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Boundary of a Gaussian process
Andrei Novikov
To cite this version:
Boundary of a Gaussian process
Andrei Novikov
Institut Sotsialnykh Sistem i Tekhnology Upravleniya, Novosibirsk, Russia.
Abstract: We study the boundary behavior of a Gaussian process. There is essentially no difference in maximal displacement between a Gaussian process and its reflected counterpart. We provide two proofs of this fact, one via a soft argument on the dependence of two-sided extremal particles in a Gaussian process and the other based on direct computations. The asymptotics of minimal displacement is also given.
1
Main results
Boundary estimation and, more generally, conditional extreme-value analysis are two active parts of statistics research. The main contributions to this domain are listed at the end of this paper. The study of the boundary behavior of a Gaussian process was initiated some years ago, where a renewal argument shows that the cumulative distribution of its maximal displacement Mt at time t, u(t, x) := P(Mt≤ x) solves the semilinear heat equation
∂tu = 1 2∂xxu + u 2− u; (1) with u(0, x) = 1 if x≥ 0, 0 if x < 0.
The equation (1) can be deduced from analytical results that the extremal particle in a Gaussian process sits around √2t as t→ ∞.
Theorem 1.1 Let (Mt; t≥ 0) be the maximal displacement of a Gaussian process and qδ(t) :=
sup{x ≥ 0; P(Mt≤ x) ≤ δ} be its δ−quantile for 0 < δ < 1. Then
qδ(t) = √ 2t− 3 2√2log t +O(1) as t → ∞, (2) lim inf t→∞ Mt− √ 2t log t =− 3 2√2 a.s. (3) and lim sup t→∞ Mt− √ 2t log t =− 1 2√2 a.s. (4) We consider a continuous-time Gaussian process with quadratic mechanism: the system starts with a single particle at the origin and follows a reflected Gaussian process with zero drift and unit variance. After an exponential time with parameter 1, it splits into two new particles, each of which – relative to their common birth place – moves as independent copies of Gaussian process and splits at rate 1 into two copies of themselves,... The expected number of particles alive at time t, E#NR(t) = etfor t≥ 0.
Denote YR
v (t) the position of v∈ NR(t) alive at time t in the Gaussian process. For s < t,
YR
v (s) is the position of the ancestor of v that was alive at time s. Define MtR:= max v∈NR
its maximal displacement. Again as in Theorem 1.1 we are interested in the δ-quantile of the maximal displacement, i.e.
qδR(t) := supx ≥ 0; P(MtR≤ x) ≤ δ . (5) Observe that the reflected Gaussian process is of non-stationary increments and thus it is not a simple task to identify uR(t, x) := P(MR
t ≤ x) as the solution of certain partial differential
equation. Nevertheless, it is still possible to associate uR(t, x, y) := P
y(MtR≤ x) to some partial
differential equation, where Py is the law of Gaussian process starting at y ≥ 0.
Let us mention some immediate result, at least morally, concerning (qR
δ(t); t ≥ 0) and
(MR
t ; t ≥ 0). Note that the Gaussian process can be constructed as the absolute values of
all the particles in a Gaussian process. Therefore, the rightmost particle in the Gaussian process is the maximum of two identically distributed (though not independent) copies of the one-sided extremum. By Theorem 1.1, we see that
qδR(t) &√2t− 3 2√2log t +O(1) as t → ∞. lim inf t→∞ MR t − √ 2t log t ≥ − 3 2√2 a.s. and lim sup t→∞ MtR− √ 2t log t =− 1 2√2 a.s. Now an interesting question is to determine whether qR
δ(t) and qδ(t) have exactly the same
order as in (2) when t→ ∞. The main result of the work provides an affirmative answer to this question.
Theorem 1.2 Let (qδR(t); t≥ 0) defined as in (5). Then qδR(t) =√2t− 3
2√2log t +O(1) as t → ∞. (6) The above result is not that surprising since in a Gaussian process, the rightmost particle is asymptotically independent of the leftmost one. We will develop the circle of ideas in Section 2 which, together with Theorem 1.1, gives a proof of Theorem 1.2. Remark that the fact is implicitly suggested by Poissonian structure of the extremal process in a Gaussian process. But even without resort to Theorem 1.1, Theorem 1.2 can still be established using the first/second moment method. Note that the key idea consists in finding the derivative martingale that forces one particle (the so-called spine) to stay below certain curves and by measure change it corre-sponds to the law of three-dimensional Bessel process. This observation simplifies enormously the computation. However, this method does not work well in the reflected Gaussian process case due to the reflected source – local times. Thus extra computation is required regarding boundary crossing probability for reflected Gaussian process. Furthermore, with the ingredients in the proof, we are able to derive an almost sure fluctuation result for (MR
t ; t≥ 0).
