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Consider $ X^{*}_1,..., X^{*}_{m_n}$, $m_n\\geq 1$, a bootstrap sample, resulting from re-sampling or stochastically re-weighing a random sample $X_1,...,X_n$, $n\\geq 1$. Put $\\bar{X}_n= \\sumn X_i/n$, the original sample mean, and define $\\bar{X^*}_{m_n}=\\sumn v_i^{(n)} X_i/m_n$, the bootstrap sample mean. 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Let ${v_1^{(n)},...,v_n^{(n)}}_{n=1}^\\infty$ be vectors of non-negative random variables (weights), independent of the data sequence ${X_1,...,X_n}_{n=1}^\\infty$, and put $m_n=\\sumn v_i^{(n)}$. Consider $ X^{*}_1,..., X^{*}_{m_n}$, $m_n\\geq 1$, a bootstrap sample, resulting from re-sampling or stochastically re-weighing a random sample $X_1,...,X_n$, $n\\geq 1$. Put $\\bar{X}_n= \\sumn X_i/n$, the original sample mean, and define $\\bar{X^*}_{m_n}=\\sumn v_i^{(n)} X_i/m_n$, the bootstrap sample mean. 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