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A note on the transience of critical branching random walks on the line.

Gantert and Muller (2006) proved that a critical branching random walk (BRW BRW Business Review Weekly (business magazine; Melbourne, Victoria, Australia)
BRW Business Report Writer
BRW Barrow, AK, USA - Barrow (Airport Code)
BRW Business Requirement Worksheet
) on the integer lattice In mathematics, The n-dimensional integer lattice (or cubic lattice), denoted Zn, is the lattice in the Euclidean space Rn whose lattice points are n-tuples of integers.  is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost right·most  
Farthest to the right: in the rightmost lane of the highway.

Adj. 1. rightmost - farthest to the right; "in the rightmost line of traffic"
 particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).

Keywords: branching random walk, critical regime, recurrence recurrence /re·cur·rence/ (-ker´ens) the return of symptoms after a remission.recur´rent

, transience, minimal and maximal max·i·mal
1. Of, relating to, or consisting of a maximum.

2. Being the greatest or highest possible.
 position, random weighted location measure, renewal theory

1 Introduction

Consider a cloud of particles which moves on the line as follows. Initially there is one particle sitting at the origin which after one unit of time splits into a random number of new particles having distribution [([p.sub.j]).sub.j [greater than or equal to] 0], where [p.sub.0] = 0. The daughter particles are then independently displaced relative to their mother's site in accordance with the same step size distribution Q, say. This process continues indefinitely, i. e., each new born particle splits after one unit of time in accordance with [([p.sub.j]).sub.j [greater than or equal to] 0], and the relative displacement of each daughter particle with respect to its mother's site has distribution Q and is independent of the relative displacements of its siblings as well as of the history of the process. This model describes a special nonexfinctive BRW the specialization being that the relative displacements of siblings (given their total number) are i.i.d. rather than chosen from a general point process on R. Likewise, one may adopt the viewpoint as in Gantert and Muller (2006) that any new born particle lives forever and performs a random walk with step size distribution Q. Right before each jump it produces j - 1 daughter particles with probability [p.sub.j] (j [greater than or equal to] 1) which start independent random walks of the same kind at initial positions relative to their mother's site chosen in accordance with Q. Note that the cloud size evolves as a simple nonextinctive Galton-Watson process [([Z.sub.n]).sub.n [greater than or equal to] 0] with one ancestor ANCESTOR, descents. One who has preceded another in a direct line of descent; an ascendant. In the common law, the word is understood as well of the immediate parents, as, of these that are higher; as may appear by the statute 25 Ed. III. De natis ultra mare, and so in the statute of 6 R.  and offspring distribution [([p.sub.j]).sub.j [greater than or equal to] 0. A more detailed specification of the model will be given in Section 2 after having described the main problem to be addressed in this note.

Suppose that Q has positive mean but also positive mass on (-[infinity], 0). Then all particles in the cloud Refers to the operation taking place within a network. See cloud.  are moving towards [infinity] with probability one while, on the other hand, their trajectories will also have negative excursions. Hence the ever increasing number of independently moving particles might cause bounded neighborhoods of 0 be visited infinitely often if the cloud is growing fast enough. We are thus led to the question whether there exists a branching threshold [m.sup.*] such that any bounded neighborhood of 0 (or any other x [member of] R in the same irreducibility ir·re·duc·i·ble  
Impossible to reduce to a desired, simpler, or smaller form or amount: irreducible burdens.

 class) is recurrent, i.e., a.s. visited by infinitely many particles, if m > [m.sup.*], while being transient, if m < [m.sup.*], where m := [[summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) ].j [greater than or equal to] 1] [jp.sub.j] is the mean offspring (including the reproducing particle). In the case of an irreducible irreducible /ir·re·duc·i·ble/ (ir?i-doo´si-b'l) not susceptible to reduction, as a fracture, hernia, or chemical substance.

 lattice (theory) lattice - A partially ordered set in which all finite subsets have a least upper bound and greatest lower bound.

This definition has been standard at least since the 1930s and probably since Dedekind worked on lattice theory in the 19th century; though he may not
 random walk, the positive answer has been given by Comets et al. (1998) in their analysis of the more general BRW in random environment. Moreover, transience holds true in the boundary case The term boundary case is frequently used in software engineering to refer to the behavior of a system when one of its inputs is at or just beyond its maximum or minimum limits. It is frequently used when discussing software testing.  m = [m.sup.*], as recently been proved by Gantert and Muller (2006) within the more general framework of branching Markov chains where the random walk is replaced with an arbitrary irreducible Markov chain (probability) Markov chain - (Named after Andrei Markov) A model of sequences of events where the probability of an event occurring depends upon the fact that a preceding event occurred.

A Markov process is governed by a Markov chain.
 on a countable state space, see Benjamini and Peres (1994) for the basics and also the classification of possible regimes of such models as to their recurrence behavior. Further results on the recurrence or transience of various generalizations of the classical BRW may be found in Machado and Popov (2000, 2003); Machado et al. (2001); Menshikov and Volkov (1997). An essential tool in these works is the use of Lyapunov test functions, and this constitutes the main difference to the present note. Our main purpose is in fact to demonstrate how certain random weighted location measures to be defined in Section 2 and their connection to renewal and fluctuation theory for classical random walks (cf. Section 3) may be utilized as an alternative tool in order to not only reproduce the afore-mentioned results for lattice BRWs but to provide also without much additional effort an extension to the situation where Q is nonlattice. Our main result will be stated in Section 2 after the necessary formal details including a definition of recurrence and transience for BRWs. The proof will be given in Section 4. Finally, Section 5 provides some fairly sharp information on the position of the leftmost left·most  
Farthest to the left: in the leftmost lane of traffic.