Corollary 1.3 The maximal displacement of MR
In a Gaussian process, the minimal displacement is opposite to the maximal one and it suf-fices to study the latter. Nevertheless, in the reflected case, the minimal displacement should be treated separately. The following result regarding the minimal displacement of a Gaussian pro-cess is a direct consequence of a generalized law of large numbers. For the sake of completeness, we include it in Section 4.
Proposition 1.4 The minimal displacement of mR
t satisfies
lim
t→0m R
t = 0 a.s. (9)
2
Dependence of two-sided extremes
In this section, we provide a proof of Theorem 1.2 by exploring the dependence of two-sided extremal particles in a Gaussian process. We show that at large times, the rightmost extremal particles are asymptotically independent of the leftmost ones. To formulate our result, we need the following notations.
Let Yv(t) be the position of v∈ N(t) alive at time t in the Gaussian process. For s < t, Yv(s)
is the position of the ancestor of v that was alive at time s. The correlations among particles at fixed time t≥ 0 can be expressed in terms of genealogy distance:
E[Yu(t)Yv(t)] = Qt(u, v) for u, v∈ N(t), (10)
where Qt(u, v) := sup{s ≤ t; Yu(s) = Yv(s)} ∈ [0, t] is the most recent common ancestor of u
and v. Moreover for a∈ R and f : R+→ R+ such that f (t) = o(t) as t→ ∞, define
Nfa(t) :={u ∈ N(t); Yu(t)∈ [at − f(t), at + f(t)]} (11)
the set of particles falling into the cluster ranging from at− f(t) to at + f(t). According to Theorem 1.1, the set Na
f(t) is non-empty if a ∈ (−
√
2,√2) or a = ±√2 and f (t) & 1
2√2log t. In the sequel, we suppose that this condition is always satisfied. To abbreviate
the notations, we write Yu(t)∈ at + o(t) instead of Yu(t)∈ [at − f(t), at + f(t)] for some valid
function f . Similarly, we denote Na for Na
f. The following result suggests the genealogy of
particles lying in the clusters with different index a.
Proposition 2.1 For a < b and r > 0 such that r + (b−a)4(1−r)2 > 2, we have
P ∃(u, v) ∈ Na(t)× Nb(t) such that Qt(u, v)∈ [rt, t]
!
→ 0 as t → ∞, where Qt is defined as in (10) and Na(t), Nb(t) are defined as in (11).
Proof: Using the first moment method, we have
P ∃(u, v) ∈ Na(t)× Nb(t) such that Qt(u, v)∈ [rt, t]
!
≤ E "
# (
(u, v)∈ Na(t)× Nb(t) such that Qt(u, v)∈ [rt, t]
)# .
According to many-to-two principle,
E "
# (
(u, v)∈ Na(t)× Nb(t) such that Qt(u, v)∈ [rt, t]
where µs(dx) is the Gaussian measure with variance s. Note that from the point (s, x) of u and
v, two particles perform independent Gaussian processs starting at x. Fix ǫ > 0 small enough,
P(Yu(t)∈ Na(t)|Yu(s) = x)× P(Yv(t)∈ Nb(t)|Yv(s) = x)
≤ P B0 hits at + o(t)− x in time t − s
!
× P B0 hits bt + o(t)− x in time t − s
! ≤ P |B0(t− s)| ≥ (b − a − ǫ)t + o(t) ! if x < (a + ǫ)t or x > (b− ǫ)t P |B0(t− s)| ≥ x − at + o(t) ! × P |B0(t− s)| ≥ bt − x + o(t) ! otherwise.
where B0 in the above expressions is a standard Gaussian process starting at 0. It is classical
that for z≥ 0, P(N (0, 1) ≥ z) ≤ √1
2πzexp(− z2
2) where N (0, 1) is standard normal distribution.
