Lecture 5: LD in many dimensions and Markov chains
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چکیده
1 Sn lim log P( ∈ A). n→∞ n n Given θ ∈ Rd, define M(θ) = E[exp((θ, X1))] where (·, ·) represents the inner p product of two vectors: (a, b) = i aibi. Define I(x) = supθ∈Rd ((θ, x) − log M(θ)), where again I(x) = ∞ is a possibility. Theorem 1 (Cram ́ Suppose M(θ) < ∞ er’s theorem in multiple dimensions). for all θ ∈ Rd. Then (a) for all closed set F ⊂ Rd , 1 Sn lim sup log P( ∈ F ) ≤ − inf I(x) n→∞ n n x∈F
منابع مشابه
Math 484/884 Data Networks
Supplemental Course Notes Please note that there are likely many typos in the file. These will be corrected over time. The notes are mainly based on the instructor's lecture notes as well as some references that are particularly acknowledged. Further references will be included in the bibliography. Standard material on probability measures and Markov Chains are available in many texts, and no p...
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