Showing papers by "Pietro Rigo published in 2004"

Limit theorems for a class of identically distributed random variables

Abstract: A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (Xn)n≥1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration $(\mathcal{G}_{n})_{n\geq 0}$ , if it is adapted to $(\mathcal{G}_{n})_{n\geq 0}$ and, for each n≥0, (Xk)k>n is identically distributed given the past $\mathcal{G}_{n}$ . In case $\mathcal{G}_{0}=\{\varnothing,\Omega\}$ and $\mathcal{G}_{n}=\sigma(X_{1},\ldots,X_{n})$ , a result of Kallenberg implies that (Xn)n≥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 (Xn)n≥1 is exchangeable if and only if (Xτ(n))n≥1 is c.i.d. for any finite permutation τ of {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)∑k=1nXk converges a.s. and in L1 whenever (Xn)n≥1 is (real-valued) c.i.d. and E[|X1|]<∞. As to the CLT, three types of random centering are considered. One such centering, significant in Bayesian prediction and discrete time filtering, is $E[X_{n+1}\vert \mathcal{G}_{n}]$ . For each centering, convergence in distribution of the corresponding empirical process is analyzed under uniform distance.

Convergence in distribution of nonmeasurable random elements

Abstract: A notion of convergence in distribution for non (necessarily) measurable random elements, due to Hoffmann-Jorgensen, 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-Jorgensen'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.

Limit theorems for a class of identically distributed random variables

Abstract: 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,\Omega} and G_n=\sigma(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_{\tau(n)})_{n\geq 1} is c.i.d. for any finite permutation \tau of {1,2,...}, and that the distribution of a c.i.d. sequence agrees with an exchangeable law on a certain sub-\sigma-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.