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## Main Question or Discussion Point

In W. Hoeffding's 1963 paper

[itex]P(\bar{x}-\mathrm{E}[x_i] \geq t) \leq \exp(-2t^2n) \ \ \ \ \ \ (1)[/itex],

where [itex]\bar{x} = \frac{1}{n}\sum_{i=1}^nx_i[/itex], [itex]x_i\in[0,1][/itex]. [itex]x_i[/itex]'s are independent.

Following this theorem he gives a corollary for the difference of two sample means as:

[itex]P(\bar{x}-\bar{y}-(\mathrm{E}[x_i] - \mathrm{E}[y_k]) \geq t) \leq \exp(\frac{-2t^2}{m^{-1}+n^{-1}}) \ \ \ \ \ \ (2)[/itex],

where [itex]\bar{x} = \frac{1}{n}\sum_{i=1}^nx_i[/itex], [itex]\bar{y} = \frac{1}{m}\sum_{k=1}^my_k[/itex], [itex]x_i,y_k\in[0,1][/itex]. [itex]x_i[/itex]'s and [itex]y_k[/itex]'s are independent.

My question is: How does (2) follow from (1)?

-Jan

^{*}he gives the well known inequality:[itex]P(\bar{x}-\mathrm{E}[x_i] \geq t) \leq \exp(-2t^2n) \ \ \ \ \ \ (1)[/itex],

where [itex]\bar{x} = \frac{1}{n}\sum_{i=1}^nx_i[/itex], [itex]x_i\in[0,1][/itex]. [itex]x_i[/itex]'s are independent.

Following this theorem he gives a corollary for the difference of two sample means as:

[itex]P(\bar{x}-\bar{y}-(\mathrm{E}[x_i] - \mathrm{E}[y_k]) \geq t) \leq \exp(\frac{-2t^2}{m^{-1}+n^{-1}}) \ \ \ \ \ \ (2)[/itex],

where [itex]\bar{x} = \frac{1}{n}\sum_{i=1}^nx_i[/itex], [itex]\bar{y} = \frac{1}{m}\sum_{k=1}^my_k[/itex], [itex]x_i,y_k\in[0,1][/itex]. [itex]x_i[/itex]'s and [itex]y_k[/itex]'s are independent.

My question is: How does (2) follow from (1)?

-Jan

^{*}http://www.csee.umbc.edu/~lomonaco/f08/643/hwk643/Hoeffding.pdf (equations (2.6) and (2.7))