[Solution Library] In the setting of the previous question, prove if lim _n \rightarrow ∞ S_n exists in probability then the limit exists a.s. if
Question: In the setting of the previous question, prove
- if \(\lim _{n \rightarrow \infty} S_{n}\) exists in probability then the limit exists a.s.
- if the \(\left(X_{i}\right)\) are identically distributed and if \(n^{-1} S_{n} \rightarrow 0\) in probability then \(n^{-1} \max _{m \leq n} S_{m} \rightarrow 0\) in probability.
Deliverable: Word Document 
![[Step-by-Step] Let (X_i) be i.i.d. taking values](/images/solutions/MC-solution-library-69388.jpg "[Step-by-Step] Let (X_i) be i.i.d. taking values in -1,1,3,7,15,")
[Step-by-Step] Let (X_i) be i.i.d. taking values in -1,1,3,7,15,
![[Solution] Suppose S and T are stopping](/images/solutions/MC-solution-library-69389.jpg "[Solution] Suppose S and T are stopping times. Are the following")
[Solution] Suppose S and T are stopping times. Are the following
(Steps Shown) Let (X_i) be i.i.d. with E X_i^2 var;(S_T)=(var;(X_1))(E
(See Solution) Let (X_i) be a sequence of random variables, and
![[Solution Library] Let S_n=∑_i=1^n X_i, where (X_i)](/images/solutions/MC-solution-library-69392.jpg "[Solution Library] Let S_n=∑_i=1^n X_i, where (X_i) are i.i.d.")
[Solution Library] Let S_n=∑_i=1^n X_i, where (X_i) are i.i.d.
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