A note on non-regular martingales

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.