I have two questions related to the stopping time.
Suppose we have a iid sequence $X_n$, where $P(X<0)=1$. And we have a stopping time N, N=$\inf${n:$\sum_{i=1}^nX_i>0$}.
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My First question is what the value of E($\sum_{i=1}^NX_i$) is.
For one side, because we set the stopping rule such that $\sum_{i=1}^NX_i$ has to be greater than $0$, so E($\sum_{i=1}^NX_i$) should be greater than 0.
On the other hand, because $X_i$ is always less than $0$, and the stopping rule will never be satisfied, so N=$\infty$, and therefore E($\sum_{i=1}^NX_i$)=-$\infty$.
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Me second question is actually related to the first question: What is E($\sum_{i=1}^NX_i$|N)? Is that $0$ or a positive number or negative infinity?
Because if it is $0$, We can use the expectation of the conditional expectation to solve E($\sum_{i=1}^NX_i$), so E($\sum_{i=1}^NX_i$) =$E_N$($E_X$($\sum_{i=1}^NX_i$|N))=$E_N(0)$=$0$.
If it is a positive number, we can use the same technique to solve E($\sum_{i=1}^NX_i$) and find E($\sum_{i=1}^NX_i$)$>0$.
From different aspect one can end up with different result, but there must be only one result. Could any one see the problem for the above solutions? I would really appreciate any help. Thanks.
Since $\mathbb P(X_1<0)=1$, it is clear that $$\mathbb P\left(\bigcap_{n=1}^\infty \left\{\sum_{i=1}^n X_i < 0\right\} \right) = 1. $$ This implies that $\mathbb P(N=\infty)=1$, and so by monotone convergence (applied to $-X_n$) $$\mathbb E\left[\sum_{i=1}^N X_i \right] = \mathbb E\left[ \sum_{i=1}^\infty X_i \right] = \sum_{i=1}^\infty \mathbb E[X_i] = -\infty. $$ (This is assuming $\mathbb E[X_1]>-\infty$ - else the sum is clearly $-\infty$.)
Since $\mathbb P(N=\infty)=1$, we have $\sigma(N) = \{\varnothing, \Omega\}$ - so conditioning on $N$ does not change anything, and $$\mathbb E\left[\sum_{i=1}^N X_i\mid N\right] = \mathbb E\left[\sum_{i=1}^N X_i\right] =-\infty.$$