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68 Chapter 2. On a conjecture about addition modulo 2 k − 1where c ∉ −N and (x) n = x(x + 1)(x + 2) · · · (x + n − 1) is the Pochhammer symbol and representsthe rising factorial.It should be first remarked that, as all the β i ’s go to infinity, the laws of the γ ′ i ’s and the δ′ i ’sconverge towards laws of independent geometrically distributed variables with parameter 1/2.From now on, let G 1 , . . . , G d and H 1 , . . . , H d be 2d such independent random variables. ThenP t,k = P [ ∑ γ ′ < ∑ δ ′ ] converges towards[ d∑P G i 0 and the above discussion thereforeproves that the conjecture is asymptotically true. We have just proved the following theorem.Theorem 2.6.2. Let d be a strictly positive integer. There exists a constant K d such that if• ∀i, β i ≥ K d and• min i α i ≥ B − 1,i=1.thenP t,k < 1 2 .We now look for an explicit expression of this limit.Definition 2.6.3. Let X d be the random variabled∑ d∑X d = G i − H i ,i=1 i=1and let P d denoteP d = P [X d = 0] .With this notation,f d (∞, . . . , ∞) = P 0 d = 1 2 (1 − P d) ,whence the importance of the random variable X d .First, it is readily seen that X d is symmetric, i.e. P [X d = k] = P [X d = −k]. So studyingP [X d = k] for k a positive integer is sufficient.Second, to get an explicit expression for P [X d = k], we need the following easy lemma givingthe law of a sum of d independent geometrically distributed variables with parameter 1/2.Lemma 2.6.4. For j ≥ 0,[ d∑] ( ) d − 1 + j 1P G i = j =d − 1 2 j+1 .i=1It is then possible to express P [X d = k] as a hypergeometric series.