A note on large deviation probabilities for empirical distribution of branching random walks

Accepted Manuscript A note on large deviation probabilities for empirical distribution of branching random walks Wanlin Shi

PII: DOI: Reference:

S0167-7152(18)30384-5 https://doi.org/10.1016/j.spl.2018.11.029 STAPRO 8391

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Statistics and Probability Letters

Received date : 18 April 2018 Revised date : 18 November 2018 Accepted date : 22 November 2018 Please cite this article as: W. Shi, A note on large deviation probabilities for empirical distribution of branching random walks. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.11.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A NOTE ON LARGE DEVIATION PROBABILITIES FOR EMPIRICAL DISTRIBUTION OF BRANCHING RANDOM WALKS WANLIN SHI∗

Abstract. We consider a branching random walk on R started from the origin. Let Zn (·) be the counting measure which counts the number of individuals at the n-th generation located in a given set. For any √ Z ( nA) interval A ⊂ R, it is well known that nZ (R) converges a.s. to ν(A) under some mild conditions, where n ν is the standard Gaussian measure. In this note, we study the convergence rate of  √   ¯n P Z nσ 2 A − ν(A) ≥ ∆ , for a small constant ∆ ∈ (0, 1 − ν(A)). Our work completes the results in [3] and [6], where the step size of the underlying walk is assumed to have Weibull tail, Gumbel tail or be bounded.

1. Introduction and Main results 1.1. Branching random walk and its empirical distribution. Let T be a Galton-Waston tree rooted at ρ with offspring distribution {pk , k ≥ 0}. For any u, v ∈ T , we write u  v if u is an ancestor of v or u = v. For each node v ∈ T \ {ρ}, we attach a real-valued random variable Xv to it. Moreover, for the given tree T , {Xv , v ∈ T \ {ρ}} are i.i.d. copies of some random variable X, which is called step size. The position of v is given by X Sv := Xu ρ≺uv

with Sρ = 0 for convenience. Then, {Su , u ∈ T } is the branching random walk we want to study. We assume that the number of children and the motions are independent. For any n ∈ N, we introduce the following counting measure X Zn (·) = 1{Sv ∈·} , |v|=n,v∈T

where |v| denotes the generation of node v, i.e., the graph distance between v and ρ.

In this article, we always assume p0 = p1 = 0, that is the B¨ ottcher case. We also take b := min{k ≥ 2; pk > 0} and B := sup{k ≥ 2; pk > 0} ∈ [b, ∞]. One can see that P {Zn (R), n ≥ 0} is a supercritical Galton-Waston process. Besides, it is assumed that the mean m := k≥1 kpk is finite.

(·) Let Z¯n (·) := ZZnn(R) be the corresponding empirical distribution. A central limit theorem on Zn (·), conjectured by Harris [4], was proved by Asmussen-Kaplan [1] and further extended by Klebaner [5] and Biggins [2]. More precisely, if the step size X has zero mean and finite variance σ 2 , for any A ∈ A0 := {(−∞, x], x ∈ R}, √  (1.1) lim Z¯n nσ 2 A = ν(A), a.s. n→∞ R where ν(A) := A dΦ(x) is the standard Gaussian measure on R, and Φ(·) is the standard normal distribution function.

In this paper, what we concern is the convergence rates of (1.1). We analyze large deviation probabilities of the form:  √   (1.2) P Z¯n nσ 2 A − ν(A) ≥ ∆ ,

for a large class of measurable sets A ⊆ R and a small constant ∆ > 0. In fact, this question has been investigated by Louidor and Perkins [6] in B¨ ottcher case with bounded step size. Later, Louidor and Tasiri Date: November 27, 2018. 2010 Mathematics Subject Classification. 60J80, 60F10. Key words and phrases. empirical distribution, branching random walk, B¨ ottcher case, large deviation. 1

WANLIN SHI∗

2

[7] extended the result of [6] by allowing dependence between the motion of children and their numbers. However, Chen and He [3] found that if X is not bounded, the asymptotic behaviour of (1.1) depends on α xα the tail distribution of step size X. In particular, in [3], it is assumed that P (X > x) ∼ e−λx or e−e , as x → ∞. In this paper, we shall consider more general cases. That is P (X > x) ∼ e−f (x) for some function f , as x → ∞. Our results complete Theorem 1.3 and Theorem 1.4 in [3]. 1.2. Main results. Let A be the algebra generated by {(−∞, x], x ∈ R}. For p ∈ (ν(A), 1) with any A ∈ A \ ∅ such that ν(A) > 0, define IA (p) = inf{|x|; ν(A − x) ≥ p}. Note that either 0 < IA (p) < ∞ or IA (p) = ∞ if p > ν(A). When 0 < IA (p) < ∞, there exists x ∈ R with |x| = IA (p) such that ν(A − x) ≥ p, see Proposition 3 in [6]. In what follows, we only consider the case when p ∈ (ν(A), 1) such that IA (p) < ∞ and IA (·) is continuous at p. In addition, we always assume that X is symmetric with σ 2 = E[X 2 ] = 1 and X (1.3) E(eθZ1 (R) ) = pk eθk < ∞, for some θ > 0. k≥0

Now we are ready to present our results. Theorem 1.1. Assume P (X > x) = e−f (x)(1+o(1))

and

f (x) = xβ L(x),

for some β ≥ 1 as x → ∞, where f is a continuous nondecreasing, convex function and L(x) is a slowly varying function. If B > b, then  1−β √ β(log B − log b) (log n)β−1 ¯ √ (1.4) IA (p)β , log P(Zn ( nA) ≥ p) = − lim n→+∞ nβ/2 L( n ) 2 log B log b log n where we make convention that

log B−log b log B

= 1 if B = ∞.

