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Pietro Rigo

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Clark University

University of Durham

Sayantan Choudhury

SGT UNIVERSITY GURGAON

Mariafelicia De Laurentis

Università degli Studi di Napoli "Federico II"

Vassilios Bogiatzis/ Βασίλης Μπογιατζής

Panteion University of Social and Political Sciences

Adebayo D Agunbiade

Olabisi Onabanjo University, Ago-Iwoye, Nigeria

Konrád Miklós

Institute of History of Hungarian Academy of Sciences

Stefano Cavazza

Università di Bologna

Patrick Dasgupta

University of Delhi

Nikos Mantzakouras

National & Kapodistrian University of Athens

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Papers by Pietro Rigo

Some Skorohod-type results

Decisions in economics and finance, Jul 1, 2024

Some duality results for equivalence couplings and total variation

Electronic communications in probability, 2024

Let (Ω, F) be a measurable space and E ⊂ Ω × Ω. Suppose that E ∈ F ⊗ F and the relation on Ω defi... more Let (Ω, F) be a measurable space and E ⊂ Ω × Ω. Suppose that E ∈ F ⊗ F and the relation on Ω defined as x ∼ y ⇔ (x, y) ∈ E is reflexive, symmetric and transitive. Following [7], say that E is strongly dualizable if there is a sub-σ-field G ⊂ F such that min P ∈Γ(µ,ν) (1 − P (E)) = max A∈G |µ(A) − ν(A)| for all probabilities µ and ν on F. This paper investigates strong duality. Essentially, it is shown that E is strongly dualizable provided some mild modifications are admitted. Let G0 be the E-invariant sub-σ-field of F. One result is that, for all probabilities µ and ν on F, there is a probability ν0 on F such that ν0 = ν on G0 and min P ∈Γ(µ,ν 0)

Quantitative bounds in the central limit theorem for m-dependent random variables

Alea, 2024

For each n ≥ 1, let X n,1 ,. .. , X n,Nn be real random variables and S n = Nn i=1 X n,i. Let m n... more For each n ≥ 1, let X n,1 ,. .. , X n,Nn be real random variables and S n = Nn i=1 X n,i. Let m n ≥ 1 be an integer. Suppose (X n,1 ,. .. , X n,Nn) is m n-dependent, E(X ni) = 0, E(X 2 ni) < ∞ and σ 2 n := E(S 2 n) > 0 for all n and i. Then, d W S n σ n , Z ≤ 30 c 1/3 + 12 U n (c/2) 1/2 for all n ≥ 1 and c > 0, where d W is Wasserstein distance, Z a standard normal random variable and U n (c) = m n σ 2 n Nn i=1 E X 2 n,i 1 |X n,i | > c σ n /m n. Among other things, this estimate of d W S n /σ n , Z yields a similar estimate of d T V S n /σ n , Z where d T V is total variation distance.

Convergence in Total Variation to a Mixture of Gaussian Laws

Mathematics, Jun 11, 2018

It is not unusual that X n dist −→ VZ where X n , V, Z are real random variables, V is independen... more It is not unusual that X n dist −→ VZ where X n , V, Z are real random variables, V is independent of Z and Z ∼ N (0, 1). An intriguing feature is that P VZ ∈ A = E N (0, V 2)(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 V 2. In this paper, conditions for d TV (X n , VZ) → 0 are given, where d TV (X n , VZ) is the total variation distance between the probability distributions of X n and VZ. To estimate the rate of convergence, a few upper bounds for d TV (X n , VZ) are given as well. Special attention is paid to the following two cases: (i) X n is a linear combination of the squares of Gaussian random variables; and (ii) X n is related to the weighted quadratic variations of two independent Brownian motions.

Coherent Inferences and Improper Priors

Annals of Statistics, Sep 1, 1994

A Glivenko-Cantelli theorem for exchangeable random variables

Statistics & Probability Letters, Apr 1, 1997

ELSEVIER Statistics Probability Letters 32 (1997) 385391 A GlivenkoCantelli theorem for exchangea... more ELSEVIER Statistics Probability Letters 32 (1997) 385391 A GlivenkoCantelli theorem for exchangeable random variables Patrizia Berti *, Pietro Rigo Dipartimento di Statistica "G. Parenti "', Universit Deffli Studi di Firenze, Viale Morgaffni, 59, 50134 Firenze, Italy Received March 1995; ...

