Suppose X is a random walk with probability(adsbygoogle = window.adsbygoogle || []).push({});

[itex]P(X_k=+1)=p [/itex] and [itex]P(X_k=-1)=q=1-p [/itex]

and [itex]S_n=X_1+X_2+...+X_n [/itex]

Can anyone explain why does line 3 equal to line 4?

[itex]P(S_k-S_0≠0 ,S_k-S_1≠0 ,…,S_k-S_{k-1}≠0)[/itex]

[itex]=P(X_k+X_{k-1}+⋯+X_1≠0 ,X_k+X_{k-1}+⋯+X_2≠0 ,…,X_k≠0)[/itex]

[itex]=P( X_k≠0 ,X_k+X_{k-1}≠0 ,…,X_k+X_{k-1}+⋯+X_1≠0 )[/itex]...............Line 3

[itex]=P( X_1≠0 ,X_2+X_1≠0 ,…,X_k+X_{k-1}+⋯+X_1≠0 )[/itex]..................Line 4

[itex]=P( X_1≠0 ,X_1+X_2≠0 ,…,X_1+X_2+⋯+X_k≠0 )[/itex]

The above comes from a book on random walk, I attached a link here (page 36),

http://books.google.com/books?id=7suiLOKqeYQC&printsec=frontcover#v=onepage&q&f=false

Thanks

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# Markov Chain - Random Walk

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