Theorem 1.2. For x → ∞, assume that

P (X > x) = e−e

f (x)(1+o(1))

,

where f is a continuous, positive and strictly increasing function on (0, ∞) with f (x)  log x. In addition, we suppose that x 7→ ef (x) is a convex function on [M, ∞) for some constant M > 0. (1) If B < ∞, then (1.5)

(1.6)

lim

n→+∞

√ 1 log[− log P(Z¯n ( nA) ≥ p)] = log b, t(n)

where x 7→ t(x) satisfies the following equation log b Z log B t(x) √ log b −1 )t(x)f (t(x) log b) + f −1 (u log B)du = xIA (p), (1 − M0 log B log B

for some constant M0 > 0 and ∀x ≥ 1. (2) If B = ∞, then √ 1 lim log[− log P(Z¯n ( nA) ≥ p)] = log b, n→+∞ t(n) (1.7)

where x 7→ t(x) is determined by the equation

t(x)f −1 (t(x) log b) =

√

xIA (p),

∀x ≥ 1.

Remark 1.3. The continuity and strict monotonicity of f ensure that (1.6) has one unique solution t(x) for x sufficiently large. R logBb t(x) −1 Remark 1.4. When B < ∞, we take M0 > 0 such that f −1 (M0 ) exists and log f (u log B)du is M0

convergent.

log B

Remark 1.5. If one takes f (x) = xα for α > 0, then we obtain Theorem 1.4 in [3]. Remark 1.6. The assumption f (x)  log x ensures that t(x)  log x. It suffices to show that t(x) determined √ √ (p) x (p) x by (1.7) satisfies t(x)  log x. In fact, by (1.7), we have t(x) log b = f ( IAt(x) )  log( IAt(x) ) = 12 log x + √ log IA (p) − log t(x). Consequently, t(x)  log x. Moreover, (1.6) and (1.7) imply that t(x) = o( x).

LARGE DEVIATION BOTTCHER

3

√ The proof strategy in this paper is similar to the idea in [3]. To obtain {Z¯n ( nA) ≥ p}, we take an intermediate generation √ t(n) and suppose√that most of the individuals at this generation are located near √ n (A − x)) & p. x n, so that finally Z¯n ( nA) ≈ νn−t(n) ( n(A − x)) ≈ ν( n−t(n) The paper is organised as follows. In Section 2 we first present some preliminary results which will be used in the proof. Then we prove the main result in Section 3. Let C1 , C2 · · · and c1 , c2 · · · denote positive constants which might change from line to line. As usual, we write fn = on (1)gn if fn /gn converges to 0 as n tends to infinity. fn = O(gn ) means that fn ≤ Cgn for some C > 0 and all n ≥ 1. fn = Θ(gn ) means that fn is bounded both above and below by a positive and finite constant multiple of gn . The proof of Theorem 1.1 is parallel to Theorem 1.3 in [3]. We shall omit all details and only give the proof of Theorem 1.2 in this paper. 2. Preliminary results In this section, we recall some useful lemmas, which will be applied in the next section. Recall that ν represents the standard normal distribution on the real line. Denote by νn := PX ∗ · · · ∗ PX , the distribution {z } | n times

of a X-random walk at the n-th step. The following Lemma 2.1, which can be found in [6] and [3], states the uniform convergence of νn towards ν.

Lemma 2.1. Let A ∈ A \ ∅ and p ∈ (0, 1). For ν(∂A) = 0 and any l > 1, we have the following uniform convergence, √ lim sup sup |νn ( n(aA + b)) − ν(aA + b)| = 0. n→∞ a∈[l−1 ,l] b∈R

Let M be the collection of locally finite counting measures on R. For any finite measure ζ ∈ M, we can P|ζ| write ζ = i=1 δxi with xi ∈ R and the total mass |ζ| < ∞ . We write x ∈ ζ if ζ{x} ≥ 1 for convenience. Let {Znζ } be the counting measure of branching random walk started from Z0ζ = ζ. Similarly, let Z¯nζ (·) be the corresponding empirical distribution. We then have the following lemma, borrowed from [3]. Lemma 2.2. Let assumption (1.3) hold. There exist C1 , C2 > 0 such that for all ∆ > 0 sufficiently small and n ≥ 1, for any finite ζ ∈ M,   X 2 1 (2.1) P Z¯nζ (A) > νn (A − x) + ∆ ≤ C1 e−C2 ∆ |ζ| . |ζ| x∈ζ

The same holds if >, +∆ are replaced by <, −∆, respectively.