Adj. 1. leftmost - farthest to the left; "the leftmost non-zero digit"
 and the rightmost particle in a critical BRW as time goes to infinity. Our theorems This is a list of theorems, by Wikipedia page. See also
  • list of fundamental theorems
  • list of lemmas
  • list of conjectures
  • list of inequalities
  • list of mathematical proofs
  • list of misnamed theorems
  • Existence theorem
 stated there follow without much ado Ado (ä`dō), city (1987 est. pop. 287,000), SW Nigeria. Located in a region where rice, corn, cassava, and yams are grown. Traditionally an important cotton-weaving town, Ado also manufactures bricks, tile, and pottery.  from a recent result by Hu and Shi (2008).

2 Model description and main results

Let V be the infinite Ulam-Harris tree with vertex A corner point of a triangle or other geometric image. Vertices is the plural form of this term. See vertex shader.  set [[union].sub.n [greater than or equal to] 0] [N.sup.n] where N = {1, 2, ...} denotes the set of positive integers and [N.sup.0] := {[empty set]} by convention. Each vertex v = ([v.sub.1], ..., [v.sub.n]) of length [absolute value of v] = n, shortly written as [v.sub.1][v.sub.2] ... [v.sub.n] hereafter In the future.

The term hereafter is always used to indicate a future time—to the exclusion of both the past and present—in legal documents, statutes, and other similar papers.
, is uniquely connected to the root [empty set] by the path [empty set] [right arrow] [v.sub.1] [right arrow] [v.sub.1] [v.sub.2] [right arrow] ... [right arrow] [v.sub.1] [v.sub.2] ... [v.sub.n]. If w = [w.sub.1] ... [w.sub.m] denotes another vertex, we write vw for the concatenation of v and w, i.e., for [v.sub.1] ... [v.sub.n] [w.sub.1] ... [w.sub.m]. In the present context, each v is interpreted as a (potential) particle of the n-th generation. It is the mother of the successors [vi := [v.sub.1] ... [v.sub.n] i, i [member of] N, and an ancestor of any vw, w [member of] V. In places where it occurs [v.sub.1] ... [v.sub.n] := [empty set] is stipulated whenever n = 0.

The following weighted branching model assigns a random weight L(v) [member of] {0,1} and a random position S(v) in R to each node v of the tree, where L(v) = 1 means that particle v is actually alive. Let ([OMEGA 1. (programming) Omega - A prototype-based object-oriented language from Austria.

["Type-Safe Object-Oriented Programming with Prototypes - The Concept of Omega", G. Blaschek, Structured Programming 12:217-225, 1991].
], U, P) be a given probability space In probability theory, the definition of the probability space is the foundation of probability theory. It was introduced by Kolmogorov in the 1930s. For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.  which carries i.i.d. random sequences

T(v) [cross product] X(v) := [([T.sub.i](v), [X.sub.i](v)).sub.i [greater than or equal to] 1] : [OMEGA] [right arrow] [({0, 1} x R).sup.N], v [member of] V.

Furthermore, [([T.sub.i](v)).sub.i [greater than or equal to] 1] and [([X.sub.i](v)).sub.i [greater than or equal to] 1] are independent as well for each v [member of] V with

P([T.sub.1](v) = ... = [T.sub.j](v) = 1,[T.sub.j+1](v) = [T.sub.j+2](v) = ... = 0) = [p.sub.j]

for j [greater than or equal to] 1 and [X.sub.1](v), [X.sub.2](v), ... being i.i.d. having common distribution Q. Put L([empty set]) := 1, S([empty set]) := 0, and define recursively

L(vi) := L(v)[T.sub.i](v) and S(vi) := S(v) + [X.sub.i](v)

for v = [v.sub.1] ... [v.sub.n] [member of] V and i [member of] N, thus

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]

The total size of the n-th generation (number of particles alive at time n) is now given by

[Z.sub.n] := [summation over ([absolute value of v=n])] L(v)

for n [greater than or equal to] 0 and forms a simple Galton-Watson process with offspring distribution [([p.sub.j]).sub.j [greater than or equal to] 0]. Throughout this article, we will make the standing assumption that [([Z.sub.n]).sub.n [greater than or equal to] 0] is supercritical Adj. 1. supercritical - (especially of fissionable material) able to sustain a chain reaction in such a manner that the rate of reaction increases
critical - at or of a point at which a property or phenomenon suffers an abrupt change especially having enough mass
, i.e., m > 1 or, equivalently, [p.sub.1] < 1 (as [p.sub.0] = 0). In order to describe the positions of all living particles at time n, we introduce the random location measures

[[PI].sub.n] := [summation over ([absolute value of v]=n])] L(v)[[delta].sub.S(v)], n (greater than or equal to) 0

and call [([[PI]].sub.n]).sub.n [greater than or equal to] 0] a BRW on R with offspring distribution [([p.sub.j]).sub.j [greater than or equal to] 0] and increment To add a number to another number. Incrementing a counter means adding 1 to its current value.  distribution Q. Put also

[PI] := [summation over n ([greater than or equal to] 0]) [[PI].sub.n] = [summation over (v[member of]V)] L(v)[[delta].sub.S(v)]

which is the associated overall empirical occupation measure and will be called branching renewal measure of [([[PI].sub.n]).sub.n [greater than or equal to] 0].

Definition 2.1 A BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] with increment distribution Q is called d-arithmetic if Q is d-arithmetic, i.e., if

d := sup{c > 0 : Q(cZ) = 1} > 0,

and is called nonarithmetic (0-arithmetic) otherwise.

So the lattice-span of [([[PI].sub.n]).sub.n [greater than or equal to] 0] is just the lattice-span d of its increment distribution Q (= 0 in the nonarithmefic case). Notice that d = [infinity] if Q = [[delta].sub.0]. Excluding this case, we may and will assume hereafter w.l.o.g. that d = 1 whenever d > 0. It is convenient to further define [G.sub.0] := R and [G.sub.1] := Z.

