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The Markov chain defined by \( \ X_n = | X_{n-1} - Y_n | \) , where \( \ (Y_n)_{n\ge 1} \) is a sequence of i.i.d. random variables taking values in the set of non-negative real numbers, admits a unique stationary measure if the distribution function F of \(Y_n\) is absolutely continuous with respect to the Lebesgue measure. If this distribution has an infinite mean, the stationary measure is unbounded. In the case where the distribution is regularly varying and non-arithmetic, with an index \(-\alpha \) , where \( \frac{1}{2} < \alpha < 1\) the recurrence of the Markov chain is established using an approximation of the renewal function and martingale techniques.

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A Case of Recurrence in the Absolute Difference Chains with Heavy Tailed Increments

  • 会议论文

  • 发表日期：

    2025年

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  • Fetima Ladjimi

摘要

The Markov chain defined by \( \ X_n = | X_{n-1} - Y_n | \) , where \( \ (Y_n)_{n\ge 1} \) is a sequence of i.i.d. random variables taking values in the set of non-negative real numbers, admits a unique stationary measure if the distribution function F of \(Y_n\) is absolutely continuous with respect to the Lebesgue measure. If this distribution has an infinite mean, the stationary measure is unbounded. In the case where the distribution is regularly varying and non-arithmetic, with an index \(-\alpha \) , where \( \frac{1}{2} < \alpha < 1\) the recurrence of the Markov chain is established using an approximation of the renewal function and martingale techniques.