Table Of ContentComment on “Towards a large deviation theory for strongly correlated systems”
Hugo Touchette∗
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK
(Dated: September 13, 2012)
I comment on a recent paper by Ruiz and Tsallis [Phys. Lett. A 376, 2451 (2012)] claiming to
have found a “q-exponential” generalization of the large deviation principle for strongly correlated
random variables. I show that the basic scaling results that they find numerically can be reproduced
with a simple example involving independent random variables, and are not specifically related to
the q-exponential function. In fact, identical scaling results can obtained with any other power-law
deformations of the exponential. Thus their results do not conclusively support their claim of a
q-exponential generalization of the large deviation principle.
Ruiz and Tsallis [1] have published a recent a paper β >1. HencethedensityofS concentratesinsomelimit,
n
2 in which they claim to have found a generalized large which is the first ingredient that we need to define an
1 deviation principle (LDP) for aprocess involvingstrongly LDP. However, since
0
correlated random variables. This LDP supposedly takes
2 γ
the form not of an exponential, as in standard large devi- p(Sn =s)∼ nβ−1s2 (4)
p
ation theory [2], but of a so-called q-exponential function,
e
which is one possible generalization of the exponential as n→∞, the concentration to the delta function is not
S
function. My purpose here is to show that the main re- exponential in n, which implies that p(Sn =s) does not
2 sults of [1], which take the form of scaling plots, are not satisfy an LDP. In fact, if we try to define an LDP for Sn
1
specifically related to q-exponentials or correlated ran- using the standard large deviation limit,
] dom variables, and do not conclusively point, as claimed, 1
ch to a q-exponential generalization of the LDP and large I(s)=nl→im∞−nlnp(Sn =s), (5)
deviation theory. I illustrate these points with a simple
e
m examplethatshowshowonemightbemisledintodefining we find that the rate function I(s) is 0 for all s. This
- “q-exponential LDPs” for distributions that do not have meansthatSndoesnotsatisfyanLDP:onecouldsaythat
t in fact the form of a q-exponential. ithasazeroratefunction,butthisgivesusnoinformation
a
t The example is a classic one in large deviation theory about Sn apart from knowing that the concentration of
.s showing that not all sums of random variables have an p(Sn) is slower than exponential. The same conclusion
at LDP [2, 3]. Consider the sample mean applies to any sums of (symmetric) L´evy-stable random
m variables.
- 1 (cid:88)n From this it is tempting to change the definition of the
S = X , (1)
d n nβ i LDP in the following way. Since we know that p(Sn =s)
n i=1 decays to zero for s(cid:54)=0 as a polynomial in n, we could
o
and assume that the X ’s are independent random vari- try to quantify this decay by a rate function defined on
c i
[ ables identically distributed according to the Cauchy dis- a scale different from logarithm. This is in essence the
tribution: proposal of Ruiz and Tsallis [1]. In particular, we could
1
replace, as they do, the normal log with the so-called
v 1 γ
1 p(Xi =x)= πx2+γ2, (2) q-logarithm function defined by
1
x1−q−1
6 whereγ isapositiveparameterthatcontrolsthewidthof ln (x)= , (6)
2 q 1−q
thedistribution. AtthispointIshouldsaythat,sincethe
.
9 Xi’s are assumed to be mutually independent, one might which converges to the normal logarithm for q →1.
0
be tempted to discard this example as being unrelated For the Cauchy example, it is easy to see that
2
to the results of Ruiz and Tsallis [1]. However, it will
1
: becomeclearthatthepropertiesofSn areinfactrelevant ln p(S =s)∼ n(1−β)(1−q) (7)
v to discuss their claims. q n 1−q
i
X Since Cauchy random variables are stable [4], the prob-
as n→∞, so we can choose
r ability density of Sn is also a Cauchy distribution:
a β
1 γ(cid:48) γ q = >1 (8)
p(S =s)= , γ(cid:48) = . (3) β−1
n πs2+γ(cid:48)2 nβ−1
to obtain
Withthisresultwecanseethatp(S =s)concentratesto
n
theDirac-deltafunctionδ(s)inthelimitn→∞whenever ln p(S =s)∼−c n, (9)
q n s
Typeset by REVTEX
2
0 0
(cid:45)200000
(cid:45)500000
(cid:72)(cid:76)ps (cid:72)(cid:76)ps (cid:45)400000
lnq (cid:45)1.0(cid:180)106 n(cid:56)(cid:60)k
l
(cid:45)600000
(cid:45)1.5(cid:180)106
(cid:45)800000
0 5000 10000 15000 20000 0 5000 10000 15000 20000
n n
FIG. 1: Plot of the q-log of the probability density p(s) of Sn FIG. 2: Plot of the κ-log of the probability density p(s) of Sn
for β =1.5 as a function of n. Different lines correspond to for β =1.5 as a function of n for different values of s (from
different values of s (from 0 to 2 in steps of 0.1, from top to 0 to 2 in steps of 0.1, from top to bottom). The plot is for
bottom). The plot is for q=3, the only value of q for which κ=2, the only value of κ for which this plot yields straight
this plot yields straight lines. lines.