Since our interest lies in those BRWs that do not move in one direction only we make the standing assumption hereafter that the increment distribution Q puts mass on (-[infinity], 0) as well as (0, [infinity]), that is,

Q((-[infinity], 0)) [conjunction] Q((0, [infinity])) > 0.

Such a Q as well as an associated BRW is called genuinely two-sided hereafter. The definitions of recurrence and transience for a genuinely two-sided BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] are preceded by the following classification result (a zero-one law In probability theory, a zero-one law is a result that states that an event must have probability 0 or 1 and no intermediate value.

It may refer to:
  • the Hewitt-Savage zero-one law;
  • Kolmogorov's zero-one law.
) for its branching renewal measure [PI]. For intervals I [subset] R, let [H.sub.I] : R [right arrow] [0, 1] be the function given by

[H.sub.I](t) := P([PI](t + I) < [infinity]), t [member of] R.

Then put H := [H.sub.(-[infinity],0)], thus H(t) = P([PI]((-[infinity],t)) < [infinity]), and [H.sub.[epsilon]] := [H.sub.(-[epsilon], [epsilon])], thus [H.sub.[epsilon]](t) = P([PI]((t - [epsilon], t + [epsilon])) < [infinity]).

Proposition 2.2 Let [([[PI].sub.n]).sub.n [greater than or equal to] 0] be a genuinely two-sided, d-arithmetic BRW, d [member of] {0, 1}, with increment distribution Q. Suppose also [p.sub.0] = 0 and [p.sub.1] < 1. Then either [H.sub.[epsilon]] [equivalent to] 0 for all [member of] > 0, or [H.sub.[epsilon]] [equivalent to] 1 for all [epsilon] > 0. Similarly, either H [equivalent to] 0 or H [equivalent to] 1.

Remark 2.3 (a) Plainly, in the 1-arithmetic case the dichotomy di·chot·o·my  
n. pl. di·chot·o·mies
1. Division into two usually contradictory parts or opinions: "the dichotomy of the one and the many" Louis Auchincloss.
 for the [H.sub.[epsilon]] reduces to the statement that the [PI]({n}), n [member of] Z, are either all a.s. finite or all a.s. infinite. Moreover, a reflection argument (replace X(v) with -X(v) for each v [member of] V) shows that the zero-one dichotomy holds for [H.sub.(0,[infinity])] as well.

(b) If Q is concentrated on one halfline and having an atom at 0, then Proposition 2.2 may fail. For instance, if Q([N.sub.0]) = 1, p := Q({0}) [member of] (0,1) and mp > 1, then it is easily seen that P([PI]({0}) < [infinity]) (= [H.sub.[epsilon](0) for [epsilon] [member of] (0,1)) equals the extinction probability [q.sup.*] [member of] (0, 1) of the supercritical Galton-Watson process defined as [Z.sup.*.sub.n] := [[summation].sub.[absolute value of v]=n] L(v)[1.sub.{S(v)=0}], n [greater than or equal to] 0. The function H in this situation equals 1 on (-[infinity], 0], takes values in (0, [q.sup.*]] on (0, [infinity]) and converges to 0 as t [right arrow] [infinity].

Definition 2.4 (a) A genuinely two-sided 1-arithmetic BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] is called recurrent if

[PI]({k}) = [summation over n ([greater than or equal to] 0)] [[PI].sub.n]({k}) = [infinity] a.s. (1)

for some (and then all) k [member of] Z, and transient otherwise.

(b) A genuinely two-sided nonarithmetic BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] is called (topologically to·pol·o·gy  
n. pl. to·pol·o·gies
1. Topographic study of a given place, especially the history of a region as indicated by its topography.

) recurrent if

[PI](I) = [summation over (n [greater than or equal to] 0)] [[PI].sub.n](I) = [infinity] a.s. (2)

for some (and then all) nonempty bounded open intervals I, and transient otherwise.

In order to present our main result, we now confine to the case where Q has finite positive mean [mu](Q).

Defining the Laplace transform Laplace transform

In mathematics, an integral transform useful in solving differential equations. The Laplace transform of a function is found by integrating the product of that function and the exponential function ept
 of Q

[PSI]([theta Theta

A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option.
]) := [integral] [e.sup.-[theta]x] Q(dx)

with domain [D.sub.[PSI]] := {[theta] : [PSI]([theta]) < [infinity]}, we make the additional assumption that there exists a (necessarily unique) v > 0 such that

[integral][absolute value of x][e.sup.-vx] Q(dx) < [infinity] and [PSI]'(v) = -[integral] x[e.sup.-vx] Q(dx) = 0. (3)

Positivity of v follows from [PSI]'(0) = -[mu](Q) < 0 and the convexity Convexity

A measure of the curvature in the relationship between bond prices and bond yields.

Positive convexity corresponds to curvature that opens upward. Negative convexity corresponds to curvature that opens downward.
 of [PSI] on [D.sub.[PSI]] (which contains at least [0, v]). If v is not an interior point of [D.sub.[PSI]], then [PSI]'(v) is actually the left-hand derivative of [PSI] at v

Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance.  2.5 Let [([[PI].sub.n]).sub.n [greater than or equal to] 0] be a genuinely two-sided, d-arithmetic BRW, d [member of] {0, 1}, with increment distribution Q. Suppose also [p.sub.0] = 0, [p.sub.1] < 1, [mu](Q) [member of] (0, [infinity]), and that (3) holds true. Then [([[PI].sub.n]).sub.n [greater than or equal to] 0] is recurrent, if m[PSI](v) > 1, and transient otherwise.

In view of this result, the BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] is called critical, if m[PSI](v) = 1, subcritical sub·crit·i·cal  
1. Having a mass of fissionable material that is less than that needed for a chain reaction.

2. Of less than critical importance.
, if m[PSI](v) < 1, and supercritical, if m[PSI](v) > 1.

3 Random weighted location measures and an associated random walk

This section is devoted to the necessary prerequisites in order to prove our main results in the next section. We start by defining the random weighted location measures (rw.l.m.)

