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I am given $X_t^2$ to be checked, where $X_t$ is Poisson process. How do I check? What are the properties that need to be checked?

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Let $Y_t$ be a Poisson process. For what values of $k$ is $P(Y_t=k)\ne 0$?

For what values of $k$ is $P(X_t^2=k)\ne 0$?

Added: The above hint leads to probably the simplest way to see that $X_t^2$ is not a Poisson process. However, almost all of the characteristic features of the Poisson process fail for $X_t^2$.

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    Sir, you're genious :O Just that easily... Amazing!2012-05-22