PHYSICAL REVIEW E 89, 052111 (2014)

Sparre-Andersen theorem with spatiotemporal correlations

Roberto Artuso,1,2,* Giampaolo Cristadoro,3,† Mirko Degli Esposti,3,‡ and Georgie Knight3,§

1Center for Nonlinear and Complex Systems, Universita dell’ Insubria, Via Valleggio 11, Como 22100, Italy2I.N.F.N. Sezione di Milano, Via Celoria 16, Milano 20133, Italy

3Dipartimento di Matematica, Universita di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy(Received 22 January 2014; published 8 May 2014)

The Sparre-Andersen theorem is a remarkable result in one-dimensional random walk theory concerning theuniversality of the ubiquitous first-passage-time distribution. It states that the probability distribution ρn of thenumber of steps needed for a walker starting at the origin to land on the positive semiaxes does not depend onthe details of the distribution for the jumps of the walker, provided this distribution is symmetric and continuous,where in particular ρn ∼ n−3/2 for large number of steps n. On the other hand, there are many physical situationsin which the time spent by the walker in doing one step depends on the length of the step and the interestconcentrates on the time needed for a return, not on the number of steps. Here we modify the Sparre-Andersenproof to deal with such cases, in rather general situations in which the time variable correlates with the stepvariable. As an example we present a natural process in 2D that shows that deviations from normal scaling arepresent for the first-passage-time distribution on a semiplane.

DOI: 10.1103/PhysRevE.89.052111 PACS number(s): 05.40.Fb, 02.50.Ey, 05.60.Cd

For more than a century (see, for instance, [1]) randomwalks have played a crucial role as a theoretical tool to modelan impressive number of physical (and not only) problems.Fundamental questions in the theory of stochastic processesare related to the problem of when a variable in a systementers some a priori specified state for the first time: thefirst-passage time. Knowledge of the first-passage-time distri-bution (FPTD) finds application in many diverse areas of thenatural sciences and economics—from the spike distributionin neuronal dynamics, the meeting time of two molecules indiffusion-controlled chemical reactions, the cluster density inaggregation reactions, to the price of a stock reaching a specificvalue and the ruin problem of actuarial science (see [2] for anextensive treatment of the problem, and early references toapplications); in the last decades, relevance of such a theoryto nonequilibrium problems has been exploited [3]. In thisframework the Sparre-Andersen theorem (SA) [4–6] plays anoutstanding role: in physics it has been invoked in the studyof persistence in stochastic spin models and random walks[7], the study of polymer dynamics [8], and in the analysis ofscattering from a Lorentz slab [9]. In particular, SA states thatthe probability that a random walker who starts at the originenters the positive semiaxis for the first time (its first-passagetime) after n steps is independent of the particular details of thejump length distribution, provided that it is symmetric aboutthe origin and continuous: in such a case the decay of theFPTD has the universal asymptotics n−3/2. Here the conceptualimport of SA is apparent: it provides an outstanding example ofuniversality in the realm of stochastic processes, with a hugeuniversality class. The origin of such universality is subtlerthan other examples in probability: to our knowledge it cannot

*roberto.artuso@uninsubria.it†giampaolo.cristadoro@unibo.it‡mirko.degliesposti@unibo.it§georgiesamuel.knight@unibo.it

be encompassed by simple renormalization, like the centrallimit theorem (see, for example, [10]).

However, in most physical situations, spatiotemporal cor-relations exist in the time taken to perform a jump of a certainlength, a noteworthy example being represented by Levyglasses [11]: an optical (quenched) Levy walk pinball, wherethe number of scattering events n is not an accessible quantity,while the relevant variable is the (continuous) physical timet . Here we reconsider SA in order to take into accountsuch correlations in a continuous-time-random-walk (CTRW)jump model [12] (not relying on subordinations schemes;see [6,13]). We hope and expect that this will foster widerapplications of this beautiful result from probability theory.

The time costs associated with each jump event we considerare quite general and include the case where they correlate onlyto the length of a jump (as in a velocity model). Extending toour setting the method in Feller [5] (Chap. XII) we derive theLaplace transform of the distribution of first entry time intothe positive semiaxis. To illustrate the importance of taking thereal time of the process into account, we introduce a simpleexample of a random walk in the plane where the universal SAscaling does not hold in continuous time, while being valid forthe discrete time. In particular the FPTD does not decay liket−3/2. We finally comment on the results for Levy walks.

