Suppose $X_1(t), \cdots, X_n(t)$ are random variables of a continuous time stochastic process.

Suppose for any $p>1$, $\sup_{t \geq 0} E\left[\sum_{i=1}^n X_i(t)^p \right] < K_p$ where $K_p$ is some constant that depends on the value of $p$.

Does it automatically follow that for each $i$, $\sup_{t \geq 0} E\left[X_i(t)^p\right] \leq K_p$? I would think it does since it should simply follow by linearity of expectations and $\sup$ is also linear. I just want to verify if my thought process is correct.

  • $\begingroup$ The assertion is true if your process is non-negative almost surely. It is not necessarily true otherwise. $\endgroup$ – Alex R. Jun 25 '15 at 20:48
  • $\begingroup$ @AlexR. In my case, they are non-negative a.s. Thank you! $\endgroup$ – Brenton Jun 25 '15 at 20:49

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