A major mathematical advance after I wrote the materials in the parent page focuses on additional stochastic dependence among particles (processes) arising from position depndence of intensity functions. This page is a very brief introduction on the subject.
(Warning:-) This page is more for enthusiast on mathematical analysis technique than for practical, social or economical, interest. This page assumes background on basic technique on mathematical analysis.
For a handy access, here is a pdf of slide for a talk `Hydrodynamic limit of stochastic ranking process' in the meeting `Stochastic analysis on large scale interacting systems' held at Osaka Univ., on 2019.11.
If your terminal allows Japanese Kanji and Kana characters, here is a link to corresponding introduction in Japanese. This page contains a smaller subset of the corresponding Japanese page, assuming that a reader of this page has easier linguistic accessto the original papers than a reader of Japanese pages:
The stochastic ranking process (denoted hereafter by SRP) considers the position (or rank) of each particle as a stochastic process. a particle stays in the queue of the particles between the `jumps to the top, and the relative position of a particle, in the spatial interval in which the queue is contained, is determined, in the time interval between the jumps to the top, by the jumps of other particles behind (at the lower ranks) in the queue. This implies that the processes of the particle positions have dependences (in the sense of stochastic variables) that they mutually determine the random time developement of other particles.
While the positions of the particles have dependence, the jump times to the top are defined to be independent among different particles in the prototype SRP. To be a little more specific, note first that the accumulated number of jumps to the top for each particle can be seen as a point process, and the time of jump corresponds to the arrival time of the point process. The prototype SRP chooses this point process to be a Poisson point process. The expectation of the process is called the intensity, and a Poisson point process is defined to have intensity constant in time.
If the intensity has time dependence, we still have independence of jump times among different particles. The story changes when we go further and consider spatial (position, or rank) dependence for intensity functions. Then the arrival times of the point processes (the jump times of the particles) depend on particle positions, which in turn depend on positions of other particles, and every stochastic variables in the model are dependent. This fact drastically complicates the mathematical analysis of SRP with position dependent intensities, compared to the prototype SRP. This web page is intended to focus on this complication. Introducing spatial independence on the intensity function, not only is a natual generalization of the model, but also has a practical application in analyzing the effect of good ranking of a book in an online bookstore enhancing the attention of potential readers.
Before moving on, we are aware that
we have to formulate mathematically the meaning of `intensity function
determining a point process being dependent on other
stochastic processes'.
This is fortunately a standard matter in the theory of
stochastic differential equations.
See our first paper on SRP with time dependent intensities for the
formulation:
T. Hattori, S. Kusuoka,
Stochastic ranking process with space-time dependent intensities,
ALEA, Lat. Am. J. Probab. Math. Stat. 9(2) (2012) 571-607
200KB pdf file.
Sometimes there are phenomena (which is called `universality' in
some field of theoretical research)
that a local perturbation does not affect asymptotic behavior.
If introduction of stochastic dependence through position dependence
of intensity functions of the point processes associated to the particles
of SRP dissappears in the infinite particle (hydrodynamic) limit,
then things would go more or less easily.
This naive expectation is wrong for the present issue.
It would be enough just to mention that
the point process in the hydrodynamic limit
lacks independent increment property.
(Note that the Poisson process is a typical example of a process
with independent increments,
and the property is shared with an inhomogeneous Poisson process with
time dependent intensity,)
I could find no previous research of the process,
so I named it a point process with last arrival time dependent intensity
in the paper:
T. Hattori,
Point process with last-arrival-time dependent intensity and 1-dimensional incompressible fluid system with evaporation,
Funkcialaj Ekvacioj 60 (2017) 171-212.
240KB pdf file.
See the paper for the definition and some interesting and useful properties.
(for analysis of SRP with position dependent intensities).
In short, the process is difficult but tractable;
difficult because the process lacks the independent increment property,
and tractable in the sense that, for example, there is an explicit
functional equation for the time dependence of the probabilities,
and series expansion formulas,
degenerating to the simple properties of the Poisson process in
the constant intensity limit.
This balance of difficulty and tractability suggests there would
be still more to be explored on this process. See also
T. Hattori,
Open problems to an infinite system of quasi-linear partial differential equations with non-local terms,
Symposium on Probability Theory, Kyoto, 2014/12,
RIMS Kokyuroku 1952 (2015) 9-16.
120KB pdf file.
We aim at proving that the motion of a particle in the hydrodyamic limit is a (deterministic) position as a function of time between the arrival times (jump times to the top) of the corresponding point process. This deterministic function of time could be viewed as a `stream line', with the domain of time variable an interval starting at either 0 or positive, and for the former case the value at time 0 represents the (given) initial condition, while for the latter case, the value of the stream line function at the starting time is 0, the upper stream end of the spatial interval.