Consider a random walker on the real line starting atthe origin and let (X1,T1), (X2,T2), . . . , (Xj,Tj ), . . . be asequence of independent identically distributed (iid) pairsof random variables corresponding to the steps X and theassociated times T . More precisely, at the event i the randomwalker waits for a time Ti after which an instantaneous jump oflength Xi takes place. Denote the joint probability density ofX and T by p(x,t). After n steps the walker will be in positionSn at time Cn where

S0 = 0, Sn =n∑

k=0

Xk, (1)

C0 = 0, Cn =n∑

k=0

Tk. (2)

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ARTUSO, CRISTADORO, ESPOSTI, AND KNIGHT PHYSICAL REVIEW E 89, 052111 (2014)

We can think of the random variables C as a “time cost”associated with the process. Note that while we assume thepairs to be iid, we do not impose a priori that Ti is independentof Xi [i.e., in general p(x,t) does not factorize], as per usualin CTRW models.

We are interested in the probability distribution of thefirst entrance into the positive axis, i.e., what is termedin the mathematical literature as the first ladder time. Inthe (discrete-time) watch ruled by events this will be theprobability ρn that n is the first index such that Sn > 0. In thisframework, SA theorem states that if the distribution of jumpsis continuous and symmetric, then the generating function ofρn is

∞∑n=1

ρnzn = 1 − √

1 − z. (3)

Here we are interested in a continuous-time setting, so themain quantity under investigation will be π (t), namely theprobability density for the first ladder time,

π (t) =∞∑

n=1

πn(t), (4)

where πn(t) denotes the probability density to land in thepositive semiaxis at time t for the first time after the nth “event”took place. We will provide a closed formula for the Laplacetransform of such a distribution:

π (s) =∫ ∞

0π (t) e−st dt

=∞∑

n=1

∫ ∞

0πn(t) e−st dt =

∞∑n=1

πn(s). (5)

Before entering into the derivation of a general expressionfor π (s) we first consider the simplified case p(x,t) =r(x)ψ(t), where step lengths and time costs are uncorrelated,

πn(s) = ρnψ(s)n, (6)

and thus, provided r(x) is continuous and symmetric, Eq. (3)implies

π (s) = 1 −√

1 − ψ(s). (7)

This result is not surprising in such a case: when the time isindependent of the jump, the effective time cost of a stepis ruled by a random (subordinated) clock that effectivelyreplaces the role of the simple clock in the standard SA result(see also [14]).

In the general case the factorization in Eq. (6) is not apriori valid and more care should be taken. There exist anumber of techniques by which the general problem can behandled, such as the Wiener-Hopf factorization scheme; itturns out that for our purposes a particularly simple procedureis via a combinatorial lemma (closely following the approachof [5]). Indeed the crucial observation is that, thanks to itscombinatorial nature, such a lemma can be equally appliedto the subset of all paths that have the same time cost, oncesuch a time cost is invariant under permutations of the jumpsin the given path, as in our setting. A derivation of a generalexpression for π (s) thus passes through grouping the set of

exiting paths into subsets with fixed exit time where the lemmastill applies. In the following we will show how this idea leadsto resumming the contributions of each subset and finally toan explicit expression for π(s).

The combinatorial lemma takes into account a genericsequence consisting of m events, a corresponding set ofm cyclical permutations (Xp, . . ,Xm,X1, . . ,Xp−1) and theirpartial sums S

(p)j , and states that, if Sm > 0, then, if r denotes

the number of cyclically permuted sequences for which m is aladder index, then r � 1, and for each such a sequence m is therth ladder index. An auxiliary set of binary random variablesis then defined {ε(v)

[k];m}, in such a way that ε(v)[k];m = 1 if m is the

kth ladder index for the vth cyclic permutation, and ε(v)[k];m = 0

otherwise.It is also instrumental to consider the probability densities

that t is the kth ladder time (kth absolute maximum in thesequence of partial sums): π[k](t) = ∑

m π[k];m(t), if, onceagain, we partition the physical time into “events.” Such akth ladder time distribution is useful since it appears whenconsidering powers of the generating function (5):

π (s)j =∞∑

m=1

∫ ∞

0dT e−sT π[j ];m(T ), (8)

that follows from the fact that

π[j ];m(T ) =∑

m1+···+mk=m

∫dt1 · · · dtkδ

(t −

k∑i=1

ti

)

×∏j

πmj(tj ).

