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in text the formula for scaled random walk is:

W^(n) (t) = (1/√n) M_nt

in the example it says that:

set t=0.25, n=100 and consider the set of possible values of W^(100) (0.25) = 1/10 M_25. This random variable is generated by 25 coin tosses, and since the unscaled random walk M_25 can take the value of any odd integer between -25 and 25.My first question is that why unscaled random walk takes only the odd integer?

The scaled random walk W^(100) (0.25) can take any of the following values:

-2.5, -2.3, -2.1,......,-0.3,-0.1,0.1,0.3,.......,2.1,2.3,2.5

My second question is that why only this range?

In order for W^(100) (0.25) to take the value 0.1, we must get 13 heads and 12 tails in the 25 coin tosses. The probability of this is

P{W^(100) (0.25) = 0.1} = {(25!)/(13! *12!)} * (1/2)^25 = 0.1555

by drawing a histogram bar centered at 0.1 with area 0.1555, since this bar has width 0.2, its height must be 0.1555/0.2 = 0.7775.

My last question is that why 13 heads and 12 tails, and why we did the factorial part "{(25!)/(13! *12!)}"?

Thanks in advance.

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# Limiting Distribution of Scaled Random Walk

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