Al. a. TCHOITPROFF 
157 
tend to zero, as N increases, when i = 3, 4, 5, 2r — ], 2r. Noting that when 
these conditions are satisfied the fraction J^^'zhS^^^^ tends to zero, we arrive at 
the well-known result (cf. A. A. Markoff, Theory of Probahilitij, Srd Edition, 
pp. 329-330). 
The probability of the fullihnent of the inequality 
1 
1 
1 
ft. 
tends with increasing N to the limit — - e~^' dt, whatever be the law of dis- 
tribution of values of the variable X, if only it satisfies the condition, that 
Niii 
[iATTp^ should tend with increasing N to zero, when t = 3, 4, 5, ... oo , and if at 
the same time the law of distribution of values remains unaltered, and the separate 
experiments are mutually independent*. 
(2) From (22) we find : 
1.3.5...(2'«-l);t./,(A.) ^^N\ 2 U/^/ ) ' 9 ya/j 
Thus we see that ^"'^'.'^'^^ tends the more slowly to 1.3.5 ...(2r— 1), the more 
the law of distribution of values deviates from the Gauss-Laplace law, and the 
greater r is. If fx.^ > 3/x.,-, then for sufficiently large values of N, 
^^55^)>1.3.5 ...(2r- 1). 
* The condition tends, with increasing N, to zero for i = 3, 4, 5, ... oo,'' while sufficient, 
is, as is well known, not necessary. From the form to which Liapounoff succeeded in reducing the 
condition (see Proc. Imp. Acad. Sci. viii series, vol. xii) it follows, among other consequences, that 
the law of distribution of values of X{]\t) tends with increasing N to the Gaussian, if ^ tends to 
zero. Noting that 
3 = 
we see that this condition is satisfied if '"f^'^' tMids with increasing N to 3. It is in this way that 
Liapounoff's results .justify arguments based ou the examination of the two moments only /U3( y) and 
pi-iiN) in deciding the question, whether the law of distribution of the values of X^^) tends to the 
Gaussian with increasing N or not. The assumption usually underlying such a procedure, viz. that 
the law of distribution is the Gauss-Laplace law, if '"% = 0 and — i = 3, is clearly inexact: the coincidence 
of the values of two or even more moments does not guarantee the identity of the laws, but merely 
compresses the possible divergence between them into limits which become narrower as the number of 
coinciding njoments increases (cf. the investigation by Chebysheff "On the Integral..., forming Approxi- 
mations to the Value of an Integral" in Ofiivres, t. ii, and the related papers by A. A. Markoff; 
cf. also T. T. Stieltjes, " Recherche s sur les fractions continues" in Aiiiuiles de la faculte de Toulouse 
(1894). 
