Limit theorems for a class of identically distributed random variables
Limit theorems for a class of identically distributed random variables
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DOI：
10.1214/009117904000000676
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发表时间：
2004-07-01
影响因子：
2.3
通讯作者：
Rigo, P
中科院分区：
文献类型：
--
作者：
Berti, P;Pratelli, L;Rigo, P
引入并研究了随机变量序列的一类新的随机相依性。确切地说，（X-n）n大于或等于1被称为条件同分布（c.i.d.），关于过滤（g（n））（n大于或等于0），如果它适合于（g（n））（n大于或等于0），并且对于每个n大于或等于0，给定过去的g（n），（X-k）（k>n）是相同分布的。在g（0）= {θ，Ω}且g（n）= σ（X-1，...，X-n），Kallenberg的一个结果暗示（X-n）（n ≥ 1）是可交换的当且仅当它是平稳的且c.i.d.在给出了一些自然的不可交换c.i.d.的例子之后。序列，证明了（X-n）（n ≥ 1）是可交换的当且仅当（X-tau（n））（n ≥ 1）是c.i.d.对于{1，2，...}的任何有限置换τ，而C.I.D.序列符合某个次σ域上的交换律。此外，（1/n）Sigma（k=1）（n）X-k收敛于a. S.在L-1中，当（X-n）（n大于或等于1）是（实值）c.i.d.而E[\X-1\]
A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X-n)ngreater than or equal to1 is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (g(n))(ngreater than or equal to0), if it is adapted to (g(n))(ngreater than or equal to0) and, for each ngreater than or equal to0, (X-k)(k>n) is identically distributed given the past g(n). In case g(0) = {theta, Omega} and g(n) = sigma(X-1,...,X-n), a result of Kallenberg implies that (X-n)(ngreater than or equal to1) 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)(ngreater than or equal to1) is exchangeable if and only if (X-tau(n))(ngreater than or equal to1) 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) Sigma(k=1)(n) X-k converges a. s. and in L-1 whenever (X-n)(ngreater than or equal to1) is (real-valued) c.i.d. and E[\X-1\]