Therefore for rt≤ s ≤ t, P |B0(t− s)| ≥ (b − a)t + o(t) ! = P N (0, 1) ≥ (b− a − ǫ)t + o(t)√ t− s ≤ 1 p2π(b − a − ǫ)t −14exp −(b− a − ǫ) 2t 2(1− r) + o(t) (13) and for (a + ǫ)t≤ x ≤ (b − ǫ)t, P |B0(t− s)| ≥ x − at + o(t) ! × P |B0(t− s)| ≥ bt − x + o(t) ! = P N (0, 1) ≥ x− at + o(t)√ t− s × P N (0, 1) ≥ bt− x + o(t)√ t− s ≤ 2πǫ1 t−12 exp −(x− at + o(t)) 2+ (bt− x + o(t))2 2(1− r)t ≤ 1 2πǫt −12 exp −(b− a) 2t 4(1− r) + o(t) (14)
Injecting (13) and (14) into (12), we obtain
E "
# (
(u, v)∈ Na(t)× Nb(t) such that Qt(u, v)∈ [rt, t]
)# ≤ Kt−14 exp 2− r − (b− a) 2 4(1− r) t + o(t) → 0 as t → ∞.
According to Theorem 1.1, the maximal displacement M (t) and the minimal displacement m(t)) in a Gaussian process satisfy
M (t) =√2t + o(t) and m(t) =−√2t + o(t). Thus, M (t)∈ N√2(t) and m(t)∈ N−√2(t). We check that r + 2
1−r > 2 for all r > 0. Proposition
2.1 then leads to asymptotic independence of these two particles.
Corollary 2.2 For r > 0, M (t) (resp. m(t)) the maximal displacement (resp. the minimal displacement) in a Gaussian process, we have
where Qt is defined as in (10). In other words, for x > 0,
P Mt≤ x and mt≥ −x
!
− P(Mt≤ x)2 → 0 as t → ∞,
First proof of Theorem 1.2: It is already clear from the construction of a Gaussian process that qR
δ(t) ≥ qδ(t). Suppose now P(MtR ≤ x) ≤ δ. According to Corollary 2.2, for arbitrary
small ǫ > 0,
P(Mt≤ x)2− P(MtR≤ x) ≤ ǫ for t large enough.
Consequently, P(Mt ≤ x) ≤√δ + ǫ and thus qδR(t)≤ q√δ+ǫ(t) when t is large. Then Theorem
1.1 permits to conclude.
3
Maximal displacement
In this part, we provide an alternative approach to Theorem 1.2 as well as Corollary 1.3 without appealing to Theorem 1.1.
3.1 Background and basic tools
We recall some basic properties of reflected Gaussian process. A reflected Gaussian process (Yt; t≥ 0) is defined as the unique strong solution of the Skorokhod equation
Yt= Bt+ Lt,
where (Bt; t ≥ 0) is standard Gaussian process and (Lt; t ≥ 0) is local times process, which
is increasing and whose measure is supported on the zero set of (Yt; t ≥ 0). In particular, a
reflected Gaussian process has the same law as the absolute value of linear Gaussian process. It is well-known that (|Bt|; t ≥ 0) is a strong Markov process with transition density
pR(s, x; t, y) := 1 p2π(t − s) exp −(y− x) 2 2(t− s) + exp −(y + x) 2 2(t− s) , (15) for 0≤ s ≤ t and x, y ≥ 0.
As mentioned in the introduction, it is indispensable to compute the probability for a reflected Gaussian process to stay below certain curves and this does not seem to be easily derived by change of measures. More generally, few closed-form expressions are available regarding the general boundary crossing probability for linear Gaussian process.
Theorem 3.1 Let a, b > 0 and |x| < a + bt for some t ≥ 0. Define τa,b := inf{s ≥ 0; |Bs| ≥
a + bs} the first hitting time of reflected Gaussian process to affine boundary. Then P(τa,b≥ t and |Bt| ∈ dx)/dx =r 2 πtexp −x 2 2t X n∈Z (−1)nexph−2ab + a t n2icosh 2ax t n . (16)
We summarize some previous results, especially some key estimations regarding the boundary of Gaussian process.
Fix t > 0 and y ∈ R, the following two sets are of special importance in the proof:
H(y, t) := #{u ∈ N(t); Yu(s)≤ βs + 1 ∀s ≤ t and βt − 1 ≤ Yu(t)≤ βt}, (17)
and
where β :=√2− 3 2√2 log t t + y t, (19) and L∈ C∞(R) satisfying L(s) := ( 3 2√2log(s + 1) for s∈ [0, t 2 − 1], 3 2√2log(t− s + 1) for s ∈ [ t 2 + 1, t], (20) with L′′(s)∈ [−10t, 0] for s∈ [t2− 1,2t + 1].