[[LAMBDA The Greek letter "L," which is used as a symbol for "wavelength." A lambda is a particular frequency of light, and the term is widely used in optical networking. Sending "multiple lambdas" down a fiber is the same as sending "multiple frequencies" or "multiple colors. ].sub.n] := [m.sup.-n][[PI].sub.n] = [m.sup.-n] [summation over ([absolute value of v]=n)] L(v)[[delta].sub.S(v)], n [greater than or equal to] 0

as well as their multivariate extensions

[[LAMBDA].sub.0:n] := [m.sup.-n] [summation over ([absolute value of v]=n)] L(v)[[delta].sub.S(v)], n [greater than or equal to] 0


S(v) := ([S.sub.0](v), [S.sub.1](v), ..., [S.sub.n-1](v), [S.sub.n](v)),

with [S.sub.k](v) := S([v.sub.1]... [v.sub.k]) if v = [v.sub.1] ... [v.sub.n] and 0 [greater than or equal to] k [greater than or equal to] n. Since [[LAMBDA].sub.0:n]([R.sup.n+1]) = [[LAMBDA].sub.n](R) = [m.sup.-n][Z.sub.n], n [greater than or equal to] 0, forms a positive martingale martingale

a leather strap running from the girth to the reins or the noseband for the purpose of restricting the movements of the horse's head. There are many designs. The common ones are the standing martingale, which is attached to the noseband, and the running martingale, which
 with mean one, we see that

[[bar.[LAMBDA]].sub.n] := E[[LAMBDA].sub.n] and [[bar.[LAMBDA]].sub.0:n] := E[[LAMBDA].sub.0:n]

are both probability distributions Many probability distributions are so important in theory or applications that they have been given specific names. Discrete distributions
With finite support
  • The Bernoulli distribution, which takes value 1 with probability p
 for each n. Of course, E[[LAMBDA].sub.n] and E[[LAMBDA].sub.0:n] are the measures more explicitly given by

(E[[LAMBDA].sub.n])(A) = E[[LAMBDA].sub.n](A) and (E[[LAMBDA].sub.0:n])(B) = E[[LAMBDA].sub.0:n](B)

for measurable A [subset] R and B [subset] [R.sup.n+1].

For any [theta] [member of] [D.sub.[PSI]], let us further define the r.w.l.m.


as well as the probability distribution Probability distribution

A function that describes all the values a random variable can take and the probability associated with each. Also called a probability function.

probability distribution 

[Q.sub.[theta]](dx) := [PSI][([theta]).sup.-1][e.sup.-[theta]x]Q(dx).

We point out that the [[LAMBDA].sup.[theta].sub.n] and [[LAMBDA].sup.[theta].sub.0:n] are the counterparts of [[LAMBDA].sub.n], respectively [[LAMBDA].sub.0:n] for the weighted branching model based upon [([T.sup.[theta]](v) [cross product] X(v)).sub.v[member of]V], where


Plainly, [[LAMBDA].sub.n] = [[LAMBDA].sup.0.sub.n], [[LAMBDA].sub.0:n] = [[LAMBDA].sup.0.sub.0:n], T(v) = [T.sup.0](v) and Q = [Q.sub.0].

The following two lemmata provide the connection to random walks. The first of them has been given in various places, see Lemma lemma (lĕm`ə): see theorem.

(logic) lemma - A result already proved, which is needed in the proof of some further result.
 4.1 in Biggins and Kyprianou (1997), Proposition 11 in Biggins and Kyprianou (2005), Lemma 1 in Bingham and Doney (1975), or p. 289 in Durrett and Liggett (1983).

Lemma 3.1 For each [theta] [member of] [D.sub.[PSI]], let [P.sub.[theta]] (= P if [theta] = 0) be a probability measure on ([OMEGA],U) and [([bar.S].sub.n]).sub.n [greater than or equal to] 0] a sequence of random variables on ([OMEGA],U) which, under [P.sub.[theta]], constitutes a random walk with [[bar.S].sub.0] := 0 and increment distribution [Q.sub.[theta]]. Then

[[bar.[LAMBDA]].sup.[theta].sub.0:n] = [P.sub.[theta]](([[bar.S].sub.0], ..., [[bar.S].sub.n]) [member of] x),

in particular [[bar.[LAMBDA]].sup.[theta].sub.n] = [Q.sup.*n.sub.[theta]] for all n [greater than or equal to] 0.

We will also need an extension of the previous lemma to a certain class of stopping lines, called homogeneous stopping lines (HSL (Hue Saturation Luminosity) A color space similar to HSB. See HSB. ) hereafter. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[]([S.sub.0], [S.sub.1], ...) := inf{n [greater than or equal to] 0 : ([S.sub.0], ..., [S.sub.n]) [member of] [B.sub.n]}

be any formal stopping rule In probability theory, in particular in the study of stochastic processes, a stopping time is a specific type of "random time".

The theory of stopping rules and stopping times can be analysed in probability and statistics, notably in the optional stopping theorem.
 where [B.sub.n] [member of] B([R.sup.n+1]) for n [greater than or equal to] 0 and inf 0 := [infinity], and let

[Y.sub.n] := [[pi].sub.0:n]({[] = n}),

for n [greater than or equal to] 0 where [[pi].sub.0:n] denotes the projection [([S.sub.k]).sub.k [greater than or equal to] 0] [??] ([S.sub.0], ..., [S.sub.n]). For v = ([v.sub.1], [v.sub.2], ...) [member of] [N.sup.N] (viewed as the boundary of V), we further define

[[sigma].sub.v] := [](S(v)), S(v) = [([S.sub.n](v)).sub.n [greater than or equal to] 0] := (S([empty set]), S([v.sub.1]), S([v.sub.1][v.sub.2]), ...),

and then

S := {v|[[sigma].sub.v] : v [member of] [N.sup.N]} [intersection] V = {v|[[sigma].sub.v] : v [member of] [N.sup.N], [[sigma].sub.v] < [infinity]},

where v[absolute value of 0 := [empty set], v]n := [v.sub.1] ... [v.sub.n] for n [member of] N, and v|[infinity] := v. We call S the HSL associated with []. It consists of all nodes v [member of] V that are obtained as stopping places when applying the same rule [] to the random walks S(v) along all infinite paths v of the tree. Notice that S may be empty and that S = {v : [absolute value of v] = n} in the case [] [equivalent to] n. Stopping lines, also called optional lines, have been defined in varying generality gen·er·al·i·ty  
n. pl. gen·er·al·i·ties
1. The state or quality of being general.