The connection with the starting issue is that

π[j ];m(t) = Prob(ε

(1)[j ];m = 1 | Cm = t

), (9)

where Prob(·) denotes the probability density function ofthe corresponding variable. Since all the variables {ε(v)}have a common distribution, then (here 〈·〉 will indicate thecorresponding expectation value)

π[j ];m(t) = 1

m

⟨ε

(1)[j ];m + · · · ε(m)

[j ];m

∣∣Cm = t⟩, (10)

and by taking into account the combinatorial lemma, we seethat ∑

j

1

jπ[j ];m(t) = 1

mProb(Sm > 0 | Cm = t). (11)

If we now take a sum over m and Laplace transform (11) weobtain

− ln[1 − π (s)] =∞∑

m=1

1

m

∫ ∞

0Prob(Sm > 0 | Cm = t) e−st dt.

(12)Equation (12) is the main general result.

Under the further assumption that p(x,t) is continuous inx and spatially symmetric p(x,t) = p(−x,t), then

Prob(Sm > 0 | Cm = t) = 12 Prob(Cm = t).

As the variables Tj are identically distributed (i.e., they donot depend on j ), by the convolution theorem Eq. (12)

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SPARRE-ANDERSEN THEOREM WITH SPATIOTEMPORAL . . . PHYSICAL REVIEW E 89, 052111 (2014)

reduces to

π(s) = 1 −√

1 − c(s), (13)

where

c(s) =∫ ∞

−∞

∫ ∞

0p(x,t) e−st dtdx (14)

is the cost-associated characteristic function. Note that ifp(x,t) factorizes then c(s) = ψ(s), thus recovering Eq. (7).

A particularly interesting case is when the time length isdirectly correlated to the step length so that we have p(x,t) =r(x)δ(t − |x|) (as in the Levy walk, note also the discussion atthe end of the paper). Indeed it is straightforward to observethat

(1) if steps have a finite average length 〈l〉 then for small s

we have c(s) ∼ 1 − 〈l〉s and thus π (s) ∼ 1 − (〈l〉s)1/2 givingthe conventional SA asymptotics π (t) ∼ t−3/2 via Tauberiantheorems;

(2) if steps have an infinite average length and theirdistribution asymptotically decays as r(x) ∼ |x|−(1+α) α ∈(0,1), then for small s we have c(s) ∼ 1 − Bsα and thusτ (s) ∼ 1 − (Bsα)1/2 giving the asymptotic t−(1+α/2), similarlyto the subdiffusive uncorrelated case (see for example [15,16],and [17,18]).

Now, we will use Eq. (13) to discuss a process in 2D thatshows deviations from the SA scaling for the first-passage-timedistribution on a semiplane. At each step, a vertical barrier ischosen at a random x position drawn from a distribution r(x)on R. When hitting the barrier the particle is scattered withuniform outgoing angle ψ , while maintaining unit velocity;then a new random barrier is placed. We will derive theFPTD on the negative semiplane for a particle starting at theorigin. Note that such a problem is solved by just projectingon the x axis. The physical time associated with each stepis l/ cos(ψ); we can use Eq. (13) to study the distributionof first entry times into the region x > 0 in this setting.From a purely one-dimensional perspective this is a modelwith a fluctuating velocity; other types of fluctuating velocityhave been discussed in [19]. The outgoing angle introduces afurther random variable ξ = 1

|vx | = 1cos ψ

so our starting objectis (assuming unit velocity)

pξ (x,t) = φ(|x|)δ(t − |x|ξ ). (15)

In order to compute the asymptotic behavior of the FPTD wehave to evaluate

c(s) =∫ ∞

0dl φ(l)

∫ ∞

1dξ ℘(ξ )

∫ ∞

0dt e−st δ(t − lξ ).