Coherent Statistical Inference and Bayes Theorem

Annals of Statistics, Mar 1, 1991

Generating knockoffs via conditional independence

arXiv (Cornell University), Dec 19, 2022

Let X be a p-variate random vector and X a knockoff copy of X (in the sense of [9]). A new approa... more Let X be a p-variate random vector and X a knockoff copy of X (in the sense of [9]). A new approach for constructing X (henceforth, NA) has been introduced in [8]. NA has essentially three advantages: (i) To build X is straightforward; (ii) The joint distribution of (X, X) can be written in closed form; (iii) X is often optimal under various criteria. However, for NA to apply, X 1 ,. .. , Xp should be conditionally independent given some random element Z. Our first result is that any probability measure µ on R p can be approximated by a probability measure µ 0 of the form µ 0 A 1 ×. .. × Ap = E p i=1 P (X i ∈ A i | Z). The approximation is in total variation distance when µ is absolutely continuous, and an explicit formula for µ 0 is provided. If X ∼ µ 0 , then X 1 ,. .. , Xp are conditionally independent. Hence, with a negligible error, one can assume X ∼ µ 0 and build X through NA. Our second result is a characterization of the knockoffs X obtained via NA. It is shown that X is of this type if and only if the pair (X, X) can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti's theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of X given X is obtained in a few cases. In one of such cases, it is assumed X i ∈ {0, 1} for all i.

Finitely additive mixtures of probability measures

Journal of Mathematical Analysis and Applications, Aug 1, 2021

Concentration Curve and Index, Zenga's

Trivial intersection of σ-fields and Gibbs sampling

Annals of Probability, Nov 1, 2008

Let (, F , P) be a probability space and N the class of those F ∈ F satisfying P (F) ∈ {0, 1}. Fo... more Let (, F , P) be a probability space and N the class of those F ∈ F satisfying P (F) ∈ {0, 1}. For each G ⊂ F , define G = σ (G ∪ N). Necessary and sufficient conditions for A ∩ B = A ∩ B, where A, B ⊂ F are sub-σ-fields, are given. These conditions are then applied to the (twocomponent) Gibbs sampler. Suppose X and Y are the coordinate projections on (, F) = (X × Y, U ⊗ V) where (X, U) and (Y, V) are measurable spaces. Let (X n , Y n) n≥0 be the Gibbs chain for P. Then, the SLLN holds for (X n , Y n) if and only if σ (X)

A Skorohod representation theorem for uniform distance

Probability Theory and Related Fields, Mar 11, 2010

Let μ n be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distanc... more Let μ n be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X n such that X n ∼ μ n for all n ≥ 0 and X n − X 0 → 0 in probability, where • is the sup-norm. Such conditions do not require μ 0 separable under •. Applications to exchangeable empirical processes and to pure jump processes are given as well.

Skorohod representation on a given probability space

Probability Theory and Related Fields, Jul 26, 2006

Let (, A, P) be a probability space, S a metric space, µ a probability measure on the Borel σ-fie... more Let (, A, P) be a probability space, S a metric space, µ a probability measure on the Borel σ-field of S, and X n : → S an arbitrary map, n = 1, 2,. . .. If µ is tight and X n converges in distribution to µ (in Hoffmann-Jørgensen's sense), then X ∼ µ for some S-valued random variable X on (, A, P). If, in addition, the X n are measurable and tight, there are S-valued random variables ∼ X n and X, defined on (, A, P), such that ∼ X n ∼ X n , X ∼ µ, and ∼ X n k → X a.s. for some subsequence (n k). Further, ∼ X n → X a.s. (without need of taking subsequences) if µ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to σ (A ∪ {H}) for some H ⊂ with P * (H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken ((0, 1), σ (U ∪ {H}), m H), for some H ⊂ (0, 1) with outer Lebesgue measure 1, where U is the Borel σ-field on (0, 1) and m H the only extension of Lebesgue measure such that m H (H) = 1.

