Limit Distributions of Random Variables Associated with Long Duplications in a Sequence of Independent Trials

Generalized Coupon Collection: The Superlinear Case

“…For S ≡ 1, the result was established in Zubkov and Mikhaȋlov (1974); the extension to S ≡ s is attributed to Kolchin et al (1978, p. 221).…”

Convergence to the Poisson Process in a Scheme of Cumulative Sums of Dependent Random Variables

“…In [3] for the case k 2 and for identically distributed random variables X, it was shown that, under conditions similar to those of Theorem 3, the distribution of the random variable (n, N) r/(i, i2) Part (ii) is proved similarly to part (ia). This completes the proof of Theorem 3.…”

“…From (1),(4) and the assumption A(P) > 0 it follows that the powers of the sets {qg(i)e Ps} tend to infinity; denoting by ns the sum of the powers of the sets {p(i) e Pj}, j 1,..., s, for s 1,..., m, we get" A= min (n-n_l)--..l_s_m Reindex the variables r/i in such a way that {p(i)Ps} {ns_ + 1, ..., n,}. Now condition (5) follows from (4), and (1') from (1); if we set I,,..., (n 1,"" nm) B,, .,,(n nm) f'l then conditions (2') and (3') will follow from(2)and(3). Whence, by Theorem 2, we obtain the required convergence of joint distributions.…”

“…Whence, by Theorem 2, we obtain the required convergence of joint distributions. Theorem is proved.Let us apply Theorem to the problem on the number of long duplications considered in[3] and[4]. Let X0, X1,... be a sequence of independent random variables taking il, ..., ik) e(i 1, ..., 1)r/(il, ..., ik).Associate with each collection of indices a (il, "'", ik) the point u(a) (il/N,..., ik/N) Rk, where N is some natural number, and set for every A Let k const.…”

Genes, information and sense: complexity and knowledge retrieval

“…Actually, d * value is proportional to lnN for a sequence from any (finite) alphabet, while the proportionality factor k (r *kln N) depends on the alphabet cardinality. k = 1 for four-letter alphabet with equal frequency of each letter occurrence (Zubkov and Mikhailov 1974). Table 1 shows the pattern of r observed for some human genes, before and after the intron excision.…”