with a decreasing trend to break a record value that increases with k : suppose the k ∗ -th clone is dominant and W k ∗ > 0 ; then, informally, in order for the k -th clone ( k > k ∗ ) to become dominant, we must have log W k - log W k ∗ ⪆ α ( T k - T k ∗ ) ≃ R ( T k ) - R ( T k ∗ ) ; in particular, if the immigration process is time-homogeneous, the increment between log W k ∗ and log W k must grow linearly with k - k ∗ , requiring the distribution of log W k (given W k > 0 ) to be heavy tailed, which is not possible since W k has finite moments (Harris 1963 ).
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