I didn't quite get that yet. I'll have to go back and watch the movie again but what was the significance of his character and his part in Darjeeling Limited?
Murray simply makes the best cameos. Little Shop of Horrors, The Darjeeling Limited, and Zombieland...I'd say he rates as the king of cameos, and one in a movie of his favorite collaborator was a jewel in his crown.
"...if that was off, I'd be whoopin' your ass up and down this street." ~ an irate Tarantino
I like the "bookends" theory as well as the last person that mentioned the "Billy Murray is the father" theory...
I actually had to watch the last half hour again because i fell asleep (long day not because the movie was bad) and woke up to camera scanning the train and there's Bill Murray again...I went "$hit i just missed Bill Murray back in the movie, what's his significance?" so i rewatched only to realize i woke up right when they showed him again.
Bill Murray is obviously their deceased father, hence his running alongside Peter (Peter, being very attached to the father's belongings, is the one still grieving the most or at the very least having the hardest time, of the three of them, coping with it the death). Peter looks at him briefly and continues to run for the train. Their father is the connection (in part) that led them to start their journey. Bill Murray's appearance is symbolic for the emotional/psychological that the brothers are taking on their journey. There are many examples of why this symbolism is shown with Peter.