2. An observation or principle having general application; a generalization.

 in the literature, the most general one appearing in lagers (1989), which also provides the basic framework. We mention further Chauvin (1991), Kyprianou (2000), and Biggins and Kyprianou (2004) and note that the last reference contains the definition that is closest to that of an HSL and called very simple line there.

Lemma 3.2 Given any HSL S associated with a stopping rule [], put [sigma] := []([[bar.S].sub.0],[[bar.S].sub.1], ...). Then the following assertions hold true for each [theta] [member of] [D.sub.[PSI]]:


for all n [greater than or equal to] 0 and B [member of] B([R.sup.n+1]), in particular

[P.sub.[theta]]([sigma] = n) = 1/[(m[PSI]([theta])).sup.n] E ([summation over (v[member of]S,[absolute value]=n] L(v)[e.sup.-[theta]S(v)]) = [[bar.[LAMBDA]].sup.[theta].sub.0:n]([Y.sub.n]) (5)

for each n [greater than or equal to] 0 and


Finally, putting [Z.sup.[theta].sub.S] := [[summation].sub.v[member of]S] L(v)[e.sup.-[theta]S(v)] and [Z.sub.S] := [Z.sup.0.sub.S] = [[summation].sub.v[member of]S] L(v),

E[Z.sup.[theta].sub.S] = [E.sub.[theta]][(m[PSI]([theta])).sup.[sigma]][1.sub.{[sigma]<[infinity]}] and E[Z.sub.S] = E[m.sup.[sigma]][1.sub.{[sigma]<[infinity]}] (7)

for all [theta] [member of] [D.sub.[PSI]].

Proof: Noting {[sigma] = n} = {([[bar.S].sub.0], ..., [[bar.S].sub.n]) [member of] [Y.sub.n] and the fact that v [member of] S, [absolute value of v] = n holds iff S(v) [member of] [Y.sub.n], we infer from Lemma 3.1


which proves (4). The assertions (5) and (6) being direct consequences, let us directly turn to (7). But for [theta] [member of] [D.sub.[PSI]], we infer with the help of (5)


and thus the first half of (7). For the second choose [theta] = 0.

In view of the previous result a HSL S associated with [] will be called hereafter HSL associated with [sigma], where [sigma] = []([[bar.S].sub.0], [[bar.S].sub.1], ...).

4 Proofs of Proposition 2.2 and Theorem 2.5

The following common partial order relations [??] and [??] on V will be needed hereafter: Write v [??] w if v [not equal to] w and v belongs to the ancestral ANCESTRAL. What relates to or has, been done by one's ancestors; as homage ancestral, and the like.  line of w, while v [??] w also allows v = w. Moreover, v [??] ([??]) C for any C [subset] V shall mean that w [??] ([??])v for all w [member of] C. Now we can define the pre-S random location measure as

[[PI].sup.[??].sub.S] := [summation over (v[??]S)] L(v)[[delta].sub.S(v)]

for an arbitrary HSL S. Finally, put also

T := {v [member of] V : L(v) = 1} and [T.sub.n] := {v [member of] T : [absolute value of v] = n}.

The following lemma provides the key to the proof of Proposition 2.2.

Lemma 4.1 In the situation of Proposition 2.2, the function [H.sub.I] is constant for each nonempty open interval open interval
A set of numbers consisting of all the numbers between a pair of given numbers but not including the endpoints.

open interval 

Proof: Note that [H.sub.I](t) = [H.sub.t+I](0) for all intervals I and t [member of] R. Hence it suffices to verify that, fixing any nonempty open I with [H.sub.I](0) < 1 and any t [member of] [G.sub.d], we have [H.sub.I](t) = [H.sub.I](0).

Case 1. I bounded.

Since Q is genuinely two-sided, the associated random walk [([[bar.S].sub.n]).sub.n [greater than or equal to] 0] is topologically irreducible on [G.sub.d], that is,


for all x [member of] [G.sub.d] and [member of] > 0. For the subsequent argument, we restrict ourselves to the nonarithmetic case, the arithmetic one being even simpler. If I = (x, x + 4[epsilon]) for some x [member of] R and [epsilon] > 0, put [I.sub.1] := (x, x+2[epsilon]), [I.sub.2] := (x + [epsilon], x + 3[epsilon]), and [I.sub.3] := (x + 2[epsilon], x + 4[epsilon]). By (8), there are [k.sub.1], [k.sub.2], [k.sub.3], such that


Consequently, with k := [k.sub.1] [disjunction disjunction /dis·junc·tion/ (-junk´shun)
1. the act or state of being disjoined.

2. in genetics, the moving apart of bivalent chromosomes at the first anaphase of meiosis.
] [k.sub.2] [disjunction] [k.sub.3],


Next, since [H.sub.I](0) < 1, the event {[PI](I) = [infinity]} has positive probability 1 - [H.sub.I](0), and on this event the stopping times [[sigma].sub.0] := 0,

[[sigma].sub.i] := inf{n > [[sigma].sub.i-1] + k : S(v) [member of] I for some v [member of] [T.sub.n]}, i [greater than or equal to] 1,

are all a.s. finite. Let [v.sup.i] be the leftmost (with respect to lexicographic lex·i·cog·ra·phy  
The process or work of writing, editing, or compiling a dictionary.