3. Proof of Theorem 1.2 In this section we suppose that all assumptions in Theorem 1.2 hold. The proof can be seen as follows. f (x)

Remark 3.1. The assumption that P (X > x) ∼ e−e as x → ∞ in Theorem 1.2 is made for convenience. f (x)(1+o(1)) The result still holds under the assumption that P (X > x) = e−e as x → ∞. That is to say, the proofs can also carry through, albeit with some extra epsilons and deltas. In what follows, we are going to prove that √ 1 (3.1) log[− log P(Z¯n ( nA) ≥ p)] = log b. lim n→+∞ t(n) (1) If B < ∞, x 7→ t(x) satisfies (1 −

log b )t(x)f −1 (t(x) log b) + log B

Z

log b log B t(x) M0 log B

f −1 (u log B)du =

for some constant M0 > 0 and ∀x ≥ 1. (2) If B = ∞, x 7→ t(x) is determined by the following equation √ (3.2) t(x)f −1 (t(x) log b) = xIA (p), ∀x ≥ 1. Firstly, we state the following lemma which will be used in the proof.

√

xIA (p),

WANLIN SHI∗

4

Lemma 3.2. Let {Xi }i≥1 be independent identically distributed random variables following the same law as f (x) X, and suppose that P(X > x) ∼ e−e as x → ∞, where f is a nondecreasing function with f (x)  log x. Meanwhile, assume that x 7→ ef (x) is a convex function on [M 0 , ∞) for some constant M 0 > 0 . Then there exists a sequence of integers t(n) determined by (3.2) satisfies Pt(n) √ log[− log P( i=1 Xi ≥ a n)] √ (3.3) ≥ 1, for any a > 0. lim inf n→∞ ) f ( a n−t(n) t(n) Pt(n) Pt(n) √ √ Proof. Notice that P( i=1 Xi ≥ a n) ≤ P( i=1 Xi+ ≥ a n), where Xi+ := Xi ∨ 0. Thus it suffices to prove the desired results when X is non-negative. In this proof, we always assume X ≥ 0 a.s.. Observe that     ! t(n) t(n) X X √ √ (3.4) P Xi ≥ a n ≤ P sup Xi ≥ n + P  Xi ≥ a n, sup Xi < n . 1≤i≤t(n)

i=1

1≤i≤t(n)

i=1

Note that there exists c1 > 0 such that for any x ≥ 0,

P(X ≥ x) ≤ c1 e−e

It is easy to see that P

(3.5)

sup 1≤i≤t(n)

Xi ≥ n

!

f (x)

.

≤ t(n)P(X ≥ n) ≤ c1 t(n)e−e

f (n)

.

In addition,  t(n) X √ P Xi ≥ a n, i=1

sup 1≤i≤t(n)

X

=

Xi < n

xi ∈[0,n)∩N,i=1,··· ,t(n)

≤

≤ ≤

X





P

t(n) X i=1

sup 1≤i≤t(n)



Xi < n, Xi ∈ [xi , xi + 1)

P(Xi ∈ [xi , xi + 1], ∀1 ≤ i ≤ t(n))

√ ≥ a n − t(n) xi ∈ [0, n) ∩ N, i = 1, · · · , t(n) t(n) Σi=1 xi

√ Xi ≥ a n,

X

c1

X

c1

t(n)

exp{−

√ t(n) Σi=1 xi ≥ a n − t(n) xi ∈ [0, n) ∩ N, i = 1, · · · , t(n)

t(n) X i=1

t(n)

ef (xi ) }

exp{−t(n)ef (

√ a n−t(n) ) t(n)

(1 + on (1))}

√ ≥ a n − t(n) xi ∈ [0, n) ∩ N, i = 1, · · · , t(n) t(n) Σi=1 xi

≤ (nc1 )t(n) exp{−t(n)ef (

a

√

n−t(n) ) t(n)

(1 + on (1))},

where the third inequality follows from the convexity of x 7→ ef (x) on [M 0 , +∞) for some constant M 0 > 0. In fact, t(n) X

e

i=1

f (xi )

≥

t(n) X

f (xi ∨M 0 )

e

i=1

n ≥ t(n) exp f ≥ t(n)e

which gives (3.6)

− t(n)e

f (M 0 )

 t(n) X √ P Xi ≥ a n, i=1

sup 1≤i≤t(n)



f



Pt(n) i=1

t(n)

 √ a n−t(n)) t(n)

xi

!

o

− t(n)ef (M

0

)

(1 + on (1)),

   √  a n−t(n) f t(n) Xi < n ≤ (nc1 )t(n) exp −t(n)e (1 + on (1)) .

LARGE DEVIATION BOTTCHER

5

√ √ Note that t(n)f −1 (t(n) log b) = Θ( n) by (3.2), so f ( at(n)n ) = Θ(t(n) log b)  log n. By use of (3.4), (3.5) and (3.6), we have

Pt(n) √ log[− log P( i=1 |Xi | ≥ a n)]

lim inf

√

f(a

n→∞

n−t(n) ) t(n)

≥ 1,

which suffices to conclude Lemma 3.2. We are ready to prove Theorem 1.2. Lower bound of (3.1): According to the definition of IA (p), for any sufficiently small δ > 0, there exists x0 ∈ R, η = x0 −I2A (p) > 0 such that inf

y∈[x0 −3η,x0 ]

ν(A − y) ≥ p + δ.