Convergence in distribution of nonmeasurable random elements

Annals of Probability, 2004

A notion of convergence in distribution for non (necessarily) measurable random elements, due to ... more A notion of convergence in distribution for non (necessarily) measurable random elements, due to Hoffmann-Jørgensen, is characterized in terms of weak convergence of finitely additive probability measures. A similar characterization is given for a strengthened version of such a notion. Further, it is shown that the empirical process for an exchangeable sequence can fail to converge, due to the nonexistence of any measurable limit, although it converges for an i.i.d. sequence. Because of phenomena of this type, Hoffmann-Jørgensen's definition is extended to the case of a nonmeasurable limit. In the extended definition, naturally suggested by the main results, the limit is a finitely additive probability measure.

On the existence of inferences which are consistent with a given model

Annals of Statistics, Jun 1, 1996

If p is a-additive statistical model and a finitely additive prior, then any statistic T is suffi... more If p is a-additive statistical model and a finitely additive prior, then any statistic T is sufficient, with respect to a suitable inference Ä 4 Ž. consistent with p and , provided only that p T s t s 0 for all and t. Here, sufficiency is to be intended in the Bayesian sense, and consis-Ä 4 tency in the sense of Lane and Sudderth. As a corollary, if p is-additive and diffuse, then, whatever the prior , there is an inference Ä 4 which is consistent with p and. Two versions of the main result are also obtained for predictive problems.

Well Calibrated, Coherent Forecasting Systems

Theory of Probability and Its Applications, 1998

WELL-CALIBRATED, COHERENT FORECASTING SYSTEMS 1) В данной работе вводится определение предсказуем... more WELL-CALIBRATED, COHERENT FORECASTING SYSTEMS 1) В данной работе вводится определение предсказуемого вывода, основанного на последовательных наблюдениях, в свете принципа ко герентности де Финетти. Даются характеризации когерентных пред сказуемых выводов и стратегических предсказуемых выводов. Про водится тщательный анализ понятий правильно калиброванной (wellcalibrated) и финитно правильно калиброванной систем прогнозирова ния. В связи с данной тематикой устанавливаются некоторые новые законы больших чисел относительно конечно-аддитивных распреде лений вероятностей. Эти результаты используются, чтобы реши тельно опровергнуть некоторые возражения, высказываемые против принципа когерентности. Ключевые слова и фразы: калибрование, когерентность, услов ная вероятность, конгломеративность (conglomerability), продолже ние, конечная аддитивность, предсказуемый вывод, стратегия. According to a prevailing view, the goal of statistical inference is a state ment about the probability law governing a given observable phenomenon. Consequently, statistical methods are conceived just as tools to single out that law. This point of view presupposes that the existence of probability laws, as well as the existence of laws governing observable phenomena, is viewed as an objective fact. Hence, the above mentioned idea of statistical inference might clash with the pragmatic position according to which the essential role of any scientific theory lies in making previsions about future possible facts. On the other hand, the process of inferring values for unknown observable facts, based on current observations and other information, is immune to the criticism from those who refuse to assume the existence of «true» probability laws (subjectivistic interpretation of probability). This kind of process, which is known as predictive statistical approach, was shared by almost all the pioneers of statistical induction; see [17]. It can be carried out directly, without passing through the mediation of a family of parametric probability laws (statistical model), under the sole guidance of de Finetti's

A central limit theorem for some generalized martingale arrays

Electronic Communications in Probability, 2023

Let (Xn,j) and (Yn,j) be two arrays of real random variables and f : R → R a Borel function. Defi... more Let (Xn,j) and (Yn,j) be two arrays of real random variables and f : R → R a Borel function. Define Dn = j f j−1 i=1 Yn,i Xn,j and D = Z 1 0 f 2 (B G(t)) dF (t) where B is a standard Brownian motion, Z a standard normal random variable independent of B, and F and G are distribution functions. Conditions for Dn → D, in distribution or stably, are given. Among other things, such conditions apply to certain sequences of stochastic integrals, when the quadratic variations of the integrand processes converge in distribution but not in probability. An upper bound for the Wasserstein distance between the probability distributions of Dn and D is obtained as well.

