Examples of martingales goes to $-\infty$

Question

One example I can find is that set $X_{0}=0$,

Let $X_{j}=\left\{\begin{array}{ll}{j^{2}} & {\text { with probability } \frac{1}{j^{2}}} \\ {\frac{-j^{2}}{j^{2}-1}} & {\text { with probability } 1-\frac{1}{j^{2}}}\end{array}\right.$,

then this is a martingale tends to $-\infty$ with probability 1.

I wonder if there are more examples? Preferably more concise and straightforward.

Also, can you please give an example of a $L^{1}$-bounded martingale $\left\{X_{n}\right\}_{n \geq 0}$ for which $E M=\infty$, where \begin{aligned} M_{n} &=\max _{k \leq n}\left|X_{k}\right| \quad \text { and } \\ M &=\sup _{n \geq 1} M_{n} \end{aligned}

score 1 · Accepted Answer · answered Apr 22 '19 at 00:43

1

For the second part define $X_n=nI_{(0,\frac 1 n)}$ on $(0,1)$ with Lebesgue measure. This is an $L^{1}$ bounded martingale and $M(x) \geq \frac 1 x$ so $EM=\infty$.

answered Apr 22 '19 at 00:43

Kavi Rama Murthy

359,332

Can you remind me of the definition of $I_{(0,\frac 1 n)}$ please? – The R Apr 22 '19 at 00:58
It is the characteristic function of the interval $(0,\frac 1 n)$; it is $1$ for points in this interval and $0$ for points outside the interval. – Kavi Rama Murthy Apr 22 '19 at 05:09

Examples of martingales goes to $-\infty$

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