Al. a. Tchouproff 
287 
tends, as N increases, to the limit zero, for /• = 3, 4, 5, ... oo , then the law of dis- 
tribution of the values of X{n) tends to become the Laplace-Gauss law as iV" 
becomes infinitely great, whatever may be the law of distribution of the individual 
variates*. 
V 
If all the variates follow the Laplace-Gauss law then for all positive integral 
values of r, 
^2r + l, (A') = 0, 
If ^ {J\ i\ -1 J.\ r-i I'Zr^i 
N-1 iV )•-! 
[2r!] 
X 1 . 3 . 5 . . . (2n - 1 ) . 1 . 3 . 5 . . . ( 2r - 2/-, - 1 ) [f^'y- 
N-r + lN-r+2 N ., ,. 
... + 1.3.5...(2r- 1).1.2.3...?- S 1 ... ^ /t^'V! •••/^.. 
', = 1 = (,.= '1-1 + 1 
1.3.5...(2r-l) 
N ... N-l N r-\ I ,. , ... 
N 
-f- . . . -f- 2, 2, ... Z r\fji„ fj,„ ■■■ fJ-.^ 
1.3.5...(2r-l) 
1.3..5...(2r- 1 ) . 
and consequently 
,^-^'<^=1.3.5...(2r-l). 
Thus in this case, the arithmetic mean of the random values taken by the variates 
follows the Laplace-Gauss law whatever may be the constants of the laws of 
distribution of the individual variatesf. 
Putting! 
we find 
CHAPTER II 
I 
1 * 
771 / " V 
-C-Zi [,., AT] = ^ .-i^ /^^. = f^[r, N] 
1 
«■ = 1 
?•(?•- 1) 1 
....(1). 
* Cf. Biometrika, Vol. xii, pp. 1.56—157. t Cf. Biometrika, Vol. xii, p. 158. 
+ Fide Tschupioff, " Zur Theorie dei' Stabilitat statistischer Reihen " (Skandinavisk Aktiiarietid- 
skrift, 1919, pp. 82—83). 