Exchangeable Sequences Driven by an Absolutely Continuous Random Measure

RePEc: Research Papers in Economics, Mar 1, 2011

Let S be a Polish space and (X n : n ≥ 1) an exchangeable sequence of Svalued random variables. L... more Let S be a Polish space and (X n : n ≥ 1) an exchangeable sequence of Svalued random variables. Let α n (•) = P (X n+1 ∈ • | X 1 ,. .. , X n) be the predictive measure and α a random probability measure on S such that α n weak −→ α a.s. Two (related) problems are addressed. One is to give conditions for α λ a.s., where λ is a (nonrandom) σ-finite Borel measure on S. Such conditions should concern the finite dimensional distributions L(X 1 ,. .. , X n), n ≥ 1, only. The other problem is to investigate whether α n − α a.s. −→ 0, where • 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].

On the almost sure convergence of sums

arXiv (Cornell University), Sep 7, 2020

Two counterexamples, addressing questions raised in [1] and [3], are provided. Both counterexampl... more Two counterexamples, addressing questions raised in [1] and [3], are provided. Both counterexamples are related to chaoses. Let Fn = Yn +Zn. It may be that Fn a.s. −→ 0, Fn L 2+δ −→ 0 and E sup n |Fn| δ < ∞, where δ > 0 and Yn and Zn belong to chaoses of uniformly bounded degree, and yet Yn fails to converge to 0 a.s.

A probabilistic view on predictive constructions for Bayesian learning

arXiv (Cornell University), Aug 14, 2022

Given a sequence X = (X 1 , X 2 ,. . .) of random observations, a Bayesian forecaster aims to pre... more Given a sequence X = (X 1 , X 2 ,. . .) of random observations, a Bayesian forecaster aims to predict X n+1 based on (X 1 ,. .. , Xn) for each n ≥ 0. To this end, in principle, she only needs to select a collection σ = (σ 0 , σ 1 ,. . .), called "strategy" in what follows, where σ 0 (•) = P (X 1 ∈ •) is the marginal distribution of X 1 and σn(•) = P (X n+1 ∈ • | X 1 ,. .. , Xn) the n-th predictive distribution. Because of the Ionescu-Tulcea theorem, σ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence X is determined. The strategies concern generalized Pólya urns, random change points, covariates and stationary sequences.

Some Skorohod-type results

Decisions in economics and finance, Jul 1, 2024

Some duality results for equivalence couplings and total variation

Electronic communications in probability, 2024

Let (Ω, F) be a measurable space and E ⊂ Ω × Ω. Suppose that E ∈ F ⊗ F and the relation on Ω defi... more Let (Ω, F) be a measurable space and E ⊂ Ω × Ω. Suppose that E ∈ F ⊗ F and the relation on Ω defined as x ∼ y ⇔ (x, y) ∈ E is reflexive, symmetric and transitive. Following [7], say that E is strongly dualizable if there is a sub-σ-field G ⊂ F such that min P ∈Γ(µ,ν) (1 − P (E)) = max A∈G |µ(A) − ν(A)| for all probabilities µ and ν on F. This paper investigates strong duality. Essentially, it is shown that E is strongly dualizable provided some mild modifications are admitted. Let G0 be the E-invariant sub-σ-field of F. One result is that, for all probabilities µ and ν on F, there is a probability ν0 on F such that ν0 = ν on G0 and min P ∈Γ(µ,ν 0)

Quantitative bounds in the central limit theorem for m-dependent random variables

Alea, 2024

For each n ≥ 1, let X n,1 ,. .. , X n,Nn be real random variables and S n = Nn i=1 X n,i. Let m n... more For each n ≥ 1, let X n,1 ,. .. , X n,Nn be real random variables and S n = Nn i=1 X n,i. Let m n ≥ 1 be an integer. Suppose (X n,1 ,. .. , X n,Nn) is m n-dependent, E(X ni) = 0, E(X 2 ni) < ∞ and σ 2 n := E(S 2 n) > 0 for all n and i. Then, d W S n σ n , Z ≤ 30 c 1/3 + 12 U n (c/2) 1/2 for all n ≥ 1 and c > 0, where d W is Wasserstein distance, Z a standard normal random variable and U n (c) = m n σ 2 n Nn i=1 E X 2 n,i 1 |X n,i | > c σ n /m n. Among other things, this estimate of d W S n /σ n , Z yields a similar estimate of d T V S n /σ n , Z where d T V is total variation distance.