We choose an integer d ≥ b such that pd > 0. (If B < ∞, then we choose d = B. If B = ∞, we will let d → ∞ later.) Let j log log t− (n) + t− (n) log b k ∧ t− (n), s− (n) = log d where x 7→ t− (x) is determined by the following equation log b − Z log d t (x) √ log b − (3.7) (1 − f −1 (u log d)du = xIA (p), )t (x)f −1 (t− (x) log b) + M 0 log d log d for some constant M0 > 0 and ∀x ≥ 1.

Recall that T is the embedded Galton-Waston tree. Let Tt− (n) = {u ∈ T : |u| ≤ t− (n)} the subtree before generation t− (n). We construct a tree t of height t− (n) ( see FIGURE 1) in the following way. Firstly, let tt− (n)−s− (n) := {v ∈ t : |v| ≤ t− (n) − s− (n)} be a b-regular tree. Then we take the individuals at the (t− (n) − s− (n))-th generation for consideration. Here denote u∗ = (1, · · · , 1) to be the first individual of the (t− (n) − s− (n))-th generation in the lexicographic order. t(u∗ ) is taken as a d-regular tree and t(u) are all bregular trees for all u 6= u∗ in the (t− (n) − s− (n))-th generation, where for any u ∈ t, t(u) := {v ∈ t : u  v} is the subtree of t rooted at u.

Figure 1. The tree t Define the following event (3.8)

√ √ Γt− (n),b,d = {Tt− (n) = t, Su ∈ [(x0 − 3η) n, x0 n], for all |u| = t− (n) with u ∈ t(u∗ )}.

It follows that (3.9)

√ √ P(Z¯n ( nA) − ν(A) ≥ ∆) ≥ P(Z¯n ( nA) − ν(A) ≥ ∆; Γt− (n),d,b ).

Define −

M0 = {ζ ∈ M : |ζ| = ds

(n)

−

+ bt

(n)

−

− bs

(n)

√ √ − , ζ([(x0 − 3η) n, x0 n]) ≥ ds (n) }.

WANLIN SHI∗

6

For any ζ ∈ M0 , we have −

1 X x ds (n) ν(A − √ ) ≥ t− (n) |ζ| n b − bs− (n) + ds− (n) x∈ζ

inf

y∈[x0 −3η,x0 ]

ν(A − y) ≥

log t− (n) (p + δ), 1 + log t− (n)

which, together with Lemma 2.1, implies that for all n sufficiently large, √ 1 X νn−t− (n) ( nA − x) ≥ p + δ/2. |ζ| x∈ζ

Given Γt− (n),d,b , Zt− (n) ∈ M0 . Consequently, √ P(Z¯n ( nA) ≥ p; Γt− (n),d,b ) X √ ζ P(Zt− (n) = ζ; Γt− (n),d,b )P(Z¯n−t ≥ nA) ≥ p) − (n) ( ζ∈M0

≥

X

ζ∈M0

P(Zt− (n) =



√ ζ ζ; Γt− (n),d,b )P Z¯n−t nA) − (n) ( −C2 |ζ|δ 2 /4

≥ P(Γt− (n),d,b )(1 − C1 e

),

where the last inequality follows from Lemma 2.2. As |ζ| ≥ bt (3.10)

 √ 1 X νn−t− (n) ( nA − x) − δ/2 ≥ |ζ| x∈ζ

−

(n)

, for n large enough,

√ 1 P(Z¯n ( nA) − ν(A) ≥ ∆) ≥ P(Γt− (n),d,b ). 2

Define Pt = P(·|Tt− (n) = t). It remains to consider P(Γt− (n),d,b ). In fact, √ √ s− (n) pdc6 d Pt (Su ∈ [(x0 − 3η) n, x0 n], ∀u  u∗ , |u| = t− (n)). √ √ To obtain the event {Su ∈ [(x0 − 3η) n, x0 n]}, we suppose that for some η 0 > 0 whose value will be ∗ determined later and for any ancestor of u : u  u∗ , i h η0 Xu ∈ f −1 (t− (n) log b), f −1 (t− (n) log b) + f −1 (t− (n) log b) , 4 (3.11)

P(Γt− (n),d,b ) ≥ pcb5 b

t− (n)

and suppose that for any descendant of u∗ : u∗ ≺ u s.t. t− (n) − s− (n) < |u| < h Xu ∈ f −1 (t− (n) log b − (|u| + s− (n) − t− (n)) log d),

f −1 (t− (n) log b − (|u| + s− (n) − t− (n)) log d) +

and Xu ∈ (0, M1 ],

log b+log d − M0 t (n) − s− (n) − log log d d,

i η 0 −1 − f (t (n) log b) ; 4

log b + log d − M0 t (n) − s− (n) − ≤ |u| ≤ t− (n), log d log d

where M1 is a fixed real number such that P(Xu ∈ (0, M1 ]) ≥ c2 for some constant c2 > 0, and M1 ≤ η 0 −1 − (t (n) log b) for all large n. This ensures that for any descendant u of u∗ at the t− (n)-th generation, 4f its position Su satisfies log b −

