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This paper by Dekking and Host is quite old and much has been done in this area since. Today we know that under reasonable assumptions, there are constants $a\in\mathbb{R}$, $b\ge 0$, such that $E(Y_n) = an + b \log n + O(1)$. How to get the constant $a$ was known for quite a long time, see

Biggins, J. D. (1977). Chernoff’s Theorem in the Branching Random Walk. Journal of Applied Probability, 14(3), 630. doi:10.2307/3213469

For the second term and for almost sure behaviour of $Y_n$, see

Hu, Y., & Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. The Annals of Probability, 37(2), 742-789. doi:10.1214/08-AOP419

For the definite answer for the law of $Y_n$ in the non-lattice case, see

Aïdékon, E. (2011). Convergence in law of the minimum of a branching random walk. Retrieved from http://arxiv.org/abs/1101.1810

Note that all of this was already known long before for branching Brownian motion, see the references in the respective articles.

UPDATE: I forgot to add the important reference

Addario-Berry, L., & Reed, B. (2009). Minima in branching random walks. The Annals of Probability, 37(3), 1044-1079. doi:10.1214/08-AOP428

Here, the authors show the above-mentioned result for $E[Y_n]$ in almost complete generality, and exponential tails for $Y_n−E[Y_n]$ as well

1

This paper by Dekking and Host is quite old and much has been done in this area since. Today we know that under reasonable assumptions, there are constants $a\in\mathbb{R}$, $b\ge 0$, such that $E(Y_n) = an + b \log n + O(1)$. How to get the constant $a$ was known for quite a long time, see

Biggins, J. D. (1977). Chernoff’s Theorem in the Branching Random Walk. Journal of Applied Probability, 14(3), 630. doi:10.2307/3213469

For the second term and for almost sure behaviour of $Y_n$, see

Hu, Y., & Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. The Annals of Probability, 37(2), 742-789. doi:10.1214/08-AOP419

For the definite answer for the law of $Y_n$ in the non-lattice case, see

Aïdékon, E. (2011). Convergence in law of the minimum of a branching random walk. Retrieved from http://arxiv.org/abs/1101.1810

Note that all of this was already known long before for branching Brownian motion, see the references in the respective articles.