where c is some constant that depends only on s. This have obtained similar results, i.e., straight lines that are
s
behavior is illustrated in Fig. 1 with β = 1.5 for which neither necessarily related to the q-exp nor the κ-exp, the
q = 3. Any value of q not given by Eq. (8) will yield a inverseoftheκ-log. Therefore, theirclaimthattheyhave
plot of ln p(S =s) that is not linear in n, as can easily found a generalization of the LDP linked specifically to
q n
be checked analytically and numerically. the q-exp is misleading: the only conclusion that follows
IfwearetofollowRuizandTsallis[1],thenthestraight from their plots is that they study a distribution with
lines seen on the q-log plot of p(S = s) should imply power-law tails, having the same asymptotic form as any
n
that this density has the form of a q-exponential, power-law deformation of the exponential.
Of course, the reason why the κ-log does the same job
exq =[1+(1−q)x]1−1q, (10) astheq-logisthattheybothbehaveasinversepower-laws
for small arguments, so that a function with power-law
the inverse of the q-logarithm. But, as can be seen from
tails must appear linear when transformed with either
Eq. (3), this is an incorrect conclusion: p(S = s) does
n function. Fromthissimplereasoning,anyotherpower-law
not have the form of a q-exponential, although it has the
deformation of the logarithm (there is an infinite number
same tails as this function, so the straight lines that we
of them) will do the job. In fact, we do not even need
see are not a conclusive sign of a q-exponential LDP—
to consider deformations of the logarithm: by inserting
they are only an indication of power-law tail behavior.
p(S =s) into
Any density that has power-law tails will lead to straight n
lines over a wide range of values when plotted on a q-log xr
f (x)= , (13)
scale, and this even if its center does not behave as a r r
power-law, as is the case for the Cauchy sum. Moreover,
we can choose r =1/(1−β) to obtain
it will show straight lines when plotted with any other
power-law deformations of the exponential, so there is fr(p(Sn =s))∼−n, (14)
nothing special about the q-log and the q-exp.
which is just a different way of expressing the power-law
To illustrate this point, consider the so-called κ-
behavior of p(S =s). The same trick can be applied to
n
logarithm introduced by Kaniadakis [5],
the distribution considered in [1] with the same result.
x−κ−xκ Intheend,onecouldtrytodistinguishwhetherthedis-
ln{κ}x= 2κ , (11) tribution of a random variable such as Sn exactly follows
a q-exp, a κ-exp or another power-law deformation of the
whichrecoversthenormallogasκ→0. Thisdeformation
exponential by studying its behavior for small values of n.
of the log is obviously different from the q-log; yet it can
However, this would defeat the purpose of large deviation
be used in the same way as before to transform p(S =s)
n theory, which is to derive general asymptotic results by
into a seemingly linear function of n. In this case, we
considering specific scaling limits of stochastic processes,
have to choose κ=q−1 to obtain
such as the large-n limit studied here, or the long-time or
low-noise limits [2]. It is by considering these limits that
ln p(S =s)∼−d n (12)
{κ} n s
general asymptotics are seen to arise in a broad class of
as n→∞. This is shown in Fig. 2. stochastic processes.
From this it is clear that if Ruiz and Tsallis [1] had Theresultsof[1]canbetakenasasignofaninteresting
decided to use the κ-log instead of the q-log, they would power-lawscalinginthetailsofaprobabilitydistribution.
3
But, as I have shown, they are insufficient to conclude [1] G. Ruiz and C. Tsallis, Phys. Lett. A 376, 2451 (2012).
that this scaling has the form of a q-exponential or that [2] H. Touchette, Phys. Rep. 478, 1 (2009).
it is specifically related to correlated random variables, [3] O. E. Lanford, in Statistical Mechanics and Mathematical
Problems, edited by A. Lenard (Springer, Berlin, 1973),
as implied in [1]. Moreover, as these results only concern
vol. 20 of Lecture Notes in Physics, pp. 1–113.
one specific process, they are obviously not enough to
[4] B. V. Gnedenko and A. N. Kolmogorov, Limit Distribu-
conclude that one has generalized large deviation theory
tionsforSumsofIndependentRandomVariables (Addison-
to strongly correlated random variables. As of now, no Wesley, Cambridge, MA, 1954).
such generalization exists. [5] G. Kaniadakis, Physica A 296, 405 (2001).
∗ Electronic address: [email protected]
