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August 29th, 2017, 12:26 AM  #1 
Newbie Joined: Aug 2017 From: Singapore Posts: 2 Thanks: 0  Prove a large number of samplings converge to a set of small number of samples
In a $k$dimensional space, given a set of $n$ points: $X=\{x_1, x_2,..., x_n\}$ which is sampled from an known distribution $P$. Given a set of $m$ points ($m << n$): $Y=\{y_1,y_2,...,y_m\}$ which is also drawn from $P$. Suppose $\{z_1, z_2,...,z_m\} \subset X$ are the closest points (in term of Euclidean distance) of $\{y_1,y_2,...,y_m\}$ respectively. A gap $g$ is defined as follows: $\displaystyle g = max\{z_1  y_1^2, z_2  y_2^2,..., z_m  y_m^2\}$ where $ab^2$ denotes the Euclidean distance between a, b. Prove that if $m$ is fixed, $g \rightarrow 0$ as $n \rightarrow \infty$. I don't know how to start this problem. Thank you. Last edited by thanhvinh0906; August 29th, 2017 at 01:00 AM. 
August 29th, 2017, 01:11 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,309 Thanks: 528 
The applicable theorem is the law of large numbers, but I haven't tried to work it out. I'll leave that to you.

August 30th, 2017, 01:34 AM  #3  
Newbie Joined: Aug 2017 From: Singapore Posts: 2 Thanks: 0  Quote:
Could you please check whether my below proof is correct? I do it in a uniform distribution. Suppose $g = max\{z_1  y_1^2, z_2  y_2^2,..., z_m  y_m^2\} = \ z_{max}  y_{max} \$. For any $\varepsilon > 0$, put a hypersphere radius $\varepsilon$ around $y_{max}$ in that $y_{max}$ is the centroid (*). Suppose there is no sample of $S$ exists in the hypersphere as $n \rightarrow \infty$. (**) We project $y_{max}$ to dimension $k$, denoted as $y_{max,k}$. Suppose the boundary of dimension $k$ is $[a, b]$, i.e. $a \leq y_{max,k} \leq b$. According to the law of large numbers, $E[x_k]$ (the average of $X$ projected to dimension $k$) $\rightarrow (a + b)/2$ as $n \rightarrow \infty$. Now if there is no sample in the hypersphere: \begin{equation} \begin{split} E[x_k] \rightarrow \int_a^b \mathrm{\frac{x}{ba}}\,\mathrm{d}x = \int_a^{y_{max,k}  \varepsilon} \mathrm{\frac{x}{ba}}\,\mathrm{d}x + \int_{y_{max,k} + \varepsilon}^b \mathrm{\frac{x}{ba}}\,\mathrm{d}x \\ = \frac{a+b}{2}  \frac{2\varepsilon y_{max,k}}{b  a} \neq \frac{a + b}{2} \end{split} \end{equation} Equation (1) conflicts the law of large numbers, this indicates that (**) is false; so there exists at least one sample in the hypersphere. This statement accompanied by (*) indicates that $g = \ z_{max}  y_{max} \ \rightarrow 0$ as $n \rightarrow \infty$. Please share if you have another proof. Thank you. Last edited by skipjack; August 30th, 2017 at 04:38 PM.  
August 30th, 2017, 04:27 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,309 Thanks: 528 
There may be a flaw in that as n increases, the set of x's changes and so would the set of z's. Therefore y_{max} may change with n. I had further thoughts after my first post. Qualitatively, a histogram, with narrower intervals for increasing n, of the x values should resemble the density function of the probability. The y values correspond to specific points of the density. The intervals (containing y values) of the histograms should eventually contain some x (therefore z) value close to the y value. I have never taken a course in statistics, although my specialty is probability theory, so I am not sure hot to develop a proof. Last edited by mathman; August 30th, 2017 at 04:29 PM. 

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