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I came across an interesting problem in Jacod's probability book. But have no idea how to approach it. Should I approach it using induction? Any ideas?

Let $X_1, X_2, \cdots$ be an infinite sequence of iid sequence of integrable random variables and let $N$ be a positive, integer-valued integrable random variable which is independent from the sequence. Define $S_n = \sum_{k=1}^{n} X_k$ and assume that $S_0 = 0$.

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

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

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Incidentally, (a) is a special case of Wald's equation ( – Nate Eldredge Dec 7 '10 at 22:38
up vote 4 down vote accepted

For part (a) use $E[S_N]= \sum\nolimits_n {E[S_N |N = n]P(N = n)}$. This leads straightforwardly to the result. For part (b) use $E[e^{tS_N } ] = \sum\nolimits_n {E[e^{tS_N } |N = n]P(N = n)} $. Again, this leads straightforwardly to the result.

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Note that the key in both parts is "condition on $N$". – Nate Eldredge Dec 7 '10 at 22:36
Yes, it uses conditional expectation and independence. Its not too difficult after all. – user957 Dec 8 '10 at 17:19

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