Improve doc on distributions and period

src/guide-rngs.md

@@ -175,40 +175,40 @@ not work well with all algorithms.
 
 The *period* or *cycle length* of a PRNG is the number of values that can be
 generated after which it starts repeating the same random number stream.
-Many PRNGs have a fixed-size period, but for some only an expected average
-cycle length can be given, where the exact length depends on the seed.
-
-On today's hardware, even a fast RNG with a cycle length of *only*
+Many PRNGs have a fixed-size period, while for others ("chaotic RNGs") the
+cycle length may depend on the seed and short cycles may exist.
+
+Note that a long period does not imply high quality (e.g. a counter through
+`u128` values provides a decently long period). Conversely, a short period may
+be a problem, especially when multiple RNGs are used simultaneously.
+In general, we recommend a period of at least 2<sup>128</sup>.
+(Alternatively, a PRNG with shorter period of at least 2<sup>64</sup> and
+support for multiple streams may be sufficient. Note however that in the case
+of PCG, its streams are closely correlated.)
+
+*Avoid reusing values!*
+On today's hardware, a fast RNG with a cycle length of *only*
 2<sup>64</sup> can be used sequentially for centuries before cycling. However,
-this is not the case for parallel applications. We recommend a period of
-2<sup>128</sup> or more, which most modern PRNGs satisfy. Alternatively a PRNG
-with shorter period but support for multiple streams may be chosen. There are
-two reasons for this, as follows.
-
-If we see the entire period of an RNG as one long random number stream,
-every independently seeded RNG returns a slice of that stream. When multiple
-RNG are seeded randomly, there is an increasingly large chance to end up
-with a partially overlapping slice of the stream.
-
-If the period of the RNG is 2<sup>128</sup>, and an application consumes
-2<sup>48</sup> values, it then takes about 2<sup>32</sup> random
-initializations to have a chance of 1 in a million to repeat part of an
-already used stream. This seems good enough for common usage of
-non-cryptographic generators, hence the recommendation of at least
-2<sup>128</sup>. As an estimate, the chance of any overlap in a period of
-size `p` with `n` independent seeds and `u` values used per seed is
-approximately `1 - e^(-u * n^2 / (2 * p))`.
-
-Further, it is not recommended to use the full period of an RNG. Many
-PRNGs have a property called *k-dimensional equidistribution*, meaning that
+when multiple RNGs are used in parallel (each with a unique seed), there is a
+significant chance of overlap between the sequences generated.
+For a generator with a *large* period `P`, `n` independent generators, and
+a sequence of length `L` generated by each generator, the chance of any overlap
+between sequences can be approximated by `Ln² / P` when `nL / P` is close to
+zero.
+
+*Bias and the birthday paradox!*
+Many PRNGs have a property called *k-dimensional equidistribution*, meaning that
 for values of some size (potentially larger than the output size), all
 possible values are produced the same number of times over the generator's
-period. This is not a property of true randomness. This is known as the
-generalized birthday problem, see the [PCG paper] for a good explanation.
-This results in a noticable bias on output after generating more values
-than the square root of the period (after 2<sup>64</sup> values for a
-period of 2<sup>128</sup>).
-
+period. For some uses this may be a nice property to have, but it may suppress
+duplicates expected in a truely random generator;
+this is known as the generalized birthday problem
+(see the [PCG paper] for a good explanation). In short, this can result in
+noticable bias in output after sampling more values than the square root of the
+period (after 2<sup>64</sup> values for a period of 2<sup>128</sup>).
+
+For more on this topic, please see these
+[remarks by the Xoshiro authors](http://prng.di.unimi.it/#remarks).
 
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