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That is, when is the $\underset{n \to \infty}{\lim} \max (X_i-X_{i-1})\rightarrow 0$, where $1<i\leq n$, and $X_i\geq X_{i-1}$ and when is the limit $\neq 0$? The question supposes that the maximum distance between sorted random samplings of some distribution can be made as small as one likes if that number of samples is made sufficiently large. The answer is different for different pdf's.

By way of explanation, what I want to know is when is a space covered in a fine enough mesh of values to reduce the difference between a random variate of a pdf and the pdf itself enough that they tend to the same function. Here it may help to read the partitions of an interval definition needed for this question (as well as for Riemann integration).

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  • $\begingroup$ Carl, exactly what kind of limit do you have in mind? By itself, "lim" in this context has multiple different meanings, which makes it impossible to determine what you are actually asking. And could you explain what "$\Delta$" is intended to mean? $\endgroup$ – whuber Apr 11 '17 at 19:52
  • $\begingroup$ @whuber Good point, I was being unclear, wasn't I. So, clarified as above. $\endgroup$ – Carl Apr 11 '17 at 20:04
  • $\begingroup$ Thank you--but again, what kind of limit are you considering? In probability? Distribution? Almost surely? Something else? $\endgroup$ – whuber Apr 11 '17 at 20:07
  • $\begingroup$ @whuber, well nothing, redundant the $\Delta$ was. My apologies, the question syntax I ported from was confusing. $\endgroup$ – Carl Apr 11 '17 at 20:07
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    $\begingroup$ This question, from the "related" sidebar, has a comment with the following reference: Weiss (SIAM, 1965) On the asymptotic distribution of the largest sample spacing.. This may have what you are looking for? $\endgroup$ – GeoMatt22 Apr 11 '17 at 21:04
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There are certainly distributions for which the stated limit is zero. For example, for the beta distribution (see third simulation image below). Moreover, there are distributions for which it is false, for example the Cauchy distribution. Perhaps a bit of an explanation is in order. For the beta distribution, all of the values sampled are between 0 and 1. Because the values have nowhere else to be and because there are no empty regions in the distribution function, eventually the maximum distance between adjacent sample values will go to zero. A proof of this for complete coverage bounded interval pdf's has been noted by @GeoMatt22 in @MichaelM comment to maximum gap between order statistics of normally distributed random variables.

However, we cannot reduce this maximum distance for Cauchy distribution increased samplings (see last two simulations, below). The Cauchy distribution the support is infinite. Moreover, there are no moments. That is, if we take an average value, it will not be stable and as we continue to collect values, there will be an increasing likelihood of a very large positive or very negative value that is so different from the values near the center of the distribution that an average value will tend to + and - infinity: no matter how positive a value we have for the average value, the very next datum included in the average could cause the sign to be negative and larger in magnitude than it was outrageously positive. That is, the maximum distance between sorted samples from the Cauchy distribution will increase the more samples we collect.

Perhaps the more interesting question is: Will a normal distribution have the property that the maximum distance between values tends to zero for a sufficiently large number of values? This has been asked but not answered before: maximum gap between order statistics of normally distributed random variables. I will take a guess here and say a firm "maybe" because the tails are lighter than exponential. That is, I would expect the maximum distance to go to zero very slowly, or at least decrease somewhat.

Here is a simulation of normal random sample maximum sorted differences for 2 to 10000 independent random samples.

The program (Mathematica):

z={};
Do[
   x=RandomVariate[NormalDistribution[0,1],n];
   x=Sort[x];
   y=Table[x[[i+1]]-x[[i]],{i,1,n-1}];
   z=AppendTo[z,Max[y]],{n,2,10000}];
ListPlot[z,PlotRange->All]
ListPlot[MovingAverage[z,100],PlotRange->All]

The output shows lots of noise in which it is hard to see any particular trend in the extreme values of this plot. enter image description here

On the $y$-axis are max deviations between an $x$-axis number of $N(0,1)$ samplings. However, when we make a moving average of 100 sequential maximum differences we can see that there is a trend:

enter image description here

The question for the normal distribution is then whether a decreased maximum difference for increased sample numbers is useful, and it would not appear to be useful as although there is an overall trend to decreasing maximum difference, the guarantee of it is extremely weak, as the noise envelope renders the trend largely useless even when the maximum difference is taken for 10000 samplings.

Now to show we are not blowing smoke, here is the max random difference plot for 2 to 10000 random samples of beta(0,1) distributions.

enter image description here

Note that this does appear to trend to zero. Now for the Cauchy distribution Cauchy(0,1), this tends to increasingly erratic.

enter image description here

Next, we make a moving average of 100 mean values from $x$ (where $x=n$) realizations, to better show the Cauchy trend for increased $n$.

enter image description here

I suppose that further investigation would include relative tail heaviness, and relative survival functions with comparison to exponential tails. Exponential functions are memory-less, so that may be the point at which the density/distance query is at a breakeven.

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  • $\begingroup$ You lost me in the first paragraph because your argument applies perfectly well to any distribution with unbounded support, regardless of its moments. (These are precisely the distributions for which it is true that "the very next datum included in the average could cause the sign to be negative and larger in magnitude than it was ... positive.") It sounds like you have some particular, but unstated, idea of "limit" in mind--but what is it? $\endgroup$ – whuber Apr 11 '17 at 19:55
  • $\begingroup$ BTW, after skimming the rest of this post I was unable to find an answer to your original question about characterizing the distributions with limiting values of the range (which is what your mathematical expression is). Given the origin of your posts, wouldn't you be much more interested in values of the smallest gap between values rather than the largest? $\endgroup$ – whuber Apr 11 '17 at 19:57
  • $\begingroup$ @whuber Well, no. That the largest interval goes to zero is the Riemann assumption for integration. What I want to know is when is the space covered in a fine enough mesh of values to reduce the difference between a random variate of a pdf and the pdf itself enough that they tend to the same function. $\endgroup$ – Carl Apr 11 '17 at 20:18
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    $\begingroup$ Thank you--your modification to the question cleared up this confusion. $\endgroup$ – whuber Apr 11 '17 at 20:20
  • $\begingroup$ @whuber Actually, thank you. Your perspicacity has, as usual, led to clarity of expression, and, for that it is I, who am in your debt. $\endgroup$ – Carl Apr 11 '17 at 20:44

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