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...es. We provide positive and precise answers to the aforementioned questions for a family of envelope classes that lie between the exponential envelope classes investigated in (Boucheron et al., 2009; =-=Bontemps, 2011-=-) and the classes of sources with finite alphabets. Haussler and Opper (1997) characterize the minimax redundancy of a collection of sources using the metric entropy of the class of marginal distribut...

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... ≤ R̂(Λnc·−α) ≤ ( 2cn α− 1 ) 1 α (log n)1− 1 α +O(1), (1) where C0 is a constant (function of α and c). For exponential envelopes they prove that log2 n 8α (1 + o(1)) ≤ R̂(Λnce−α·) ≤ log2 n 2α +O(1). =-=[24]-=- improve the bounds for Λnce−α· and show that R̂(Λnce−α·) = log2 n 4α +O(log n log log n). More recently, [25] extend the arguments of [24] to find tight universal codes for the larger class of sub-ex...

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...assiat[13] focused on countably infinite alphabets with an envelope condition; they used an adapted strategy and gave upper and lower bounds for pointwise minimax regret. Later on Bontemps and Gassiat=-=[14]-=- worked on exponentially decreasing envelope class and provided a minimax strategy and the corresponding regret. In this paper, we introduce a straightforward and easy to implement method for large al...

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...up to a constant factor. For the exponential-law envelope class, defined by the envelope fi = c ⋅ e−αi where c,α > 0 they proved that2 log2 n 8α log e (1 + o(1)) ≤ R̂(Enc⋅eαi) ≤ log2 n2α log e +O(1). =-=[4]-=- improved these bounds and determined the redundancy up to the first order term and showed R̂(Enc⋅eαi) = log2 n4α log e +O(logn log logn). (2) More recently, [5] extended the arguments of [4] to find ...

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...ers α and c is the class of distributions Λce−α· over N such that ∀i ∈ N, pi ≤ ce−αi. The redundancy of Λnce−α· was considered in [12] who proved log2 n 8α (1 + o(1)) ≤ R̂(Λnce−α·) ≤ log2 n 2α +O(1). =-=[13]-=- showed the precise growth rate of exponential envelopes and showed that R̂(Λnce−α·) = log2 n 4α (1 + o(1)). 50 An analysis of their algorithm shows that the o(1) term is of the form log logn logn , n...

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...up to a constant factor. For the exponential-law envelope class, defined by the envelope fi = c ⋅ e−αi where c,α > 0 they proved that2 log2 n 8α log e (1 + o(1)) ≤ R̂(Enc⋅eαi) ≤ log2 n2α log e +O(1). =-=[4]-=- improved these bounds and determined the redundancy up to the first order term and showed R̂(Enc⋅eαi) = log2 n4α log e +O(logn log logn). (2) More recently, [5] extended the arguments of [4] to find ...

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... 9 Mar 2014 C. Roadmap This paper shows that the AC-code is adaptive over the collection of envelope classes that lie between the exponential envelope classes investigated in (Boucheron et al., 2009; =-=Bontemps, 2011-=-) and the classes of sources with finite alphabets (Theorem 2). The relevant envelopes are characterized by the fact that they have non-decreasing hazard rate (see Section III). This distributional pr...