(16)Since our interest is in investigating the anomalies induced bythe ξ distribution, we take the simplest possible step-lengthdistribution φ♠(l) = δ(l − 1). The generating function thussimplifies to

c♠(s) = 2

π

∫ ∞

1dξ e−sξ 1

ξ√

ξ 2 − 1, (17)

and we are interested in the small s asymptotics; this is easilyobtained if we notice that

dc♠(s)

ds= − 2

π

∫ ∞

1dξ e−sξ 1√

ξ 2 − 1= − 2

πK0(s), (18)

where K0(s) is a modified Bessel function. Since for small s

K0(s) ∼ − ln s, (19)

we have that

c♠(s) ∼ 1 + s ln s, (20)

thus implying a logarithmic correction π (t) ∼ (ln t)12 t−

32 for

large t again via Tauberian theorems. It is easy to check that thelogarithmic correction appears for any choice of step lengthdistribution φ(l). Finally, we may verify that our calculationsremain essentially unaltered if we go to a 3D setting, whereparallel scattering planes are distributed according to φ(d),and each scattering event results in a uniformly distributedoutgoing angle. Take for instance planes parallel to πxy , sothat ξ = 1/ cos θ . Then we have

℘(θ ) = 12 sin θ, θ ∈ [0,π/2], (21)

and, by following former steps, we get, through a change ofvariable

℘(ξ ) = 1

ξ 2, ξ ∈ [1,∞), (22)

leading to

c(3)♠ (s) =

∫ ∞

1dξ e−sξ 1

ξ 2= E2(s), (23)

where E2 is an exponential integral function, and again thesmall-s expression is

c(3)♠ (s) ∼ 1 + s ln s. (24)

To conclude, we compute the FPTD for a walker not startingat the origin but at a random position X0, distributed as allother X. This is equivalent to saying that at the ith eventthe random walker first jumps a distance given by Xi andthen waits for a time given by Ti , differently from the modelsintroduced before, where the jump is considered completedonly after the associated time cost is passed. If we let the Tj

be distributed independently of the step lengths Xj we have,similarly to Eq. (6), that πn(s) = ρnψ(s)n−1 and thus π(s) =

1ψ(s)

[1 −√

1 − ψ(s)], which is somewhat surprising given thatthe Pollatzeck-Spitzer formula is strongly dependent on thestarting point x0 [20]. As an illustrative example, considera discrete-time one-dimensional random walker with jumpstaken uniformly on [−1,1] [21] and with the time associatedwith each step chosen at random, independent of the step, tobe equal to 1 or 2 with equal probability. That is, we havep(x,n) = r(x)q(n), where

r(x) = 12 for x ∈ [−1,1],

(25)q(n) = 1

2 [δ(n − 1) + δ(n − 2)].

If we start the random walker from X0 = 0, corresponding tothe wait-then-jump model, we have from Eq. (13) the gener-ating function (that takes the role of the Laplace transform indiscrete time) given by

π (z) =[

1 −√

1 − 1

2[z + z2]

], (26)

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ARTUSO, CRISTADORO, ESPOSTI, AND KNIGHT PHYSICAL REVIEW E 89, 052111 (2014)

whereas if we start the random walker at a random initialposition distributed as X, corresponding to the jump-then-waitmodel, we have

π (z) = 2

(z + z2)

[1 −

√1 − 1

2[z + z2]

]. (27)

Note that the two models presented can be used to studythe first passage time problem for the CTRW velocity model,where the random walker moves with a constant velocitybetween steps. More precisely, for this model we investigatethe first time the walker traverses the origin rather than thefirst time a scattering event occurs in the positive semiaxes.Indeed, given the distribution of jumps and velocities definingthe CTRW velocity model, there are two obvious relatedjump-then-wait and wait-then-jump models associate with it,where the corresponding walkers respectively always precedeand follow the velocity-model walker. Let tw be the time of

first passage for the wait-then-jump walker, let tj be the timeof the first passage of the jump-then-wait walker, and let tcbe the time of first crossing of the continuous-time walker.For every given walk one has tj � tc � tw, which impliesP{tj > t} � P{tc > t} � P{tw > t}.

For a CTRW velocity model with independent step lengthsand velocities (provided the corresponding average time costis finite), the time-asymptotic behavior for the two boundingprocesses is the same and of SA type (∼ t−

32 ) and thus we can

conclude that the decay of the first return time for the velocitymodel is also of SA type in this case. Whether SA scalingis found for the CTRW velocity model with spatiotemporalcorrelations (as for example for a bona fide Levy walk) or formore general systems such as those with correlations betweenthe step times (as in [22]) remains an open question.

We acknowledge partial support by the FIRB projectRBFR08UH60 (MIUR, Italy).

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