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An exponential inequality and strong limit theorems for conditional expectations. (English) Zbl 1249.60045

Let \((\Omega, \mathcal {A},P)\) be a probability space. Let \(A\in\mathcal {A}\) be a fixed event, \(P(A)>0\). Let \(P^A\) be the conditional probability measure with mean \(E^A\). Let \(\tau_i\), \(1\leq i\leq N\), be nondegenerated independent random variables with variance \(\sigma_i\), such that \(| \tau_i| <C<\infty\), \(1\leq i\leq N\) almost surely. Let \(\sigma^2=\sum^N_{i=1}\sigma_i^2\) and \(\mu=\sum^N_{i=1}\tau_i\). The main results of the paper are exponential inequalities like \[ \text{P}^A\left\{\frac{| \mu-E^A \mu| }{\sqrt{N}}\geq\varepsilon\right\}\leq \frac{\sqrt{2}}{P(A)}e^{-\frac{\varepsilon^2}{16 \sigma^2}}(1+B), \] (see Th. 2.1) where \(B\) depends on \(\varepsilon\) and \(\sigma^2\).
Section 2 contains the statement and the proof of Th. 2.1, based on three lemmas. The proofs of Lemmas 2.2 and 2.3 deal with Khintchine type inequalities. The tail probability is estimated by the \(p\)-th moment of the sum of certain random variables which are majorated using Khintchine inequalities, and the minimum in \(p\) is computed.
A generalized allocation scheme with parameters \(n\) and \(N\) and independent random variables \(\xi_i\), \(1\leq i\leq N\), is considered in Section 3. The random variables \(\xi_i\), \(1\leq i\leq N\) are identically distributed. For sums \(S_n\) of certain non-independent random variables in the probability space \(\{\Omega, \mathcal {A}, P\}\), a statement similar to the law of the iterated logarithm is proved. As corollaries, the corresponding strong laws of large numbers are obtained. The exponential inequalities of Th. 2.1 are used in the proofs.
Applications to the generalized allocation scheme are presented in Section 4 and Section 5. In Section 4, a generalized allocation scheme with parameters \(n\) and \(N\) and independent identically distributed random variables \(\xi_i\), \(1\leq i\leq N\), is considered. A nonuniform scheme of allocations of \(n\) distinguishable balls into \(N\) boxes with nonuniform probabilities of boxes is considered in Section 5. In both sections, analogues of the law of the iterated logarithm and the strong law of large numbers are proved. Actually, a.s. strong limit results are obtained for a two-indexed sequence of random variables.

MSC:

60F15 Strong limit theorems
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References:

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