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A note on non-regular martingales

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  • Iksanov, Alex
  • Marynych, Alex

Abstract

We provide examples to demonstrate that an inequality due to Pakes [Pakes, A.G., 1987. Remarks on the maxima of a martingale sequence with application to the simple critical branching process. J. Appl. Probab. 24, 768-772] cannot be improved. One of our examples may be of interest in the theory of random walks. Gundy [Gundy, R.F., 1981. On a theorem of F. and M. Riesz and an equation of A. Wald. Indiana Univ. Math. J. 30, 589-605] constructed a non-negative non-regular martingale with a specific tail behaviour of its maximal function. We point out a simpler example of such a martingale.

Suggested Citation

  • Iksanov, Alex & Marynych, Alex, 2008. "A note on non-regular martingales," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 3014-3017, December.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:17:p:3014-3017
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