M0

b log d t (n)− log d c h i X − − −1 − − (t (n) − s (n))f (t (n) log b) + f −1 (t− (n) log b − k log d) Su k=1

 M + log log t− (n)  η √ η0 0 (3.12) ≤ t− (n)f −1 (t− (n) log b) + M1 ≤ n. 4 log d 2 √ The last inequality follows from t− (n)f −1 (t− (n) log b) = therefore choose sufficiently  O( n) by− (3.7).  We √ 0 log t (n) η small η 0 > 0 such that η4 t− (n)f −1 (t− (n) log b) + M1 M0 +log ≤ log d 2 n. Furthermore, as n →

LARGE DEVIATION BOTTCHER

∞, s− (n) ∼

log b − log d t (n),

7

by (3.7), we have M

−

−

(t (n) − s (n))f

−1

log b − 0 b log d t (n)− log d c

−

X

(t (n) log b) +

k=1

= (t− (n) − s− (n))f −1 (t− (n) log b) + = (1 − =

√

f −1 (t− (n) log b − k log d)

log b − b log d t (n)−1c

log b − )t (n)f −1 (t− (n) log b) + log d

Z

X

f −1 (k log d)

M k=b log0d c log b − log d t (n) M0 log d

f −1 (u log d)du + O(f −1 (t− (n)) log b)

n(IA (p) + on (1)).

It follows from the tail distribution of X that there exists c3 ≥ 1 such that P(X > x) ≥ c3 e−e

f (x)

,

x > 0.

So (3.12) implies that √ √ Pt (Su ∈ [(x0 − 3η) n, x0 n], ∀u∗ ≺ u, |u| = t− (n))  h t− (n)−s− (n) η0 ≥ P X ∈ f −1 (t− (n) log b), f −1 (t− (n) log b) + f −1 (t− (n) log b) 4 M

log b − 0 b log d t (n)− log d c

Y

×

k=1 s− (n)

×

Y

idk  h η0 P X ∈ f −1 (t− (n) log b − k log d), f −1 (t− (n) log b − k log d) + f −1 (t− (n) log b) 4

M

log b − 0 k=b log d t (n)− log d c+1

≥ c2d

s− (n)+1

−

P(X ∈ (0, M1 ])d

−

t (n)−s (n) c4 c3

c3

log b − t (n) d log d

k

exp

n

M

−

− (t− (n) − s− (n))et

−

X

dk et

−

(n) log b−k log d

k=1

o

− log b − M0 t− (n) log b o t (n)−s (n) c4 ≥ c2d t (n)et (n) log b + e c3 c3 exp − (t− (n) − s− (n))et (n) log b − log d log d log b n t− (n) s− (n)+1 t− (n)−s− (n) − M0 t− (n) log b log log t− (n) o log d ≥ c2d c3 c3c4 d exp − t− (n)et (n) log b + e + . log d log d s− (n)+1

−

n

(n) log b

log b − 0 b log d t (n)− log d c

log b − t (n) d log d

−

−

Plugging it into (3.11) implies that t− (n)

P(Γt− (n),b,d ) ≥ pcb5 b

s− (n)

pcd6 d

s− (n)+1

cd2

+ ≥ exp

n

− t− (n)et

−

log b − t (n)

t− (n)−s− (n) c4 d log d c3

c3

exp

log log t− (n) o log d

(n) log b

n

−

− t− (n)et

(n) log b

+

M0 t− (n) log b e log d

o (1 + on (1)) .

Going back to (3.10), we conclude the lower bound by letting d ↑ B: lim sup n→+∞

√ 1 log[− log P(Z¯n ( nA) − ν(A) ≥ ∆)] ≤ log b, t(n)

with t(n) as in (1.5). Upper bound of (3.1): Again, by the definition of IA (p), for any δ > 0 small enough, there exists η > 0 such that sup |y|≤IA (p)−η

ν(A − y) ≤ p − δ.

WANLIN SHI∗

8

√ √ Let Bn = [(−IA (p) + η) n, (IA (p) − η) n]. Observe that for any ζ ∈ M, 1 1 X 1 1 1 X 1 X ν(A − √ ) = ν(A − √ ) + ν(A − √ ) |ζ| |ζ| |ζ| n n n c x∈ζ

x∈ζ∩Bn

≤p−δ+

(3.13)

|ζ|

c ζ(Bn ) |ζ|

,

≤ δ/2. Further, by Lemma 2.1, for all n large enough,       √ ζ(Bnc ) 1 X M1 := ζ ∈ M : ≤ δ/2 ⊂ ζ ∈ M : νn−t+ (n) ( nA − x) ≤ p − δ/4 ,   |ζ| |ζ|

which is less than p − δ/2 as long as (3.14)

x∈ζ∩Bn

ζ(Bnc )

x∈ζ

+

where t (n) will be determined later. By conditioning on {Zt+ (n) = ζ} for any ζ ∈ M1 , we observe that √ √ P(Z¯n ( nA) ≥ p) ≥ P(Z¯t+ (n) (Bnc ) > δ/2) + P(Z¯n ( nA) ≥ p, Z¯t+ (n) (Bnc ) ≤ δ/2) X √ = P(Z¯t+ (n) (Bnc ) > δ/2) + P(Zt+ (n) = ζ)P(Z¯ ζ + ( nA) ≥ p), n−t (n)