Convergence in Total Variation to a Mixture of Gaussian Laws

Mathematics, Jun 11, 2018

It is not unusual that X n dist −→ VZ where X n , V, Z are real random variables, V is independen... more It is not unusual that X n dist −→ VZ where X n , V, Z are real random variables, V is independent of Z and Z ∼ N (0, 1). An intriguing feature is that P VZ ∈ A = E N (0, V 2)(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 V 2. In this paper, conditions for d TV (X n , VZ) → 0 are given, where d TV (X n , VZ) is the total variation distance between the probability distributions of X n and VZ. To estimate the rate of convergence, a few upper bounds for d TV (X n , VZ) are given as well. Special attention is paid to the following two cases: (i) X n is a linear combination of the squares of Gaussian random variables; and (ii) X n is related to the weighted quadratic variations of two independent Brownian motions.

Coherent Inferences and Improper Priors

Annals of Statistics, Sep 1, 1994

A Glivenko-Cantelli theorem for exchangeable random variables

Statistics & Probability Letters, Apr 1, 1997

ELSEVIER Statistics Probability Letters 32 (1997) 385391 A GlivenkoCantelli theorem for exchangea... more ELSEVIER Statistics Probability Letters 32 (1997) 385391 A GlivenkoCantelli theorem for exchangeable random variables Patrizia Berti *, Pietro Rigo Dipartimento di Statistica "G. Parenti "', Universit Deffli Studi di Firenze, Viale Morgaffni, 59, 50134 Firenze, Italy Received March 1995; ...

Coherent Statistical Inference and Bayes Theorem

Annals of Statistics, Mar 1, 1991

Generating knockoffs via conditional independence

arXiv (Cornell University), Dec 19, 2022

Let X be a p-variate random vector and X a knockoff copy of X (in the sense of [9]). A new approa... more Let X be a p-variate random vector and X a knockoff copy of X (in the sense of [9]). A new approach for constructing X (henceforth, NA) has been introduced in [8]. NA has essentially three advantages: (i) To build X is straightforward; (ii) The joint distribution of (X, X) can be written in closed form; (iii) X is often optimal under various criteria. However, for NA to apply, X 1 ,. .. , Xp should be conditionally independent given some random element Z. Our first result is that any probability measure µ on R p can be approximated by a probability measure µ 0 of the form µ 0 A 1 ×. .. × Ap = E p i=1 P (X i ∈ A i | Z). The approximation is in total variation distance when µ is absolutely continuous, and an explicit formula for µ 0 is provided. If X ∼ µ 0 , then X 1 ,. .. , Xp are conditionally independent. Hence, with a negligible error, one can assume X ∼ µ 0 and build X through NA. Our second result is a characterization of the knockoffs X obtained via NA. It is shown that X is of this type if and only if the pair (X, X) can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti's theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of X given X is obtained in a few cases. In one of such cases, it is assumed X i ∈ {0, 1} for all i.

Finitely additive mixtures of probability measures

Journal of Mathematical Analysis and Applications, Aug 1, 2021

Concentration Curve and Index, Zenga's

Trivial intersection of σ-fields and Gibbs sampling

Annals of Probability, Nov 1, 2008

Let (, F , P) be a probability space and N the class of those F ∈ F satisfying P (F) ∈ {0, 1}. Fo... more Let (, F , P) be a probability space and N the class of those F ∈ F satisfying P (F) ∈ {0, 1}. For each G ⊂ F , define G = σ (G ∪ N). Necessary and sufficient conditions for A ∩ B = A ∩ B, where A, B ⊂ F are sub-σ-fields, are given. These conditions are then applied to the (twocomponent) Gibbs sampler. Suppose X and Y are the coordinate projections on (, F) = (X × Y, U ⊗ V) where (X, U) and (Y, V) are measurable spaces. Let (X n , Y n) n≥0 be the Gibbs chain for P. Then, the SLLN holds for (X n , Y n) if and only if σ (X)

A Skorohod representation theorem for uniform distance

Probability Theory and Related Fields, Mar 11, 2010

Let μ n be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distanc... more Let μ n be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables X n such that X n ∼ μ n for all n ≥ 0 and X n − X 0 → 0 in probability, where • is the sup-norm. Such conditions do not require μ 0 separable under •. Applications to exchangeable empirical processes and to pure jump processes are given as well.

