When a power of the process is a submartingale

HardProbability Stochastic Processes~12m

Let $X_1, X_2, \dots \sim \text{Exp}(\lambda)$ IID, and define $M_0 = 1$ , $M_n = M_{n-1}\cdot\tfrac12 e^{(\lambda/2)X_n}$ . It turns out $\{M_n^{p}\}$ is a submartingale exactly for $p$ in an interval $(a, b)$ with $0 < a < b$ . Find $a + b$ .

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Submartingale $\iff \mathbb{E}[S^p]\ge1$ for the step $S=\tfrac12 e^{(\lambda/2)X}$ ; $\mathbb{E}[S^p]=\tfrac{(1/2)^p}{1-p/2}$ for $p<2$ .
Solve $2^{-p}=1-\tfrac p2$ : crossings at $p=1$ and $p=2$ , so the interval is $(1,2)$ .

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When a power of the process is a submartingale

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