ζ∈M1

which, by (3.14), is bounded by   X √ √ 1 X ζ P(Zt+ (n) = ζ)P Z¯n−t P(Z¯t+ (n) (Bnc ) > δ/2) + νn−t+ (n) ( nA − x) + δ/4 . nA) ≥ + (n) ( |ζ| ζ∈M1

x∈ζ

+

Note that |Zt+ (n) | ≥ bt (n) . In view of Lemma 2.2,   X √ √ t+ (n) 1 X ζ nA) ≥ νn−t+ (n) ( nA − x) + δ/4 ≤ C1 e−c7 b P(Zt+ (n) = ζ)P Z¯n−t . + (n) ( |ζ| x∈ζ

ζ∈M1

+

Since P(Z¯t+ (n) (Bnc ) > δ/2) ≤ P(Zt+ (n) (Bnc ) ≥ δbt (n) /2), then √ + t+ (n) P(Z¯n ( nA) ≥ p) ≤ P(Zt+ (n) (Bnc ) ≥ δbt (n) /2) + C1 e−c7 b (3.15) . +

Here we need to treat P(Zt+ (n) (Bnc ) ≥ δbt

(n)

/2) separately in the following two cases. √ The case of B = ∞: x 7→ t(x) satisfies the equation t(x)f −1 (t(x) log b) = xIA (p), x ≥ 1. Set t+ (n) = IAIA(p)−η (p) t(n). By Markov inequality and symmetry of X, we have P(Zt+ (n) (Bnc ) ≥ δbt

+

(n)

2 −t+ (n) b E[Zt+ (n) (Bnc )] δ √ + 4 + = b−t (n) mt (n) νt+ (n) ((IA (p) − η) n, ∞). δ

/2) ≤

By Lemma 3.2, one has immediately, 1 c t+ (n) lim inf  /2)] ≥ 1. √  log[− log P(Zt+ (n) (Bn ) ≥ δb (IA (p)−η) n n→∞ f + t (n) From (3.15), we conclude that

√ 1 log[− log P(Z¯n ( nA) − ν(A) ≥ ∆)] ≥ log b. t(n) j + k n−log t+ (n) The case of b ≤ B < ∞: We take s+ (n) = t (n) log b−log ∧ t+ (n), t+ (n) = log B lim inf n→∞

while x 7→ t(x) is determined by the equation (3.16)

log b (1 − )t(x)f −1 (t(x) log b) + log B

for some constant M0 > 0 and ∀x ≥ 1.

For n large enough, we have δbt

+

(n)

/4 ≥ B s

P(Zt+ (n) (Bnc ) ≥ δbt

+

(n)

+

(n)

Z

log b log B t(x) M0 log B

f −1 (u log B)du =

√

xIA (p),

. Observe that

√ + /2) ≤ 2P(Zt+ (n) ((IA (p) − η) n, ∞) ≥ B s (n) ).

IA (p)−η IA (p) t(n),

LARGE DEVIATION BOTTCHER

9

Recall that up to the t+ (n)-th generation, the genealogical tree Tt+ (n) is Galton-Waston. Set I(n) = √ (IA (p) − η) n. Then   X X + + (3.17) P(Zt+ (n) (I(n), ∞) ≥ B s (n) ) = P(Tt+ (n) = t)Pt 1{Su >I(n)} ≥ B s (n) . t

Observe that (3.18)

n

X

+

|u|=t+ (n),u∈t

1{Su >I(n)} ≥ B s

|u|=t+ (n),u∈t

(n)

o

=

n

[

\

J ⊂tt+ (n) ,|J |=B s+ (n) u∈J

o {Su > I(n)} ,

where tt+ (n) = {u ∈ t : |u| = t+ (n)}. This yields that

(3.19)

X √ + P(Zt+ (n) ((IA (p) − η) n, ∞) ≥ B s (n) ) ≤ P(Tt+ (n) = t) t

X

J ⊂tt+ (n)

Pt (

\

u∈J

,|J |=B s+ (n)

{Su > I(n)}).

It remains to estimate the second sum on the right hand side of (3.19). For any t, J ⊂ tt+ (n) and + |J | = B s (n) , define tJ = {v ∈ t : ρ ≺ v  u, u ∈ J }. One sees that  X  \ |Xv | ≥ I(n), ∀u ∈ J Pt ( {Su > I(n)}) ≤ Pt u∈J

≤P

(3.20)

t



ρ≺vu

   X |Xv | ≥ I(n), ∀u ∈ J . sup |Xv | ≥ n + Pt sup |Xv | ≤ n; v∈tJ

v∈tJ

ρ≺vu

It follows from the tail distribution of X that there exists c8 ≥ 1 such that P(|X| ≥ x) ≤ c8 e−e

f (x)

∀x ≥ 0.