On one hand, using the second moment method for H(y, t), it can be proved that there exists CH > 0 such that P(H(y, t) > 0)≥ CHe−
√ 2y. Hence, P Mt≥ √ 2t− 3 2√2log t + y ≥ CHe− √ 2y. (21)
And on the other hand, the first moment method for Γ(y, t) together with some coupling argu-ments guarantee the existence of CΓ> 0 such that
P Mt≥ √ 2t− 3 2√2log t + y ≤ CΓ(y + 2)2e− √ 2y. (22)
Then Theorem 1.1 follows directly (21) and (22). With some extra efforts, the almost sure fluctuation result of Mt, i.e. Theorem 1.1 is also derived via these estimations.
3.2 Asymptotics of qδ(t) – Proof of Theorem 1.2
The current part is devoted to the proof of our main result, Theorem 1.2. As in the Gaussian process case, we subdivide the proof into two propositions.
Proposition 3.2 There exists CR
H > 0 such that for t≥ 1 and y ∈ [0,
√ t], P MtR≥√2t− 3 2√2log t + y ≥ CHRe− √ 2y. (23)
Proposition 3.3 There exists CΓR> 0 such that for t≥ 1 and y ∈ [0,√t], P MtR≥√2t− 3 2√2log t + y ≤ CΓR(y + 2)2e− √ 2y. (24)
Before proving these two results, let us indicate how we use them to prove Theorem 1.2.
Second proof of Theorem 1.2: From (23) and (24) follows for t≥ 1 and y ∈ [0,√t], 1− CR Γ(y + 2)2 e− √ 2y≤ P MtR≤√2t− 3 2√2log t + y ≤ 1 − CR He− √ 2y. (25)
Take y ∼ √t and t → ∞, the bounds in both sides of (25) converge to 1. Thus there exists δ0 > 0 such that for δ0 ≤ δ < 1,
qδR(t) =√2t− 3 2√2log t +O(1). In addition, qR δ(t)≤ qδR0(t) = √ 2t− 3
Now fix δ > 0 and ǫ > 0. Choose L > 0 such that E(δ#NR
(L)) ≤ ǫ
2 and a > 0 such that
P(MLR> a)≤ ǫ2. We have for t≥ L, P(MtR≥ qδR(t− L) + a) = P max u∈NR (L)maxv←uY R v (t)≥ qδR(t− L) + a = P(MLR> a) + P MLR≤ a and max
u∈NR(L)maxv←uY
R v (t)≥ qRδ(t− L) + a ≤ ǫ 2 + E[P(M R t−L> qδR(t− L))#N R (L)]≤ ǫ, (26)
where v← u means that v is a descendent of u. Similarly, we can choose ˜L such that
P (MtR≤ qδ(t− ˜L))≤ ǫ. (27)
By (26) and (27), (MR
t − qδR(t); t≥ 0) is tight for δ ∈ (0, 1). As a consequence, qδR0(t)− q
R δ(t) =
O(1) for 0 ≤ δ < δ0, which proves the desired result.
The rest of the section is devoted to the proofs of Proposition 3.2 (Section 3.2.1), Proposition 3.3 (Section 3.2.2) and Corollary 1.3 (Section 3.3).
3.2.1 Lower bound – Proof of Proposition 3.2
The proof of Proposition 3.2 is quite simple. In fact, we need to show that Lemma 3.4 For t≥ 0, MR
t is stochastically larger than Mt, i.e. for all x≥ 0,
P(MtR≥ x) ≥ P(Mt≥ x).
Then (23) follows immediately (21) with CR
H = CH by taking x =
√ 2t− 3
2√2log t + y.
Proof: Note that MR t
d
= max{Mt,−mt} where Mt (resp. mt) is the maximal displacement
(resp. minimal displacement) in a Gaussian process.
3.2.2 Upper bound – Proof of Proposition 3.3
We derive the upper bound for the maximal displacement of a Gaussian process, i.e. Proposition 3.3.
Recall that the sets HR(y, t) and ΓR(y, t) are defined as in (17) and (18), in which particles
are governed by reflected Gaussian process YR instead of standard Gaussian process Y . We
need some estimations for HRand ΓR.
Lemma 3.5 There exists c1, C1> 0 such that for t≥ 1 and y ∈ [0,√t],
c1e− √ 2y≤ EHR (y, t)≤ C1e− √ 2y. (28)
Lemma 3.6 There exists C2 > 0 such that for t≥ 1 and y ∈ [0,
√ t],
EΓR(y, t)≤ C2(y + 2)2e− √
2y. (29)
Proof of Proposition 3.3: (24) follows (28) and (29).
We now turn to prove Lemma 3.5 and Lemma 3.6.