[lexico(n) + -graphy.
 ordering) vertex in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that S([v.sup.i]) [member of] I (i [greater than or equal to] 1). Then it follows with the help of the strong Markov property In probability theory, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state and all past states, depends only upon the present state and not on any past states, i.e.  that


for all n [greater than or equal to] 1, where [1.sub.j] := 1 ... 1 (j times). Consequently, if [PI](I) - [infinity], then

[E.sub.i] : {[[sigma].sub.i] < [infinity] and S([v.sup.i][1.sub.j]) [member of] t + I for some 1 [less than or equal to] j [less than or equal to] k)

occurs a.s. for at least one i [greater than or equal to] 1, that is, [[upsilon up·si·lon or yp·si·lon
Symbol The 20th letter of the Greek alphabet.
].sub.1] := inf {i [greater than or equal to] 1 : [E.sub.i] occurs} < [infinity] a.s. on {[PI](I) - [infinity]}. But the previous argument can be repeated (using again the strong Markov property) for the sequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to infer [v.sub.2] := inf {i > [[upsilon].sub.1] : [E.sub.i] occurs} < [infinity] a.s. on {[PI](I) - [infinity]} and thus via induction that indeed {[PI](I) - [infinity]} - lim lim
Mathematics limit
 [sup.sub.i[right arrow][infinity]], [E.sub.i] a.s. We have thus shown that


and thereby [H.sub.I] (t) [greater than or equal to] [H.sub.I] (0) < 1. By interchanging the roles of I and t + I (now possible as [H.sub.t+I] (0) = [H.sub.I] (t) < 1), we get the reverse inequality and thus the constancy con·stan·cy  
1. Steadfastness, as in purpose or affection; faithfulness.

2. The condition or quality of being constant; changelessness.

Noun 1.
 of [H.sub.I].

Case 2. I unbounded.

Then I equals either R, in which case there is nothing to prove, or (-[infinity], x), or (x, [infinity]) for some x [member of] R. But for the last two alternatives, an even simpler geometric trials arguments than above may be employed to give the asserted result. Further details are therefore omitted.

Proof of Proposition 2.2: Let f(s) := [[summation].sub.j[greater than or equal to] 0] [p.sub.j] [S.sup.j] denote de·note  
tr.v. de·not·ed, de·not·ing, de·notes
1. To mark; indicate: a frown that denoted increasing impatience.

 the generating function of [Z.sub.1] and observe that [p.sub.0] = 0 and [p.sub.1] < 1 ensure f (S) [less than or equal to] s for all s [member of] [0, 1] with equality holding iff s [member of] {0, 1}. Put

[[[PI]].sub.v] :- [summation over (w [member of] V)] [L.sub.v](w)[[delta].sub.S(vw)-S(v)]

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any w = [w.sub.1] ... [w.sub.k] [member of] V, thus [L.sub.v] (w) = L(vw)/L(v) if L(v) > 0. Then the [[[PI]].sub.v] are just copies of [PI], obtained by looking at the subtree rooted at v with weights (T (vw) [cross product] X [(vw)).sub.w[member of]v]. Our independence assumptions further ensure that the [[PI]].sub.v] for v [member of] [N.sup.n] are independent.

For any nonempty interval open I, note the obvious identity

[PI](t+I) - [[delta].sub.0](t+I) + [[Z.sub.1].summation over (j=1)] [[[PI]].sub.j] (t - S(j)+I), t [member of] R.

By combining this with the constancy of [H.sub.I] (Lemma 4.1), it follows that


for all t [member or] R, that is,

[H.sub.I] - f o [H.sub.I]. (10)

But this shows [H.sub.I] [equivalent to] 0 or [equivalent to] 1, for {s : f (s) [greater than or equal to] s} = {0, 1}.

If I = (-[epsilon], [epsilon]) for any [epsilon] > 0, we must still verify that [H.sub.[eta]] = [H.sub.[epsilon]] for each [eta] [member of] (0, [epsilon]). If [H.sub.[epsilon]] [equivalent to] 1, then [H.sub.[eta]] [equivalent to] 1 for [eta][member of] (0, [epsilon]) is indeed a trivial consequence of [H.sub.[eta]] [greater than or equal to] [H.sub.[epsilon]]. If [H.sub.[epsilon]] [equivalent to] 0, then 1 - [H.sub.[epsilon]] (0) = P([PI]((-[epsilon], [epsilon])) = [infinity]) = 1 which in turn implies 1 - [H.sub.[eta]] (t) = P([PI](t - [eta], t + [eta])) = [infinity]) > 0 for some t [member of] [G.sub.d] and thus excludes [H.sub.[eta]] [equivalent to] 1. Hence, [H.sub.[eta]] [equivalent to] 0 by another appeal to Lemma 4.1 and the proof of the asserted dichotomy is complete.

We proceed with two lemmata relevant to the proof of Theorem 2.4. The first provides a link between the behavior of H (t) = [PI]((-[infinity], t)) and the following Galton-Watson process generated by ladder lines. Let [([[sigma].sub.n]).sub.n [greater than or equal to] 0] be the possibly terminating renewal sequence of strictly descending ladder epochs associated with [([[bar.S].sub.n]).sub.n [greater than or equal to]0], defined by [[sigma].sub.0] := 0 and


where as usual inf 0 := [infinity]. Denote by [S.sub.n] the HSL associated with [[sigma].sub.n], called ladder line Ladder line or window line is a type of transmission line similar to twin-lead for balanced connection of antennas. Ladder line is constructed as a pair of evenly spaced wires with supportive plastic webbing holding the wires apart. , and observe that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a Galton-Watson process (generated by these lines), possibly in the generalized sense that individuals have an infinite number infinite number

a number so large as to be uncountable. Represented by 8, frequently obtained by 'dividing' by zero.
 of offspring with positive probability. If so the process is trivially supercritical.