,

As a consequence, we have   + f (n) Pt sup |Xv | ≥ n ≤ |tJ |P(|X| ≥ n) ≤ c8 t+ (n)B s (n) e−e . (3.21) v∈tJ

In addition,

Pt



sup |Xv | ≤ n;

v∈tJ

≤ (3.22)

≤

X

X

ρ≺vu

Pt

 \

v∈tJ

xv ∈N∩[0,n),v∈tJ

X

|Xv | ≥ I(n), ∀u ∈ J

t+ (n)B s

c8



 {|Xv | ∈ [xv , xv + 1]} 1{minu∈J

+ (n)

e

−

xv ∈N∩[0,n),v∈tJ

P

ef (xv )

v∈tJ

1{minu∈J

P

P

(xv +1)≥I(n)}

ρ≺vu

ρ≺vu

xv ≥I1 (n)} ,

√ where I1 (n) = I(n) − t+ (n) and t+ (n) = o( n) by Remark 1.6. It remains to bound R.H.S. of (3.22). To achieve this, we follow the construction in Subpart 3 in the proof of Theorem 1.3 in [3]. The constructed tree t∗J is deterministic with labels xρ = 0 and {x∗v , v ∈ t∗J \ {ρ}} ⊂ {xv , v ∈ tJ }. Moreover, X X X X f (xv ) ≥ f (x∗v ), min ∗ x∗v ≥ min (3.23) xv ≥ I1 (n). |u|=t+ (n),u∈tJ

v∈t∗ J

v∈tJ

u∈J

ρ≺vu

ρ≺vu

t∗J contains a single branch up to generation t+ (n) − s+ (n) and also has the B-regular structure up to the generation t+ (n). See the following FIGURE 2. With the help of (3.23), we get that R.H.S. of (3.22) (3.24)

t+ (n)B s

≤ c8

Note that min|u|=t+ (n)

+ (n)

P

ρ≺vu

n|tJ |

sup

∗ ∗ x∗ v ∈N,xv
x∗v ≥ I1 (n) leads to k0 X X

k=1 |u|=k

e

P

−

P

|u|=t+ (n)

t+ (n)

x∗u

+

v∈t∗ J

X

k=k0 +1

P

∗

ef (xv )

P

1{min|u|=t+ (n)

ρ≺vu

∗ |u|=k xu B k−k0

P

ρ≺vu

x∗v ≥ I1 (n)B s

≥ I1 (n),

+

. x∗ v ≥I1 (n)}

(n)

, which means

WANLIN SHI∗

10

Figure 2. The structure of t∗J with k0 = t+ (n) − s+ (n). Note that #{u ∈ Write x ¯∗k =

P

t∗J

: |u| = k} =

∗ ¯∗k = |u|=k xu for k ≤ k0 and x

1{min|u|=t+ (n)

sup

∗ x∗ v ∈N∩[0,n),v∈tJ

e

−

P

v∈t∗ J

f (x∗ v)

1{min|u|=t+ (n)

P

1, k ≤ k0 ; B k−k0 , k0 < k ≤ t+ (n).

∗ |u|=k xu for k0 B k−k0 P ≤ x∗ v ≥I1 (n)}

ρ≺vu

Consequently, (3.25)

(

P

ρ≺vu

x∗ v ≥I1 (n)}

≤

< k ≤ t+ (n). Thus we have 1

{

Pt+ (n) k=1

x ¯∗ k ≥I1 (n)}

sup

∗ x∗ v ∈N∩[0,n),v∈tJ

e

−

.

P

v∈t∗ J

f (x∗ v)

1

{

With (3.20), (3.21) and (3.22) in hand, we plug (3.24) into (3.25) and obtain that L.H.S. of (3.20) +

≤ c8 t+ (n)B s

(3.26)

(n) −ef (n)

+

+ (c8 n)t

e

+

(n)

}=

which, together with (3.19) and (3.26), gives +

+

+ (n)



v∈t∗ J

t+ (n)−s+ (n) ∗

X

k=1

X

v∈t∗ J

x ¯∗ k ≥I1 (n)}

∗

ef (xv ) }.

s+ (n)

Note that ef (xv ) ≥

X

k=1

  t+ (n)  |tt+ (n) | + s+ (n) B ≤ ≤ B t (n)B , + (n) + (n) s s B B

P(Zt+ (n) (Bnc ) ≥ δbt (n) /2) ≤ 2B t (n)B  + f (n) + s+ (n) (3.27) × c8 t+ (n)B s (n) e−e + (c8 n)t (n)B

X

exp{−

max

∗ x∗ v ≥ 0, v ∈ tJ Pt+ (n) ∗ x ¯ ≥ I 1 (n) k k=1

Notice that #{J ⊂ tt+ (n) , |J | = B s

(n)B s

Pt+ (n)

s+ (n) ∗

ef (xv ) +

∗ x∗ v ≥ 0, v ∈ tJ Pt+ (n) ∗ x ¯k ≥ I1 (n) k=1

f (x)

|v|=k+t+ (n)−s+ (n)

exp{−

X

v∈t∗ J

 ∗ ef (xv ) } .

s+ (n)

X

∗

k=1 |v|=k+t+ (n)−s+ (n)

|v|=k

where M ≥ 0 is chosen to ensure that x 7→ e X

X

max

ef (xv ∨M ) −

X

B k ef (M ) ,

k=1

is a convex function on [M, ∞). Immediately, ∗

ef (xv ∨M ) ≥ B k ef (

P

x∗ v ∨M Bk

)

.

