## Estimates of the mean size of the subset image under composition of random mappings.(English. Russian original)Zbl 1420.60038

Discrete Math. Appl. 28, No. 5, 331-338 (2018); translation from Diskretn. Mat. 30, No. 2, 27-36 (2018).
Summary: Let $$\mathcal{X}_N$$ be a set of $$N$$ elements and $$F_1,F_2,\dots$$ be a sequence of random independent equiprobable mappings $$\mathcal{X}_N\rightarrow_N$$. For a subset $$S_0 \subset \mathcal{X}_N$$, $$|S_0|=m$$, we consider a sequence of its images $$S_t=F_t(\dots F_2(F_1(S_0))\dots)$$, $$t=1,2\dots$$. An approach to the exact recurrent computation of distribution of $$|S_t|$$ is described. Two-sided inequalities for $$\mathbf{M}\{|S_t|||S_0|=m\}$$ such that the difference between the upper and lower bounds is $$o(m)$$ for $$m,t,N\rightarrow\infty$$, $$mt=o(N)$$ are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.

### MSC:

 60F05 Central limit and other weak theorems 60E05 Probability distributions: general theory
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### References:

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