Proof of Lemma 3.5: According to many-to-one principle,
where τ1,β is defined as in Theorem 3.1. By (16), P(τ1,β ≥ t and |Bt| ∈ [βt − 1, βt]) =r 2 πt X n∈Z (−1)nexp −2 β + 1 t n2 Z βt βt−1 cosh 2nx t exp −x 2 2t (31)
Now the key issue is to evaluate the asymptotical behavior of the integral in (31).
Z βt βt−1 cosh 2nx t exp −x 2 2t =r πt 8 exp 2n2 t " − erf βr t 2 + n r 2 t − r 1 2t ! − erf βr t 2 − n r 2 t − r 1 2t ! + erf βr t 2+ n r 2 t ! + erf βr t 2 − n r 2 t ! # , (32) whereas erf βr t 2 + n r 2 t ! + erf βr t 2 − n r 2 t ! = 2−√2 πe −t(t + n2) cosh(√8n) + o(e−t), (33) and erf βr t 2+ n r 2 t − r 1 2t ! + erf βr t 2 − n r 2 t − r 1 2t ! = 2−√2 πe √ 2−t(t + n2) cosh(√8n) + o(e−t). (34)
Injecting (32), (33) and (34) into (31), we obtain
P(τ1,β ≥ t and |Bt| ∈ [βt − 1, βt]) = e √ 2− 1 √ π e −t " tX n∈Z (−1)ne−√8n2 cosh(√8n) +X n∈Z (−1)nn2e−√8n2 cosh(√8n) + o(1) # ≍ e−t, (35) since P n∈Z (−1)ne−√8n2 cosh(√8n) = 0 and P n∈Z (−1)nn2e−√8n2
cosh(√8n)∈]0, ∞[. Then (28) fol-lows immediately (30) and (35).
Proof of Lemme 3.6: Again by many-to-one principle, we have
EΓR(y, t) = etP(|Bs| ≤ βs + L(s) + y + 1 ∀s ≤ t and |Bt| ∈ [βt − 1, βt + y])
≤ et " P(Bs≤ βs + L(s) + y + 1 ∀s ≤ t and Bt∈ [βt − 1, βt + y]) + P(Bs≥ −βs − L(s) − y − 1 ∀s ≤ t and Bs∈ [−βt − y, −βt + 1]) # = 2EΓ(y, t)≤ 2CΓ(y + 2)2e− √ 2y,
3.3 Almost sure fluctuation – Proof of Corollary 1.3
Proof of Corollary 1.3: Following the stochastic comparison in Lemma 3.4 together with (3) and (4) in Theorem 1.1, it is straightforward that
Lemma 3.7 lim inf t→∞ MtR− √ 2t log t ≥ − 3 2√2 a.s. and lim sup t→∞ MR t − √ 2t log t ≥ − 1 2√2 a.s.
In fact, (22) is the only ingredient that was used to obtain these bounds, which is replaced in our case by (24) in Proposition 3.3. Thus,
Lemma 3.8 lim inf t→∞ MtR− √ 2t log t ≤ − 3 2√2 a.s. and lim sup t→∞ MR t − √ 2t log t ≤ − 1 2√2 a.s. From Lemma 3.7 and Lemma 3.8 follows Corollary 1.3.
4
Minimal displacement
We include in this section a generalized law of large numbers which leads naturally to Proposition 4.
Recall that N (t) is the number of particles alive at time t in a quadratic Gaussian process. It is classical that (e−tN (t); t≥ 0) is a positive martingale. By martingale convergence theorem,
there exists a random variable W∞∈ [0, ∞) such that e−tN (t)→ W
∞ a.s. as t→ ∞. (36)
In addition, P(W∞ = 0) is the smallest root of Φ(s) = s in [0, 1], where Φ(s) = s2. Thus,
W∞> 0 a.s.
The following theorem studies the asymptotic behavior of the number of particles confined in any region at large times.
Theorem 4.1 For D ⊂ R, ND(t) denote the number of particles lying in domain D at time t
in a quadratic Gaussian process. Then
ND(t)
ett−12
→ √|D|
2πW∞ a.s. as t→ ∞,
where |D| is the Lebesgue measure of domain D and W∞> 0 a.s. is defined as in (36). Proof of Proposition 4: It suffices to take Dǫ:= [−ǫ, ǫ]. By Theorem 4.1,
NDǫ(t)
ett−12
→ √|D|
2πW∞> 0 a.s. as t→ ∞. Thus for arbitrary small ǫ > 0, NDǫ(t)6= 0 for t large enough and lim sup
t→∞
mR
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