n} to converge stably (in particular, in distribution) are provided. Some examples, where such conditions work but those already existing fail, are given as well. Key words and phrases: Anscombe theorem, Exchangeability, Random indices, Random sums, Stable convergence

math.PR↗
arXiv2010

Finitely additive equivalent martingale measures

Let $L$ be a linear space of real bounded random variables on the probability space $(Ω,\mathcal{A},P_0)$. There is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\sim P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $c\,E_Q(X)\leq\text{ess sup}(-X)$, $X\in L$, for some constant $c>0$ and (countably additive) probability $Q$ on $\mathcal{A}$ such that $Q\sim P_0$. A necessary condition for such a $P$ to exist is $\bar{L-L_\infty^+}\,\cap L_\infty^+=\{0\}$, where the closure is in the norm-topology. If $P_0$ is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\ll P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $\text{ess sup}(X)\geq 0$ for all $X\in L$.

math.PR↗
arXiv2010

Rate of convergence of predictive distributions for dependent data

This paper deals with empirical processes of the type \[C_n(B)=\sqrt{n}\{μ_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},\] where $(X_n)$ is a sequence of random variables and $μ_n=(1/n)\sum_{i=1}^nδ_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029--2052]). By such conditions, in some relevant situations, one obtains that $\sup_B|C_n(B)|\stackrel{P}{\to}0$ or even that $\sqrt{n}\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.

math.ST↗
arXiv2009

Central Limit Theorems for Multicolor Urns with Dominated Colors

An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some statistical applications are discussed.

math.PR↗
arXiv2009

Trivial intersection of $σ$-fields and Gibbs sampling

Let $(Ω,\mathcal{F},P)$ be a probability space and $\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying $P(F)\in\{0,1\}$. For each $\mathcal{G}\subset\mathcal{F}$, define $\overline{\mathcal{G}}=σ(\mathcal{G}\cup\mathcal {N})$. Necessary and sufficient conditions for $\overline{\mathcal{A}}\cap\overline{\mathcal{B}}=\overline {\mathcal {A}\cap\mathcal{B}}$, where $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are sub-$σ$-fields, are given. These conditions are then applied to the (two-component) Gibbs sampler. Suppose $X$ and $Y$ are the coordinate projections on $(Ω,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal {U}\otimes \mathcal{V})$ where $(\mathcal{X},\mathcal{U})$ and $(\mathcal{Y},\mathcal{V})$ are measurable spaces. Let $(X_n,Y_n)_{n\geq0}$ be the Gibbs chain for $P$. Then, the SLLN holds for $(X_n,Y_n)$ if and only if $\overline{σ(X)}\cap\overline{σ(Y)}=\mathcal{N}$, or equivalently if and only if $P(X\in U)P(Y\in V)=0$ whenever $U\in\mathcal{U}$, $V\in\mathcal{V}$ and $P(U\times V)=P(U^c\times V^c)=0$. The latter condition is also equivalent to ergodicity of $(X_n,Y_n)$, on a certain subset $S_0\subsetΩ$, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated and $P$ absolutely continuous with respect to a product measure.

math.PR↗
arXiv2007

0--1 laws for regular conditional distributions

Let $(Ω,\mathcal{B},P)$ be a probability space, $\mathcal{A}\subset\mathcal{B}$ a sub-$σ$-field, and $μ$ a regular conditional distribution for $P$ given $\mathcal{A}$. Necessary and sufficient conditions for $μ(ω)(A)$ to be 0--1, for all $A\in\mathcal{A}$ and $ω\in A_0$, where $A_0\in\mathcal{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $\mathcal{A}$ is a tail sub-$σ$-field. Let $H(ω)$ denote the $\mathcal{A}$-atom including the point $ω\inΩ$. Necessary and sufficient conditions for $μ(ω)(H(ω))$ to be 0--1, for all $ω\in A_0$, are also given. If $(Ω,\mathcal{B})$ is a standard space, the latter 0--1 law is true for various classically interesting sub-$σ$-fields $\mathcal{A}$, including tail, symmetric, invariant, as well as some sub-$σ$-fields connected with continuous time processes.

math.PR↗
arXiv2007

A conditional 0-1 law for the symmetric sigma-field

Let (Ω,\mathcal{B},P) be a probability space, \mathcal{A} a sub-sigma-field of \mathcal{B}, and μa regular conditional distribution for P given \mathcal{A}. For various, classically interesting, choices of \mathcal{A} (including tail and symmetric) the following 0-1 law is proved: There is a set A_0 in \mathcal{A} such that P(A_0)=1 and μ(ω)(A) is 0 or 1 for all A in \mathcal{A} and ωin A_0. Provided \mathcal{B} is countably generated (and certain regular conditional distributions exist), the result applies whatever P is.

math.PR↗
arXiv2004

Limit theorems for a class of identically distributed random variables

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_{n\geq 1} is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (G_n)_{n\geq 0}, if it is adapted to (G_n)_{n\geq 0} and, for each n\geq 0, (X_k)_{k>n} is identically distributed given the past G_n. In case G_0={\varnothing,Ω} and G_n=σ(X_1,...,X_n), a result of Kallenberg implies that (X_n)_{n\geq 1} is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown that (X_n)_{n\geq 1} is exchangeable if and only if (X_{τ(n)})_{n\geq 1} is c.i.d. for any finite permutation τof {1,2,...}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-σ-field. Moreover, (1/n)\sum_{k=1}^nX_k converges a.s. and in L^1 whenever (X_n)_{n\geq 1} is (real-valued) c.i.d. and E[| X_1| ]<\infty. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is E[X_{n+1}| G_n]. For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.

math.PR↗