.

LARGE DEVIATION BOTTCHER

11

So, X

t+ (n)−s+ (n)

e

f (x∗ v)

v∈t∗ J

where x ¯∗k =

≥

P

B (k−t

X

∗ |v|=k xv

+ (n)+s+ (n)) +

(3.28)

s+ (n)

e

f (¯ x∗ k)

+

k=1

X

x∗ ) k f (¯ k+t+ (n)−s+ (n)

B e

k=1

s+ (n)

−

X

B k ef (M ) ,

k=1

. Let Ξn := max{f (¯ x∗k ) + (k − t+ (n) + s+ (n))+ log B; 1 ≤ k ≤ t+ (n)}. Then X ∗ + ef (xv ) ≥ eΞn − B s (n)+1 ef (M ) . v∈t∗ J

√

3 n By (3.16), there exists a constant C3 > 0 such that t+ (n) log b ≤ f ( C t+ (n) ). So, √ + + C3 n bt (n) f (n) > f ( + (3.29) ) ≥ t+ (n) log b, t+ (n)B s (n) = = o(ef (n) ). t (n) n

Plugging (3.28), (3.29) into (3.27) implies that P(Zt+ (n) (Bnc ) ≥ δbt

+

(n)

/2) ≤ e−e

+ 2(c8 Bn)t

(3.30)

+

(n)B

f (n)

(1+on (1))

s+ (n) ∗ x∗ v ≥ 0, v ∈ tJ

+

max +

Pt (n) ∗ x ¯k ≥ I1 (n) k=1

exp{−ef (Ξn ) + B s

(n)+1 f (M )

e

}.

Let Kn : = Ξn (t+ (n) log b)−1 . According to the definition of Ξn , we have Ξn ≥ f (¯ x∗t+ (n) ) + s+ (n) log B ≥ s+ (n) log B = t+ (n) log b(1 + on (1)). Thus, Kn ≥ 1 + on (1). Then

+

P(Zt+ (n) (Bnc ) ≥ δbt

(n)

/2) ≤ e−e

f (n)

≤ e−e

f (n)

≤e

(1+on (1)) (1+on (1))

−ef (n) (1+on (1))

+ 2(c8 Bn)t

+

+ 2(c8 Bn)t

+

(n)B s (n)B

t+ (n)B

+ 2(c8 Bn)

+ (n)

s+ (n)

s+ (n)

+

exp{−eΞn + c9 B s exp{−eKn t exp{−e

+

(n) log b

(n) +

t (n)} +

+ c9 B s

t+ (n) log b(1+on (1))

(n) +

t (n)} +

+ c9 B s

(n) +

t (n)}.

Going back to (3.15), we conclude that √ f (n) + s+ (n) + + P(Z¯n ( nA) − ν(A) ≥ ∆) ≤ e−e (1+on (1)) + 2(c8 Bn)t (n)B exp{−et (n) log b(1+on (1)) + c9 B s (n) t+ (n)} + C1 e−c7 e +

Since t+ (n)B s

(n)

log n =

t+ (n)

b

n

lim inf

t+ (n) log b

log n = o(bt 1

n→+∞ t+ (n)

+

(n)

.

), we have

√ log[− log P(Z¯n ( nA) − ν(A) ≥ ∆)] ≥ log b,

which ends the proof of the upper bound of Theorem 1.2 by letting η ↓ 0. Acknowledgement.

We would like to thank the referee for carefully reading our paper, spotting several inaccuracies and for making valuable suggestions to improve the presentation. Moreover, I am most grateful to my supervisor Prof. Hui He for helping me. References [1] S. Asmussen and N. Kaplan. Branching random walks I. Stochastic Processes and their Applications, 4(1):1–13, 1976. [2] J. D. Biggins. The central limit theorem for the supercritical branching random walk, and related results. Stochastic processes and their applications, 34(2):255–274, 1990. [3] X. Chen and H. He. On large deviation probabilities for empirical distribution of branching random walks: Schr¨ oder case and B¨ ottcher case. arXiv:1704.03776, 2017. [4] T. E. Harris. The theory of branching processes. Springer-Verlag, 2002. [5] C. F. Klebaner. Branching random walk in varying environments. Advances in Applied Probability, 14(2):359–367, 1982. [6] O. Louidor and W. Perkins. Large deviations for the empirical distribution in the branching random walk. Electronic Journal of Probability, no. 18, 1-19, 2015.

WANLIN SHI∗

12

[7] O. Louidor and E. Tsairi. Large deviations for the empirical distribution in the general branching random walk. arXiv preprint arXiv:1704.02374, 2017. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People’s Republic of China. E-mail address:

[email protected]