Lemma 4.2 In the situation of Proposition 2.2 the following statements are equivalent:

(i) H [equivalent to] 0, i.e., [PI]((-[infinity], x)) - [infinity] a.s. for all x [member of] R.


Proof: If (i) holds true then [S.sub.n] is a.s. nonempty and thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] an a.s. nonextinctive supercritical Galton-Watson process. Conversely, the event of nonexfinction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has positive probability under (ii), and since S(v) < 0 for infinitely many v [member of] T on this event, we infer [PI]((-[infinity], 0)) = [infinity] with positive probability which in turn implies (i) by an appeal to Proposition 2.2.

Our second lemma is obtained by a geometric trials argument similar to the one in the proof of Proposition 2.2.

Lemma 4.3 In the situation of Proposition 2.2 suppose additionally that Q has finite positive mean. Then the following statements are equivalent:

(i) H [equivalent to] 0.

(ii) [H.sub.[epsilon]] [equivalent to] 0 for all [epsilon] > 0.

Proof: Clearly, it suffices to show that (i) implies (ii), that is, if any interval (-[infinity], x) is a.s. visited infinitely often, then we have the same for any bounded open interval I.

The following argument is given for the nonarithmetic case, but the modifications in the case d = 1 are straightforward and thus omitted. For b [greater than or equal to] 0, let [tau](b) := inf{n [greater than or equal to] 1 : [[bar.S].sub.n] > b} and [R.sub.b] = [[bar.S].sub.[tau](b)] - b the associated overshoot o·ver·shoot
A change from steady state in response to a sudden change in some factor, as in electric potential or polarity when a cell or tissue is stimulated.
 (the first strictly ascending ascending /as·cend·ing/ (ah-send´ing) having an upward course.


progressing to higher levels, usually used in reference to the nervous system.
 ladder height for b = 0). As Q has finite mean, the same holds true for [R.sub.0] and [R.sub.b] converges in distribution to [R.sub.[infinity]] say, with distribution function

P([R.sub.[infinity]] [less than or equal to] r) = 1/E[R.sub.0] [[integral].sup.r.sub.0] ([R.sub.0] > x) dx, r [greater than or equal to] 0.

Consequently, we can pick some positive t and [epsilon] such that


After these observations the geometric trials argument goes as follows. Assuming (i) and thus a fortiori [Latin, With stronger reason.] This phrase is used in logic to denote an argument to the effect that because one ascertained fact exists, therefore another which is included in it or analogous to it and is less improbable, unusual, or surprising must also exist.  [PI]((-[infinity], -t)) = [infinity] a.s., we can pick a (random) sequence of nodes [v.sup.1], [v.sup.2], ... such that S ([v.sup.j]) [less than or equal to] -t for each j [greater than or equal to] 1. Start with [[??].sup.1] := [v.sup.1] and follow the path [v.sup.1], [v.sup.1]1, [v.sup.1]11, ... until the first [v.sup.1][1.sub.[upsilon]] with S([v.sup.1] [1.sub.[upsilon]]) > 0 or, equivalently, S([v.sup.1],[1.sub.[upsilon]]) - S([v.sup.1]) > -S([v.sup.1]), where [1.sub.n] := 1 ... 1 (n-times) should be recalled. But our assumptions ensure that (S ([v.sup.1][1.sub.n]) - S[([v.sup.1])).sub.n [greater than or equal to] 1] is independent of S([v.sup.1]) and having the same distribution as [([[bar.S].sub.n]).sub.n [greater than or equal to]1]. Consequently, by our choice of t, [epsilon] and the stopping time [upsilon], there is a chance of at least p that S([v.sup.1] [1.sub.[upsilon]]) [member of] (0, [epsilon]]. Now pick the first node [[??].sup.2] from [[??].sup.2], [[??].sup.3], ... of length > [absolute value of [v.sup.1]] + [upsilon], follow the path [[??].sup.2], [[??].sup.2]1, [[??].sup.2]11, ... until S([[??].sup.2] [1.sub.k]) > 0 for the first time. Again, the interval (0, [epsilon]] is hit with probability at least p and independent of the first trial, by the strong Markov property. Continuing this way, the interval (0, [epsilon]] is hit once after an a.s. finite number of rounds and then indeed infinitely often, thus showing [PI]((0, [epsilon])) = [infinity] a.s. But Proposition 2.2 now ensures [PI]((x - [eta], x + [eta])) = [infinity] a.s. for all x [member of] R and [eta] > 0, i.e., (ii) holds true.

Proof of Theorem 2.5: (a) We give first an argument which does not require the previous two lemmata but works only in the 1-arithmetic case. Let [sigma] = inf {n [greater than or equal to] 1 : [[bar.S].sub.n] = 0} and S be the associated HSL. Then [Z.sub.S] = [Z.sup.[theta].sub.S] for all [theta] together with (7) in Lemma 3.2 shows that

E[Z.sub.S] = E[Z.sup.[theta].sub.S] = [E.sub.[theta]][(m[PSI]([theta])).sup.[sigma]] [1.sub.{[sigma]< [infinity]}]

for each [theta] [member of] [D.sub.[PSI]]. Now choose [theta] = v and notice that [([[bar.S].sub.n]).sub.n[greater than or equal to] 0] has drift [PSI]'(v) = 0 under [P.sub.v] and is therefore recurrent on Z, i.e., [P.sub.v] ([sigma] < [infinity]) = 1. Consequently,


Considering once again the Galton-Watson process [([[??].sub.]).sub.n [greater than or equal to] 0], say, of all particles visiting 0 with first generation size [Z.sub.S], we thus infer this process be critical or subcritical, if m[PSI](v) [less than or equal to] 1, and supercritical otherwise. In the latter case, it has a positive chance of survival, that is, P([PI]({0}) < [infinity]) < 1. Proposition 2.2 then ensures that this probability must be 0 as claimed, in other words Adv. 1. in other words - otherwise stated; "in other words, we are broke"
put differently
, the BRW is recurrent. If m[PSI]<(v) [less than or equal to] 1, then almost certain extinction of [([[??].sub.n]).sub.n[greater than or equal to] 0] naturally gives P([PI]({0}) < [infinity]) = 1 and hence the transience of [([[PI].sub.n]).sub.n[greater than or equal to]0]. In the critical case we should mention that P([Z.sub.s] = 1) = 1 is easily excluded.

