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 January 13th, 2013, 08:13 PM #1 Senior Member   Joined: Sep 2011 Posts: 140 Thanks: 0 Statistics When proving the strong law of large numbers, there is one step where i am confused (the most first step infact...) $\widehat{X}=X_k - \mu$, and $T_n= S_n-n\mu$. To bound $E(T_{n}^4)$, note that we can write : $T_{n}^4= \displaystyle\sum\limits_{k=1}^n \widehat{X_{k}}^4+ \binom {4} {1}\displaystyle\sum\limits_{1\le i \ne j \le n} \widehat{X_{i}}^3\widehat{X_j} +\frac{1}{2}\binom {4} {2}\displaystyle\sum\limits_{1\le i \ne j \le n} \widehat{X_{i}}^2\widehat{X_{j}}^2+\frac{1}{2} \binom {4} {2,1,1}\displaystyle\sum\limits_{1\le i,j,k \le n} \widehat{X_{i}}^2\widehat{X_{j}}\widehat{X_{k}}+\f rac{1}{4!} \binom {4} {1,1,1,1}\displaystyle\sum\limits_{1\le i,j,k,l \le n} \widehat{X_{i}}\widehat{X_{j}}\widehat{X_{k}}\wide hat{X_{l}}$ Could anyone please explain clearly on what the heck is going on here? i have no idea why $T_{n}^4$ becomes that above, with all different letters i,j,k,l...... i would appreciate the help
 January 14th, 2013, 01:55 PM #2 Global Moderator   Joined: May 2007 Posts: 6,807 Thanks: 717 Re: Statistics It is simply a multinomial expansion. The terms where all the X's are different get canceled by the mean.
 January 14th, 2013, 03:12 PM #3 Senior Member   Joined: Sep 2011 Posts: 140 Thanks: 0 Re: Statistics -Sorry could you please explain what you mean by the means cancelling out?, and i just dont get how it becomes like the expression above, through expanding what exactly?
 January 15th, 2013, 01:05 PM #4 Global Moderator   Joined: May 2007 Posts: 6,807 Thanks: 717 Re: Statistics I would rathen not use Latex (I still haven't got the hang of it). I will use Yk instead of X[hat]k Mutinomial expansion gives: (Y1 + Y2 + ... +Yn)^4 = {Y1^4 + Y2^4 + ....Yn^4} + etc + .... + ?YiYjYkYl (where all Y's are different) Finally E(YiYjYkYl) (where all Y's are different) = 0, since the Y's are independent with mean 0.
 January 25th, 2013, 03:55 AM #5 Senior Member   Joined: Sep 2011 Posts: 140 Thanks: 0 Re: Statistics Sorry but could you explain how the means are zero?

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