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Is there a relation between the max of a Gaussian random walk of 10 steps vs the max of 10 Gaussian random walks? Specifics (in Mathematica notation):

  (* a Gaussian random walk with standard deviation 1 *)  a[0] := 0  a[n_] := a[n-1] + RandomReal[NormalDistribution[0, 1]]   (* the max of the walk over 10 steps *)  b := Max[Table[a[i],{i,1,10}]]   (* calculate max many times to get good sample set *)  (* Mathematica "magic" insures we're not using the same random #s each time *)  c = Table[b,{i,1,10000}]   (* distribution isn't necessarily normal, but we can still compute mu + SD *)  Mean[c] (* 3.66464 *)  StandardDeviation[c] (* 1.61321 *)   

Now, consider 10 people doing a Gaussian random walk of 1 step and we take the max of these 10 values.

  (* max of 10 standard-normally distributed numbers *)  d := Max[Table[RandomReal[NormalDistribution[0, 1]],{i,1,10}]]   (* get a good sample set *)  f = Table[d,{i,10000}]   (* and now the mean and SD *)  Mean[f] (* 1.54843 *)  StandardDeviation[f] (* 0.580593 *) 

The two means/SDs are obviously different, but I sense they're related somehow, perhaps by Sqrt[10], since the sum (not max) of 10 random walks is normal with SD of Sqrt[10], and I sense that somehow the cumulative sum of the first 9 somehow cancel out.

Are these known distributions?

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