Skorohod representation on a given probability space

Probability Theory and Related Fields, Jul 26, 2006

Let (, A, P) be a probability space, S a metric space, µ a probability measure on the Borel σ-fie... more Let (, A, P) be a probability space, S a metric space, µ a probability measure on the Borel σ-field of S, and X n : → S an arbitrary map, n = 1, 2,. . .. If µ is tight and X n converges in distribution to µ (in Hoffmann-Jørgensen's sense), then X ∼ µ for some S-valued random variable X on (, A, P). If, in addition, the X n are measurable and tight, there are S-valued random variables ∼ X n and X, defined on (, A, P), such that ∼ X n ∼ X n , X ∼ µ, and ∼ X n k → X a.s. for some subsequence (n k). Further, ∼ X n → X a.s. (without need of taking subsequences) if µ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to σ (A ∪ {H}) for some H ⊂ with P * (H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken ((0, 1), σ (U ∪ {H}), m H), for some H ⊂ (0, 1) with outer Lebesgue measure 1, where U is the Borel σ-field on (0, 1) and m H the only extension of Lebesgue measure such that m H (H) = 1.

Convergence in distribution of nonmeasurable random elements

Annals of Probability, 2004

A notion of convergence in distribution for non (necessarily) measurable random elements, due to ... more A notion of convergence in distribution for non (necessarily) measurable random elements, due to Hoffmann-Jørgensen, is characterized in terms of weak convergence of finitely additive probability measures. A similar characterization is given for a strengthened version of such a notion. Further, it is shown that the empirical process for an exchangeable sequence can fail to converge, due to the nonexistence of any measurable limit, although it converges for an i.i.d. sequence. Because of phenomena of this type, Hoffmann-Jørgensen's definition is extended to the case of a nonmeasurable limit. In the extended definition, naturally suggested by the main results, the limit is a finitely additive probability measure.

On the existence of inferences which are consistent with a given model

Annals of Statistics, Jun 1, 1996

If p is a-additive statistical model and a finitely additive prior, then any statistic T is suffi... more If p is a-additive statistical model and a finitely additive prior, then any statistic T is sufficient, with respect to a suitable inference Ä 4 Ž. consistent with p and , provided only that p T s t s 0 for all and t. Here, sufficiency is to be intended in the Bayesian sense, and consis-Ä 4 tency in the sense of Lane and Sudderth. As a corollary, if p is-additive and diffuse, then, whatever the prior , there is an inference Ä 4 which is consistent with p and. Two versions of the main result are also obtained for predictive problems.

Well Calibrated, Coherent Forecasting Systems

Theory of Probability and Its Applications, 1998

WELL-CALIBRATED, COHERENT FORECASTING SYSTEMS 1) В данной работе вводится определение предсказуем... more WELL-CALIBRATED, COHERENT FORECASTING SYSTEMS 1) В данной работе вводится определение предсказуемого вывода, основанного на последовательных наблюдениях, в свете принципа ко герентности де Финетти. Даются характеризации когерентных пред сказуемых выводов и стратегических предсказуемых выводов. Про водится тщательный анализ понятий правильно калиброванной (wellcalibrated) и финитно правильно калиброванной систем прогнозирова ния. В связи с данной тематикой устанавливаются некоторые новые законы больших чисел относительно конечно-аддитивных распреде лений вероятностей. Эти результаты используются, чтобы реши тельно опровергнуть некоторые возражения, высказываемые против принципа когерентности. Ключевые слова и фразы: калибрование, когерентность, услов ная вероятность, конгломеративность (conglomerability), продолже ние, конечная аддитивность, предсказуемый вывод, стратегия. According to a prevailing view, the goal of statistical inference is a state ment about the probability law governing a given observable phenomenon. Consequently, statistical methods are conceived just as tools to single out that law. This point of view presupposes that the existence of probability laws, as well as the existence of laws governing observable phenomena, is viewed as an objective fact. Hence, the above mentioned idea of statistical inference might clash with the pragmatic position according to which the essential role of any scientific theory lies in making previsions about future possible facts. On the other hand, the process of inferring values for unknown observable facts, based on current observations and other information, is immune to the criticism from those who refuse to assume the existence of «true» probability laws (subjectivistic interpretation of probability). This kind of process, which is known as predictive statistical approach, was shared by almost all the pioneers of statistical induction; see [17]. It can be carried out directly, without passing through the mediation of a family of parametric probability laws (statistical model), under the sole guidance of de Finetti's

