# circulation of iid series of integrable arbitrary variables

I found an intriguing trouble in Jacod is probability publication. Yet have no suggestion just how to approach it. Should I approach it making use of induction? Any kind of suggestions?

Allow $X_1, X_2, \cdots$ be a boundless series of iid series of integrable arbitrary variables and also allow $N$ be a favorable, integer - valued integrable arbitrary variable which is independent from the series. Specify $S_n = \sum_{k=1}^{n} X_k$ and also think that $S_0 = 0$.

(a) Show that $E[S_N] = E[N]E[X_1]$.

(b) Show that the particular function of $S_N$ is offered by $E[\phi_{X_{1}}(t)^N]$, where $\phi_{X_{1}}$ is the particular function of $X_1$.

For component (a) make use of $E[S_N]= \sum\nolimits_n {E[S_N |N = n]P(N = n)}$. This leads straight to the outcome. For component (b) make use of $E[e^{tS_N } ] = \sum\nolimits_n {E[e^{tS_N } |N = n]P(N = n)} $. Once more, this leads straight to the outcome.

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