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Topic: problem on record-breaking values in probability
Replies: 14   Last Post: Apr 14, 2013 11:36 PM

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David Bernier

Posts: 3,729
Registered: 12/13/04
Re: problem on record-breaking values in probability
Posted: Mar 11, 2013 12:32 PM
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On 03/11/2013 02:07 AM, James Waldby wrote:
> On Sun, 10 Mar 2013 21:47:27 -0400, David Bernier wrote:
>> On 03/10/2013 08:52 PM, David Bernier wrote:
>>> On 03/01/2013 08:41 AM, David Bernier wrote:
>>>> On 02/27/2013 10:24 PM, David Bernier wrote:
>>>>> On 02/27/2013 04:05 PM, James Waldby wrote:
>>>>>> On Wed, 27 Feb 2013 07:10:08 -0500, David Bernier wrote:
>>>>>>> On 02/27/2013 05:49 AM, David Bernier wrote:
>>>>>>>> On 02/27/2013 05:31 AM, David Bernier wrote:
>>>>>>>>> I used Marsaglia's 64-bit SUPER KISS pseudo-random number generator
>>>>>>>>> to simulate uniform r.v.s on [0, 1] that are independent, as
>>>>>>>>> X_1, X_2, X_3, ad infinitum
>>>>>>>>> For each go, (or sequence) I define its 1st record-breaking value
>>>>>>>>> as R(1) as X_1, its 2nd record-breaking value R(2) as the
>>>>>>>>> value taken by X_n for the smallest n with X_n > X_1, and in general

>>>>>> [ R(k+1) is the value taken by X_n for the smallest n with X_n > R(k)]
>>>>>> ...

> [snip]
>>>>>> [etc]
>>>>>> It would be useful to report the number of trials each simulation
>>>>>> took to find its 20th RBV. If a simulation takes m trials, the

> [snip]
>>>>>> In following, let L(n) = Pr(n'th item of n is lowest). (Distribution
>>>>>> of the lowest item should be similar to distribution of 1-(highest
>>>>>> item).) I suppose that L(n) = 1/n and that the expected value of the
>>>>>> number of record-low-values (RLV's) in m trials is sum{i=1 to m}(1/i),
>>>>>> or about H_m, the m'th harmonic number, which can be approximated by
>>>>>> log(m) + gamma, with gamma = Euler-Mascheroni constant, about 0.5772.

> [snip]
>>>> In the literature, a remarkable article, which may have
>>>> appeared in the Am. Math. Monthly, can be found by
>>>> searching for:
>>>> Breaking Records and Breaking Boards. Ned Glick

> ...
>>> I did long simulations for 12th Record-Breaking Values.
>>> With MatLab, I constructed a histogram of the natural
>>> logarithms of the 76,000 values:
>>> < > .

> ...
>> S_12 is number of trials (steps) taken to find the 12th
>> Record-Breaking Value. On Average, log(S_12) is close
>> to 12 - gamma (gamma is the Euler-Mascheroni constant).
>> A number of 76,000 sequences were generated, each being
>> continued until the 12th Record-Breaking Value for
>> that sequence was found. There is such variance from
>> one sample S_12 to another that I prefer the
>> quantities log(S_12) , for the histograms.
>> Occasionally, an unusually high record is attained
>> in the 1st, 2nd, ... or 11th Record-Breaking Value.
>> That makes breaking the record all the more difficult.
>> In the simulations, the computer would pass (say) three
>> hours or more on the same sequence, with no new output
>> to the file for three or more hours.

> You may already have done so, but if not and if you are going to
> run more simulations, consider (a) profiling the code, and
> (b) trying different compilation options. (a) allows you to find
> out which lines of code use most of the hours of CPU time, so you
> can try alternate ways of coding them. Under (b), starting with
> the same random seed in each case, try optimization options -O1,
> -O2, -O3, -Ofast, timing the execution and also verifying the
> same results. From my interpretation of URL below, those 4
> optimization options are all you need to try.
> <>

Maybe I've spent enough computer time on this distribution
problem. I'm grateful for your posts in reply.

I think I found something known about the limiting distribution
of the k'th record-breaking trial number from the article of
Ned Glick:

From Ned Glick's (UBC) 1978 article:

N_r is the r'th record-breaking time (serial number
of the trial). N_1 = 1 always.

"Also Renyi [30] stated a "central limit theorem" for the random
variables N_r: as r --> oo, the
distribution of (ln(N_r) - r)/sqrt(r) is
asymptotically normal with mean = 0 and variance= 1."
Ned Glick, "Breaking Records and Breaking Boards", 1978

In Renyi's article,

(log(nu_k) - k)/sqrt(k) ~ N(0,1) as k -> oo

"Theorie des elements saillants d'une suite d'observations",

David Bernier


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