(b) Suppose now we are in the nonarithmetic case. The following argument embarks on Lemma 4.3 by which it suffices to consider the function H so as to assess recurrence or transience of the given BRW By Lemma 4.2 and in the notation from there, this can be done by computing the mean offspring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the Galton-Watson process [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose first m[PSI]<(v) [less than or equal to] 1 and notice that S(v) < 0 for all v [member of] [S.sub.1] in combination with v > 0 implies


Consequently, by (7) of Lemma 3.2,


The process [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] thus being subcritical we infer, by Lemma 4.2, that [PI]((-[infinity], 0)) < [infinity] a.s. and thereby transience of [([[PI].sub.n]).sub.n[greater than or equal to]0] as claimed.

If [e.sup.v[epsilon]] = m[PSI]<(v) > 1 (with [epsilon] > 0 defined by this equality), consider the stopping times [[sigma].sup.[epsilon].sub.0] [equivalent to] 0,


for n [greater than or equal to] 1, and let [S.sup.[epsilon].sub.n], n [greater than or equal to] 0, be the associated HSL. Under [P.sub.v], all [[sigma].sup.[epsilon].sub.n] are a.s. finite as [([[bar.S].sub.n]).sub.n[greater than or equal to]0] has drift 0 and is therefore recurrent. By another appeal to (7) of Lemma 3.2,


which in combination with the inequality


leads to


We thus arrive at the conclusion that the Galton-Watson process [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is supercritical and therefore surviving with positive probability. As


the process [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also supercritical and therefore [PI]((-[infinity], 0)) = [infinity] a.s. by Lemma 4.2. This proves the recurrence of [([[PI].sub.n]).sub.n[greater than or equal to] 0].

5 Extremal particle positions in a critical BRW

Once knowing that the critical BRW is transient and thus drifting to [infinity], it is natural ask for its minimal speed or, equivalently, the asymptotic behavior of the leftmost particle in the cloud as time goes to infinity. Define


In the critical case, it is not surprising and in fact following from an old more general result by Biggins (1976, 1977) that

[Min.sub.n]/n [right arrow] 0 a.s.

However, by drawing on recent work of Hu and Shi (2008) (see also Alsmeyer (2007)), we can provide far more precise information about the behavior of [Min.sub.n] under some additional conditions, and also about the naturally related one on the behavior of the rightmost particle (maximal speed) in the cloud. Towards this end, we make the additional assumptions hereafter that the step size distribution Q has bounded support and that

E[Z.sup.2.sub.1] = [summation over (n [greater than or equal to] 1)] [n.sup.2][P.sub.n] < [infinity]. (11)

The following result then follows directly from the more general Theorem 1.2 in Hu and Shi (2008) (note that v = 1 in this work).

Theorem 5.1 Let [([[PI].sub.n]).sub.n[greater than or equal to]0] be a genuinely two-sided critical BRW satisfying [p.sub.0] = 0, [p.sub.1] < 1, [mu](Q) [member of] (0, [infinity]) and (11). Suppose further the step size distribution Q to be bounded. Then (with v defined by (3))


as well as


The natural way for getting a similar result for the rightmost particle is to resort to the previous one after a reflection of the given BRW at a suitable line x [??] [gamma]x. Put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is strictly decreasing and continuous on {[mu] < + [PHI phi
Symbol The 21st letter of the Greek alphabet.

n See health information, protected.
]([mu]) < [PSI] ([??])} [subset] (p (Q), [infinity]). Defining further

[gamma] := sup{[mu] : m[PHI]([mu]) [greater than or equal to] > 1}, (12)

Biggins (1976) also showed that

[Max.sub.n]/n [right arrow] [gamma] a.s.

Now, if [gamma] is given along with a [kappa Kappa

Used in regression analysis, Kappa represents the ratio of the dollar price change in the price of an option to a 1% change in the expected price volatility.

Remember, the price of the option increases simultaneously with the volatility.
] > 0 such that (12) holds together with

1 = m[PHI]([gamma]) = m[e.sup.-[kappa][gamma]] [PSI] (-[kappa]) and [gamma] = - [PHI]'(-[kappa])/[PHI](-[kappa]) (13)

(the latter identity is an equivalent statement for that the derivative of [theta] [??] [e.sup.[theta][gamma]] [PSI]([theta]) at [theta] = -[kappa] be 0), then one can easily check that the reflected [([[PI].sub.n]).sub.n[greater than or equal to] 0], defined as

[[??].sub.n] = [summation over ([absolute value of v = n)] L(v)[[delta].sub.[gamma]n-S(v)]

is again genuinely two-sided with positive drift, critical and satisfying all conditions of Theorem 5.1. As for its leftmost particle position [[??].sub.n] at time n, we can thus apply Theorem 5.1 and have also the obvious relation

[Max.sub.n] = [gamma]n - [[??].sub.n]

for all n [greater than or equal to] 0. Thus we finally arrive at the following result for the rightmost particle position.

Theorem 5.2 Under the same conditions as in Theorem 5.1 let [gamma] be defined by (12). Suppose additionally the existence of a [kappa] > 0 such that the pair ([gamma], [kappa]) satisfies (13). Then


as well as



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Gerold Alsmeyer and Matthias Meiners

Institut fur Mathematische Statistik, Fachbereich Mathematik, Einsteinstrasse 62, D-48149 Munster. Germany
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Date:Jan 1, 2008
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