[15] Suppose the sequence a1a2 ...ai ... has all the properties of a collective with p = 1 2 .

(a) The limiting relative frequency of the subsequence 01 equals the limiting frequency of subsequence 11. Compute these limiting relative frequencies.

(b) Form a new sequence b1b2 ...bi ... defined by bi = ai +ai+1 for all i.

Then the bi’s are 0, 1, or 2. Show that the limiting relative frequencies of 0, 1, and 2 are 1 4 , 1 2 , and 1 4 , respectively.

(c) The new sequence constructed in Item

(b) satisfies requirement (1) of a collective: the limiting relative frequencies constituent elements of the sample space {0, 1, 2} exist. Prove that it does not satisfy requirement

(2) of a collective.

(d) Show that the limiting relative frequency of the subsequence 21 will generally differ from the relative frequency of the subsequence 11.

Comments. Hint for Item (c): Show that the subsequences 02 and 20 do not occur. Use this to select effectively a subsequence of b1b2 ...bi ...

with limiting relative frequency of 2s equal to zero. This phenomenon was first observed by M. von Smoluchowski [Sitzungsber. Wien. Akad. Wiss., Math.-Naturw. Kl., Abt. IIa, 123(1914) 2381–2405; 124(1915) 339–368]

in connection with Brownian motion, and called probability ‘after-effect.’

Also: [R. von Mises, Probability, Statistics, and Truth, Macmillan, 1933].