Abstract
Large deviations for the local time of a process $X_i$ are investigated, where $X_i=x_i$ for $t∈[S_{i-1},S_i[$ and $(x_j)$ are i.i.d. random variables on a Polish space, S_j is the $j$-th arrival time of a renewal process depending on $(x_j)$. No moment conditions are assumed on the arrival times of the renewal process.
Highlights
Large deviations for the local time of a process Xt are investigated, where Xt = xi for t ∈ [Si−1, Si[ and are i.i.d. random variables on a Polish space, Sj is the jth arrival time of a renewal process depending on
The ergodic theorem states that πt → μas t → +∞, while the Sanov theorem yields a finer estimate for the probability that πt is found in a small neighborhood of a given Borel probability measure νon X
While for τ ≡ 1 one gets the classical Sanov theorem, we are mainly interested in the case where the law of τ under μfeatures heavy tails
Summary
Consider an i.i.d. sequence (xi)i∈N+ in a Polish space X , with marginal distribution μ. One may define a stochastic process (Xt)t≥0 on X by setting Xt = xi for t ∈ [i − 1, i[, and consider its empirical measure πt. 1 t [0,t[ ds δXs. The ergodic theorem states that πt → μas t → +∞, while the Sanov theorem yields a finer estimate for the probability that πt is found in a small neighborhood of a given Borel probability measure νon X. While for τ ≡ 1 one gets the classical Sanov theorem, we are mainly interested in the case where the law of τ under μfeatures heavy tails. In such a case the Markov process Does not have good ergodic properties, and the classical Donsker-Varadhan theorem is violated
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