We call a collection of stream lines over all the initial or upper stream boundary points, a flow. The hydrodynamic limit (to be proved) is a motion of a (self-consistent) flow, where a (tagged) particle moves along a stream line with occasional random jumps to the top given by the arrival times of a point process with last arrival time dependent intensity. (The last arrival time dependence comes from the choice of the stream line which the particle follows at the time of the next jump.)
A corresponding finite particle number `discretization' is a SRP with intensity determined by the flow in the sense of the previous paragraph, and with independence among the jump times of different particles. We refer to this model as a flow driven SRP, and consider it as an intermediate model for proving the hydrodynamic limit of the original SRP.
Since a flow driven SRP is determined by a given fixed flow, the corresponding point processes are (not of independent increment, but) independent among different particles. Thus we can apply a strong law of large numbers for an average of independent function valued stochastic variables (to an appropriate quantity), to prove existence of hydrodynami limit of a flow driven SRP.
Note here that while an application of law of large numbers for an average of independent stochastic variables is shared with the analysis of the prototype SRPs, unlike the prototype, this is not the end of story for the flow driven SRP: The flow drive SRP is an intermediate model, and we have to evaluate the difference of this model and the original SRP with position dependent intensities, to prove that the difference dissappear in the hydrodynamic limit. We apply a Gronwall type of argument (treating stochastic dependence through position dependent intensities in the original model as a perturbation) for this completion of proof. To have easier estimates in the perturbation theory, we should have stronger convergence in the non-perturbed system.
Concerning the dissappearance of fluctuations, the main theme of the law of large numbers, we adopt the complete convergence of Hsu and Robbins, which is a `strongest' version of strong convergence of a sequence of random variables, and is a natural choice for a problem of approximating an average of finite fixed number of random variables by a limit, such as thermodynamic and hydrodynaic limit, where in nature there are a finite fixed number of molecles. (Bookstores also contain a finite number of books, not infinite.)
Concerning the time dependence of processes,
we note that since our limit is expected to have dependent increments,
we cannot `parallel transport' unperturbed quantities,
in the final stage of the proof of hydrodynamic limit of the original SRP,
hence we should evaluate the increments for all time interval,
hence uniform convergence with respect to starting and ending point of
a time interval should be proved for the flow driven SRP.
We abbreviate as doubly uniform complete law of large numbers,
and refer to the original paper:
T. Hattori,
Doubly uniform complete law of large numbers for independent point processes,
Journal of Mathematical Sciences the University of Tokyo 25 (2018) 171-192.
180KB pdf file
for explicit results and proofs.
The results and proofs in the paper,
besides convergence, gives estimates on the rate of convergence
through summability of moments,
which eases Gronwall type (perturbation) argument in the
final sequence of the proof of hydrodynamic limit of the original SRP.
A key quantity to evaluate for a proof of existence of hydrodynamic limit of SRP is, the joint empirical distribution of intensity (jump rate) and position (scaled rank), which has been the key since the first paper on the prototype SRP. Meanwhile, doubly uniform complete law of large numbers are formulated for real valued processes or stochastic variables taking values in a space of functions from real interval to reals, so that we need to map a difference of distributions to a difference in reals, and there the choice of metric on the space of distributions matters.
At the time of Hattori-Kusuoka paper, a point process with last arrival time dependent intensity and flow driven SRP were not found yet. More sophisticated machinery such as submartingale inequalities were employed instead, leading to stronger assumptions (narrower in application). Among the difference in assumptions with the latest results, is the choice of total variation norm for the distribution space. Assuming convergence with respect to this choice (and henceforth proving the convergence with the same norm) implies, in particular, that the discrete distributions (such as empirical distributions) cannot converge to continuous distributions such as a Pareto distribution which is commonly used in the practical or statistical analysis on long tail phenomena.
Moreover, convergence with respect to total variation
also implies that the cancellation of stochastic fluctuations,
i.e., the mechanism of law of large numbers, are solely among the
particles (point processes) with exactly the same intensity.
This is not the true story.
In the long tail phenomena,
the particles usually have different intensities and
most of the particles have small intensities
(e.g., professional books do not sell!) so that for each particle
fluctuations do not cancel.
It is the cancellation of fluctuations among processes with
different intensities which gives a statistically stable
distribution in the ranking queue.
To incooporate this picture,
we have to choose a standard weak convergence topology for the
distributions,
which the latest paper handles:
T. Hattori,
Cancellation of fluctuation in stochastic ranking process with space-time dependent intensities,
Tohoku Mathematical Journal 71(3) (2019) 359-396.
270KB pdf file,
a `dense' paper, containing everything specific to the proof of
hydrodynamic limit of SRP with position dependent intensities.
(I made it dense and concise,
assuming that there are not much demand on details.)
Concerning the final sequence of proof, we need 2 more ingredients, for which I also refer to the above paper: