8th Vilnius Prob. Conf. B. Grigelionis et al. (Eds) 2002 Vilnius

Invariance principle under self-normalization for nonidentically distributed random variables

A. RAČKAUSKAS

Vilnius University and Institute of Mathematics and Informatics, Naugarduko 24, Lt-2006 Vilnius, Lithuania

CH. SUQUET

Mathématiques Appliquées, F.R.E. CNRS 2222, Université des Sciences et Technologies de Lille, Bât. M2, F-59655 Villeneuve d’Ascq Cedex, France

Abstract. Let V_{n}^{−1}ζ_{n} be the adaptive process of self-normalized partial sums S_{k} of independent random variables X_{i}, defined by linear interpolation between the points (V_{k}^{2}/V_{n}^{2}, S_{k}/V_{n}), k ≤ n, where V_{k}^{2} = ∑_{i≤k} X_{i}^{2}. We prove that if the X_{k}’s are symmetric, V_{n}^{−1}ζ_{n} converges weakly to the Brownian motion W in each Hölder space supporting W if and only if V_{n}^{−1} max_{k≤n} |X_{k}| = o_{P} (1). We give some partial extension to the non symmetric case.

Key words: functional central limit theorem, Hölder space, partial sums processes

Mathematics Subject Classifications (2000): 60F05, 60B05, 60G17, 60E10.

1. INTRODUCTION AND RESULTS

Let X_{1}, …, X_{n}, … be a sequence of independent random variables with null expectation. Set S(0) = V_{0} = 0 and

S(t) := ∑_{k≤t} X_{k}, V_{t}^{2} := ∑_{k≤t} X_{k}^{2}, V_{t} ≥ 0,

for 0 < t ≤ n. Define the random partition (τ(k), k = 0, …, n) of the interval [0, 1] by

τ(k) = V_{k}^{2}/V_{n}^{2}, k = 0, 1, …, n.

In this paper we consider the process (ζ_{n}(t), t ∈ [0, 1]), defined on [0, 1] by linear interpolation between the points (τ(k), S(k)), k = 0, …, n. By convention, ζ_{n} and V_{n}^{−1}ζ_{n} are defined to be the null function on the event {V_{n} = 0}. Of course we discard the degenerated case where all the X_{k}’s are a.s. null.

We examine the weak convergence of V_{n}^{−1}ζ_{n} to a standard Brownian motion W in a framework of Hölder spaces H_{ρ}^{o}. For a weight function ρ : [0, 1] → ℝ_{+}, H_{ρ}^{o} is the set of continuous x : [0, 1] → ℝ such that lim_{δ→0} ω_{ρ}(x, δ) = 0, where

ω_{ρ}(x, δ) := sup_{\begin{matrix}t,s∈[0,1], \\ 0<|t−s|<δ \end{matrix}} \frac{|x(t) − x(s)|}{ρ(|s − t|)}.

The set H_{ρ}^{o} is a separable Banach space when endowed with the norm

∥x∥_{ρ} := |x(0)| + ω_{ρ}(x, 1).

In what follows we assume that ρ ∈ Q, where the class Q will be defined in the next section. For instance, the functions

ρ(h) = ρ_{α,β}(h) := h^{α} ln^{β}(\frac{c}{h}), 0 < α < 1, β ∈ ℝ,

belong to this class for a suitable choice of the constant c. We write H_{α,β}^{o} for H_{ρ}^{o} when ρ = ρ_{α,β} and we abbreviate H_{α,0}^{o} in H_{α}^{o}. For symmetric random variables we have the following result.

Theorem 1. Assume that ρ ∈ Q and

lim_{j→∞} \frac{2^{j}ρ^{2}(2^{−j})}{j} = ∞.  (1)

If X_{1}, …, X_{n}, … are symmetric then

V_{n}^{−1}ζ_{n} ⟶_{n→∞}^{D} W, in H_{ρ}^{o}  (2)

if and only if

\frac{max_{1≤k≤n} X_{k}^{2}}{V_{n}^{2}} ⟶_{n→∞}^{P} 0.  (3)

Condition (1) hold true if lim_{h↓0} ρ(h)/ρ_{c}(h) = ∞, where the critical weight function ρ_{c}(h) is the modulus of uniform continuity of W (up to a constant factor). As follows from Lévy’s theorem, W cannot have a version in H_{ρ_{c}}^{o}, so the class of Hölder spaces involved by the convergence (2) is the biggest possible. Necessity of condition (3) is actually established in Egorov [2] where it is proved that (3) is equivalent to a central limit theorem. Theorem 1 is a complete extension of the case of i.i.d. symmetric X_{k}’s in [8]. Our motivation for investigating partial sums processes in a Hölderian framework goes back to Lamperti [4]. In this paper he obtained in the i.i.d. case the H_{α}^{o} weak convergence of the process n^{−1/2}ξ_{n}, where ξ_{n} is defined by linear interpolation of the points (k/n, S(k)), under the assumption E |X_{1}|^{p} < ∞ for p > (1/2−α)^{−1}. In fact it is proved in [7] that the convergence of n^{−1/2}ξ_{n} to W in H_{ρ}^{o} for any ρ satisfying (1) requires the finiteness of E exp(|X_{1}|^{γ}) for each 0 < γ < 2. Due to the topological inclusion of H_{ρ}^{o} in C[0, 1], (3) is also equivalent to the weak convergence of V_{n}^{−1}ζ_{n} in C[0, 1]. Let us also note that for the step partial sums process Z_{n}, Z_{n}(t) = S(k) for τ(k) ≤ t < τ(k + 1), (3) is equivalent to the weak convergence of V_{n}^{−1}Z_{n} to W in D[0, 1] equipped with the sup-norm metric. This follows immediately using the obvious estimate ∥ζ_{n} − Z_{n}∥_{∞} ≤ V_{n}^{−1} max_{k≤n} |X_{k}| and Egorov [2] necessity of (3) for the relevant central limit theorem. This result in D[0, 1] is also proved by Csörgő, Szyszkowicz and Wang [1]. In the nonsymmetric case we have a partial result.

Theorem 2. Assume that the following conditions are satisfied.

i) For each k ≥ 1, E X_{k} = 0 and for some ε > 0, sup_{k≥1} E |X_{k}|^{2+ε} < ∞.

ii) There is some T > 0 such that κ(T) := sup_{j≥1} P(|X_{j}| ≤ T) < 1.

Then with any 0 < α < 1/2,

V_{n}^{−1}ζ_{n} ⟶_{n→∞}^{D} W in H_{α}^{o}.  (4)

Due to the topological inclusions of Hölder spaces, the convergence (4) takes place also in any H_{ρ}^{o} with ρ(h) = h^{α}L(1/h) where 0 < α < 1/2 and L slowly varying at infinity. In the i.i.d. case, condition ii) is automatically satisfied and the authors proved in [8] the weak convergence in H_{1/2, β}^{o} for any β > 1/2.

2. PRELIMINARIES

For readers convenience we collected in this section some facts about the Hölder spaces H_{ρ}^{o} including the tightness criterion for distributions in these spaces. In details all these facts may be found e.g. in [6]. In what follows, we assume that the function ρ involved in the definition of the Hölder space H_{ρ}^{o} satisfies the following technical conditions (denoted by ρ ∈ Q), where c_{1}, c_{2} and c_{3} are positive constants:

ρ(0) = 0, ρ(δ) > 0, 0 < δ ≤ 1;  (5)
ρ is non decreasing on [0, 1];  (6)
ρ(2δ) ≤ c_{1}ρ(δ), 0 ≤ δ ≤ 1/2;  (7)
∫_{0}^{δ} \frac{ρ(u)}{u} du ≤ c_{2}ρ(δ), 0 < δ ≤ 1;  (8)
δ ∫_{δ}^{1} \frac{ρ(u)}{u^{2}} du ≤ c_{3}ρ(δ), 0 < δ ≤ 1.  (9)

Write D_{j} for the set of dyadic numbers of level j in [0, 1], i.e. D_{0} = {0, 1} and for j ≥ 1,

D_{j} = { (2k − 1)2^{−j}; 1 ≤ k ≤ 2^{j−1} }.

Write for r ∈ D_{j}, j ≥ 0,

r^{−} := r − 2^{−j}, r^{+} := r + 2^{−j}.

For any continuous function x : [0, 1] → ℝ, define for r ∈ D_{j}, j ≥ 1,

λ_{r}(x) := x(r) − \frac{1}{2}(x(r^{+}) + x(r^{−}))

and for r ∈ D_{0}, λ_{r}(x) := x(r). The λ_{r}(x) are the coefficients of the expansion of x in a series of triangular functions. The j-th partial sum E_{j}x of this series is exactly the polygonal line interpolating x between the dyadic points k2^{−j}(0 ≤ k ≤ 2^{j}). Under (5) to (9), the norm ∥x∥_{ρ} is equivalent to the sequence norm

∥x∥_{ρ}^{seq} := sup_{j≥0} \frac{1}{ρ(2^{−j})} max_{r∈D_{j}} |λ_{r}(x)|.

It is easy to check that

∥x − E_{j}x∥_{ρ}^{seq} = sup_{i>j} \frac{1}{ρ(2^{−i})} max_{r∈D_{i}} |λ_{r}(x)|.

Proposition 3. The sequence (Y_{n}) of random elements in H_{ρ}^{o} is tight if and only if the following two conditions are satisfied:

i) For each t ∈ [0, 1], the sequence (Y_{n}(t))_{n≥1} is tight on ℝ.

ii) For each ε > 0,

lim_{j→∞} sup_{n≥1} P (∥Y_{n} − E_{j}Y_{n}∥_{ρ}^{seq} > ε) = 0.

Remark 1. Condition ii) in Proposition 3 may be replaced by

lim_{j→∞} lim sup_{n→∞} P (∥Y_{n} − E_{j}Y_{n}∥_{ρ}^{seq} > ε) = 0.  (10)

3. PROOF OF THEOREM 1

We follow the classical steps by proving first the convergence of finite dimensional distributions and next establishing tightness.

Finite dimensional distributions convergence. Due to the continuity of the map f : ℝ^{d} → ℝ^{d}, f(x_{1}, …, x_{d}) = (x_{1}, x_{1} + x_{2}, …, x_{1} + x_{2} + ⋯ + x_{d}), the convergence

V_{n}^{−1}ζ_{n} ⟶_{n→∞}^{fdd} W  (11)

reduces to

V_{n}^{−1}(ζ_{n}(t_{1}), ζ_{n}(t_{2}) − ζ_{n}(t_{1}), …, ζ_{n}(t_{d}) − ζ_{n}(t_{d−1})) ⟶_{n→∞}^{D}

(W (t_{1}), W (t_{2}) − W (t_{1}), …, W (t_{d}) − W (t_{d−1})) (12)

for any 0 ≤ t_{1} < t_{2} < ⋯ < t_{d} ≤ 1. Define for t ∈ [0, 1]

u_{n}(t) := max{k : V_{k}^{2} ≤ tV_{n}^{2}}.  (13)

From this definition,

τ(u_{n}(t)) ≤ t < τ(u_{n}(t) + 1),  (14)

for each t ∈ [0, 1]. Hence,

|V_{n}^{−1}ζ_{n}(t) − V_{n}^{−1}S(u_{n}(t))| ≤ V_{n}^{−1}|X_{u_{n}(t)+1}| ≤ V_{n}^{−1} max_{1≤k≤1} |X_{k}| ⟶_{n→∞}^{P} 0

and (12) reduces to

V_{n}^{−1}(S(u_{n}(t_{1})), S(u_{n}(t_{2})) − S(u_{n}(t_{1})), …, S(u_{n}(t_{d})) − S(u_{n}(t_{d−1})))

⟶_{n→∞}^{D} (W (t_{1}), W (t_{2}) − W (t_{1}), …, W (t_{d}) − W (t_{d−1})). (15)

Set t_{0} := 0 and for real numbers a_{1}, …, a_{d},

T_{n,d}(X_{1}, …, X_{n}) := V_{n}^{−1} ∑_{i=1}^{d} a_{i}(S(u_{n}(t_{i})) − S(u_{n}(t_{i−1}))),

T_{d} := ∑_{i=1}^{d} a_{i}(W (t_{j}) − W (t_{j−1})).

By the Cramér-Wold device, (15) is equivalent to

T_{n,d}(X_{1}, …, X_{n}) ⟶_{n→∞}^{D} T_{d},  (16)

for any real numbers a_{1}, …, a_{d}. Evidently, (16) follows if we prove that

Δ_{n} := sup_{x∈ℝ}|P(T_{n,d}(X_{1}, …, X_{n}) ≤ x) − P(T_{d} ≤ x)| ⟶_{n→∞} 0.  (17)

In what follows, we discard the trivial case where all the a_{i}’s are zero. Let (ε_{k}, k ≥ 1) be a Rademacher sequence, independent of (X_{k}, k ≥ 1). Due to the symmetry, (X_{1}, …, X_{n}) and (ε_{1}X_{1}, …, ε_{n}X_{n}) are equally distributed, and ε_{k}^{2} = 1 a.s. Hence,

P(T_{n,d}(X_{1}, …, X_{n}) ≤ x) = P(T_{n,d}(ε_{1}X_{1}, …, ε_{n}X_{n}) ≤ x).

It is convenient to view the sequences (X_{k}, k ≥ 1), (ε_{k}, k ≥ 1) and (ε_{k}X_{k}, k ≥ 1) as defined respectively on the probability spaces (Ω_{X}, F_{X}, P_{X}), (Ω_{ε}, F_{ε}, P_{ε}) and (Ω, F, P) = (Ω_{X} × Ω_{ε}, F_{X} ⊗ F_{ε}, P_{X} ⊗ P_{ε}). We uses the decomposition

T_{n,d}(ε_{1}X_{1}, …, ε_{n}X_{n}) = ∑_{k=1}^{n} ε_{k}d_{n,k},

where denoting indicator function by I,

d_{n,k} := V_{n}^{−1}X_{k} ∑_{i=1}^{d} a_{i}I{u_{n}(t_{i−1}) < k ≤ u_{n}(t_{i})}, k = 1, …, n.

Introduce a sequence (G_{n}, n ≥ 1) of random variables

G_{n} : Ω_{X} × Ω_{ε} → ℝ, (ω_{1}, ω_{2}) ↦ σ_{n}(ω_{1})G(ω_{2}),

where G is Gaussian N(0, 1) defined on Ω_{ε} and σ_{n} ≥ 0 is given by

σ_{n}^{2}(ω_{1}) = ∑_{k=1}^{n} d_{n,k}^{2}(ω_{1}).

The limiting random variable T_{d} is centered Gaussian with variance

σ^{2} := ∑_{i=1}^{d} a_{i}^{2}(t_{i} − t_{i−1}), σ > 0.

It is easily seen that

σ_{n}^{2} = ∑_{i=1}^{d} a_{i}^{2}(τ(u_{n}(t_{i})) − τ(u_{n}(t_{i−1}))).  (18)

Recalling that

τ(u_{n}(t)) = \frac{V_{u_{n}(t)}^{2}}{V_{n}^{2}} ≤ t < \frac{V_{1+u_{n}(t)}^{2}}{V_{n}^{2}} , t ∈ [0, 1],

we get with a^{2} := ∑_{i=1}^{d} a_{i}^{2}, the estimate

|σ_{n}^{2} − σ^{2}| ≤ a^{2} max_{1≤k≤n} \frac{X_{k}^{2}}{V_{n}^{2}} .  (19)

In view of Hypothesis (3), this gives in particular

\frac{σ_{n}}{σ} ⟶_{n→∞}^{P} 1.  (20)

Now to estimate estimation of Δ_{n}, write Δ_{n} ≤ sup_{x∈ℝ} Δ′_{n}(x)+sup_{x∈ℝ} Δ′′_{n}(x),

Δ′_{n}(x) := E |I{T_{n,d}(ε_{1}X_{1}, …, ε_{n}X_{n}) ≤ x} − I{G_{n} ≤ x}|
Δ′′_{n}(x) := |E I{G_{n} ≤ x} − Φ(x/σ)|,

where Φ denotes the distribution function of N(0, 1). Convergence of Δ′_{n}. Integrating with respect to P_{ε} gives

Δ′_{n}(x) ≤ ∫_{Ω_{X}} |P_{ε}(T_{n,d}(ε_{1}X_{1}, …, ε_{n}X_{n}) ≤ x) − P_{ε}(G_{n} ≤ x)| dP_{X}. (21)

Applying Berry-Esséen’s theorem conditionally on the X_{k}’s, we get an absolute constant C > 0 such that

Y_{n} := sup_{x∈ℝ}|P_{ε}(T_{n,d}(ε_{1}X_{1}, …, ε_{n}X_{n}) ≤ x) − P_{ε}(G_{n} ≤ x)|
≤ C ∑_{k=1}^{n} |d_{n,k}|^{3}σ_{n}^{−3}
≤ C max_{1≤i≤d} |a_{i}| σ_{n}^{−1} max_{1≤k≤n} \frac{|X_{k}|}{V_{n}}.

Since by (14)

∑_{i=1}^{d} a_{i}^{2}(τ(u_{n}(t_{i})) − τ(u_{n}(t_{i−1})) ≥ ∑_{i=1}^{d} a_{i}^{2}((t_{i} − t_{i−1}) − max_{1≤k≤n} \frac{|X_{k}|}{V_{n}}), (22)

condition (3) yields P_{X}(Y_{n} > δ) → 0, for any δ > 0. Moreover Y_{n} is bounded by 1, so the convergence of E Y_{n} to zero follows from E Y_{n} ≤ P_{X}(Y_{n} > δ) + δ. In view of (21), this gives

lim_{n→∞} sup_{x∈ℝ} Δ′_{n}(x) = 0.

Convergence of Δ′′_{n}. Recalling the definition of G_{n}, we have

Δ′′_{n}(x) ≤ P_{X}(σ_{n} = 0) + ∫_{{σ_{n}>0}}|∫_{Ω_{ε}} I{σ_{n}G ≤ x} − Φ(x/σ) dP_{ε}| dP_{X}

= P_{X}(σ_{n} = 0) + ∫_{\{σ_{n}>0\}}|∫_{x/σ_{n}}^{x/σ} \frac{1}{\sqrt{2π}} exp(\frac{−t^{2}}{2} ) dt| dP_{X}.

When x = 0 this last integral vanishes. For x ≠ 0, putting t = σ_{n}^{−1}xu leads to

|∫_{x/σ_{n}}^{x/σ} \frac{1}{\sqrt{2π}} exp(\frac{−t^{2}}{2} ) dt| = |∫_{1}^{σ_{n}/σ} \frac{1}{\sqrt{2π}} exp(\frac{−x^{2}u^{2}}{2σ_{n}^{2}} ) du| ≤ \frac{1}{\sqrt{2π}} |\frac{σ_{n}}{σ} − 1|.

Hence we have the uniform bound

sup_{x∈ℝ} Δ′′_{n}(x) ≤ P_{X}(σ_{n} = 0) + \frac{1}{\sqrt{2π}} E |\frac{σ_{n}}{σ} − 1|.

By (3), (18) and (22), P_{X}(σ_{n} = 0) goes to zero. The random variables σ_{n}/σ are uniformly bounded since

\frac{σ_{n}^{2}}{σ^{2}} ≤ 1 + a^{2} max_{1≤k≤n} \frac{X_{k}^{2}}{V_{n}^{2}} ≤ 1 + a^{2}.

Together with (20), this gives the convergence to zero of E |σ_{n}/σ −1|. Finally

lim_{n→∞} sup_{x∈ℝ} Δ′′_{n}(x) = 0

and the proof of the convergence (17) is complete.

Tightness. To prove the tightness of V_{n}^{−1}ζ_{n}, we use Proposition 3. As Condition i) follows from the convergence of finite dimensional distributions, we only have to check Condition ii). Set S′(0) = 0, S′(m) = ∑_{k≤m} ε_{k}X_{k}, m = 1, …, n. Let the process Z_{n} be defined linearly between the points (τ(k), V_{n}^{−1}S′(k)), k = 0, 1, …, n. Because V_{n}^{−1}ζ_{n} has clearly the same distribution as the random process Z_{n}, it suffices to prove that for any ϵ > 0,

lim_{J→∞} sup_{n} ∑_{j>J} 2^{j} max_{0≤k<2^{j}} P(|Z_{n}((k + 1)2^{−j}) − Z_{n}(k2^{−j})| > ϵρ(2^{−j})) = 0. (23)

This lead us to estimate uniformly in n the probabilities

δ(t, h, u) := P(|Z_{n}(t + h) − Z_{n}(t)| > u), u > 0.

In the case, where 0 ≤ τ(k − 1) ≤ t < t + h ≤ τ(k), we have

0 ≤ h ≤ τ(k) − τ(k − 1) = \frac{X_{k}^{2}}{V_{n}^{2}}

and linear interpolation gives

|Z_{n}(t + h) − Z_{n}(t)| = \frac{|ε_{k}X_{k}|}{V_{n}} \frac{V_{n}^{2}}{X_{k}^{2}} h = (\frac{V_{n}}{|X_{k}|}\sqrt{h})\sqrt{h} ≤ \sqrt{h}.

Next consider the following configuration:

0 ≤ τ(k − 1) ≤ t < τ(k) ≤ τ(l) ≤ t + h < τ(l + 1).

Here we have |Z_{n}(t + h) − Z_{n}(t)| ≤ δ_{1} + δ_{2} + δ_{3}, where

δ_{1} := |Z_{n}(t + h) − Z_{n}(τ(l))| ≤ \sqrt{t + h − τ(l)} ≤ \sqrt{h},
δ_{2} := |Z_{n}(τ(l)) − Z_{n}(τ(k))| = V_{n}^{−1}|S′(l) − S′(k)| ≤ \frac{|S′(l) − S′(k)|}{\sqrt{V_{l}^{2} − V_{k}^{2}}} \sqrt{h},
δ_{3} := |Z_{n}(τ(k)) − Z_{n}(t)| ≤ \sqrt{τ(k) − t} ≤ \sqrt{h}.

Hence, for any configuration we obtain

|Z_{n}(t + h) − Z_{n}(t)| ≤ \frac{|S′(l) − S′(k)|}{\sqrt{V_{l}^{2} − V_{k}^{2}}} \sqrt{h} + 2\sqrt{h},

if we agree that |S′(l) − S′(k)|(V_{l}^{2} − V_{k}^{2})^{−1/2} := 0 when k = l. Therefore

δ(t, h, u) ≤ P\{|S′(l) − S′(k)|/\sqrt{V_{l}^{2} − V_{k}^{2}} > u/(2\sqrt{h})\},

provided u > 4\sqrt{h}. In this formula the random indexes l and k depend on t, h and on the sequence (X_{i}), but are independent of the sequence (ε_{i}). Thus conditioning on X_{1}, …, X_{n} and applying Hoeffding’s inequality leads to

δ(t, h, u) ≤ c exp{−u^{2}/(8h)}.

This estimate enables us to obtain (23), since due to Hypothesis (3) we have for every ϵ > 0,

∑_{j=1}^{∞} 2^{j} exp{−ϵ2^{j}ρ^{2}(2^{−j})} < ∞.

The proof is completed.

4. PROOF OF THEOREM 2 Our proof of Theorem 2 requires some preliminary lemmas. Let us introduce some notations. Set

σ_{i}^{2} := E X_{i}^{2} < ∞, i ≥ 1.
B_{0}^{2} := 0, B_{k}^{2} := σ_{1}^{2} + ⋯ + σ_{k}^{2}, k ≥ 1.  (24)
β(0) := 0, β(k) := \frac{B_{k}^{2}}{B_{n}^{2}} , 1 ≤ k ≤ n.  (25)

Define moreover for ε > 0,

L_{n,ε} := B_{n}^{−(2+ε)} ∑_{k=1}^{n} E |X_{k}|^{2+ε}.  (26)

Lemma 4. Assume that L_{n,ε} → 0 as n → ∞ and δ_{n} → 0 is such that δ_{n}^{−1}L_{n,ε}^{2/(2+ε)} → 0. Then

δ_{n}^{−1} max_{1≤k≤n} |β(k) − τ(k)| ⟶_{n→∞}^{P} 0.  (27)

Proof. Define

X̂_{i} := X_{i}I{X_{i}^{2} ≤ δ_{n}B_{n}^{2}}, i = 1, …, n.

Since

∑_{k=1}^{n} P(X_{k}^{2} ≥ δ_{n}B_{n}^{2}) ≤ δ_{n}^{−(2+ε)/2}L_{n,ε} → 0,

the proof of (27) reduces to the proof of

δ_{n}^{−1} max_{1≤k≤n} |β(k) − τ̂(k)| ⟶_{n→∞}^{P} 0,  (28)

where τ̂(k) := V̂_{n}^{−2}V̂_{k}^{2} and V̂_{k}^{2} := X̂_{1}^{2} + ⋯ + X̂_{k}^{2}, k = 0, 1, …, n. In view of

|β(k) − τ̂(k)| = |\frac{V̂_{k}^{2}}{B_{n}^{2}} − \frac{B_{k}^{2}}{B_{n}^{2}} + \frac{V̂_{k}^{2}}{V̂_{n}^{2}} \{\frac{V̂_{n}^{2}}{B_{n}^{2}} − 1\}| ≤ 2 max_{1≤k≤n}|\frac{V̂_{k}^{2}}{B_{n}^{2}} − \frac{B_{k}^{2}}{B_{n}^{2}} |,

the proof of (28) reduces in establishing the convergence

δ_{n}^{−1} max_{1≤k≤n}|\frac{V̂_{k}}{B_{n}^{2}} − \frac{B_{k}^{2}}{B_{n}^{2}} | ⟶_{n→∞}^{P} 0.  (29)

To center random variables observe that

\frac{B_{k}^{2}}{B_{n}^{2}} − B_{n}^{−2} ∑_{j=1}^{k} E X̂_{j}^{2} = B_{n}^{−2} ∑_{j=1}^{k} E X_{i}^{2}I\{X_{i}^{2} ≥ δ_{n}B_{n}^{2}\} ≤ δ_{n}^{−ε/2}L_{n,ε} = o(1).

Hence it remains to show that

δ_{n}^{−1}B_{n}^{−2} max_{1≤k≤n}|∑_{j=1}^{k}(X̂_{j}^{2} − E X̂_{j}^{2})| ⟶_{n→∞}^{P} 0.  (30)

By Kolmogorov’s inequality, for any η > 0,

P\{ max_{1≤k≤n}|∑_{j=1}^{k}(X̂_{j}^{2} − E X̂_{j}^{2})| ≥ ηδ_{n}B_{n}^{2}\} ≤ \frac{1}{η^{2}δ_{n}^{2}B_{n}^{4}} ∑_{k=1}^{n} E X_{k}^{4}I\{X_{k}^{2} ≤ δ_{n}B_{n}^{2}\}
≤ η^{−2}δ_{n}^{−2}δ_{n}^{1−ε/2}L_{n,ε} = o(1)

and (30) follows. Lemma 5. If the conditions of Lemma 4 are satisfied, then

δ_{n}^{−1}V_{n}^{−1} max_{1≤k≤n} X_{k}^{2} ⟶_{n→∞}^{P} 0.

Proof. First note that

\frac{X_{k}^{2}}{V_{n}^{2}} = \frac{V_{k}^{2}}{V_{n}^{2}} − \frac{V_{k−1}^{2}}{V_{n}^{2}} ