I attend a lecture about Stochastic Processes even though I have not studied mathematics and some of the basics in probability theory are missing. So I hope you can help me with the following questions and you will be patient with me.

1) We derived in a concrete example that the joint distribution function of $X_1, \dotsc, X_n$ is the product of the distribution function of $n$ Poisson random variables. Why does this imply that $X_1, \dotsc, X_n$ are independent and Poisson distributed? Couldn't it be that some other marginal distributions for $X_1, \dotsc, X_n$ (where they might also be dependent) lead to the same joint distribution?

2) We wanted to show that a continuous-time stochastic process $(X_t)_{t \geq 0}$ and a discrete-time stochastic process $(Y_n)_{n \geq 0}$ are independent. Is it true that it is enough to show for any finite set of $X_t$ and for any finite set of $Y_n$ that they are independent in order to show that the processes are independent?

3) We wanted to show that some random variables are independent. We showed that the characteristic function factorizes. But doesn't this only imply subindependence (according to Wikipedia)?

Thank you very much for your help!

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    $\begingroup$ For 3) see math.stackexchange.com/questions/287138/… $\endgroup$ – Calculon Mar 14 '15 at 18:12
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    $\begingroup$ At the risk of confusing you for 2) you may need Kolmogorov's extension theorem :) $\endgroup$ – Calculon Mar 14 '15 at 18:16
  • $\begingroup$ Yes, it indeed confuses me because I think they should have at least mentioned it if they used it and I cannot make any sense of it and have never heard about it. But I will think about it! Thank you for your comments. $\endgroup$ – user136457 Mar 14 '15 at 18:23
  • $\begingroup$ No problem. Good luck. $\endgroup$ – Calculon Mar 14 '15 at 18:24
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    $\begingroup$ What is "weak independence"? $\endgroup$ – Did Mar 14 '15 at 18:50

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