A central limit theorem for some generalized martingale arrays

Electronic Communications in Probability, 2023

Let (Xn,j) and (Yn,j) be two arrays of real random variables and f : R → R a Borel function. Defi... more Let (Xn,j) and (Yn,j) be two arrays of real random variables and f : R → R a Borel function. Define Dn = j f j−1 i=1 Yn,i Xn,j and D = Z 1 0 f 2 (B G(t)) dF (t) where B is a standard Brownian motion, Z a standard normal random variable independent of B, and F and G are distribution functions. Conditions for Dn → D, in distribution or stably, are given. Among other things, such conditions apply to certain sequences of stochastic integrals, when the quadratic variations of the integrand processes converge in distribution but not in probability. An upper bound for the Wasserstein distance between the probability distributions of Dn and D is obtained as well.

Exchangeable Sequences Driven by an Absolutely Continuous Random Measure

RePEc: Research Papers in Economics, Mar 1, 2011

Let S be a Polish space and (X n : n ≥ 1) an exchangeable sequence of Svalued random variables. L... more Let S be a Polish space and (X n : n ≥ 1) an exchangeable sequence of Svalued random variables. Let α n (•) = P (X n+1 ∈ • | X 1 ,. .. , X n) be the predictive measure and α a random probability measure on S such that α n weak −→ α a.s. Two (related) problems are addressed. One is to give conditions for α λ a.s., where λ is a (nonrandom) σ-finite Borel measure on S. Such conditions should concern the finite dimensional distributions L(X 1 ,. .. , X n), n ≥ 1, only. The other problem is to investigate whether α n − α a.s. −→ 0, where • 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].

On the almost sure convergence of sums

arXiv (Cornell University), Sep 7, 2020

Two counterexamples, addressing questions raised in [1] and [3], are provided. Both counterexampl... more Two counterexamples, addressing questions raised in [1] and [3], are provided. Both counterexamples are related to chaoses. Let Fn = Yn +Zn. It may be that Fn a.s. −→ 0, Fn L 2+δ −→ 0 and E sup n |Fn| δ < ∞, where δ > 0 and Yn and Zn belong to chaoses of uniformly bounded degree, and yet Yn fails to converge to 0 a.s.

A probabilistic view on predictive constructions for Bayesian learning

arXiv (Cornell University), Aug 14, 2022

Given a sequence X = (X 1 , X 2 ,. . .) of random observations, a Bayesian forecaster aims to pre... more Given a sequence X = (X 1 , X 2 ,. . .) of random observations, a Bayesian forecaster aims to predict X n+1 based on (X 1 ,. .. , Xn) for each n ≥ 0. To this end, in principle, she only needs to select a collection σ = (σ 0 , σ 1 ,. . .), called "strategy" in what follows, where σ 0 (•) = P (X 1 ∈ •) is the marginal distribution of X 1 and σn(•) = P (X n+1 ∈ • | X 1 ,. .. , Xn) the n-th predictive distribution. Because of the Ionescu-Tulcea theorem, σ can be assigned directly, without passing through the usual prior/posterior scheme. One main advantage is that no prior probability is to be selected. In a nutshell, this is the predictive approach to Bayesian learning. A concise review of the latter is provided in this paper. We try to put such an approach in the right framework, to make clear a few misunderstandings, and to provide a unifying view. Some recent results are discussed as well. In addition, some new strategies are introduced and the corresponding distribution of the data sequence X is determined. The strategies concern generalized Pólya urns, random change points, covariates and stationary sequences.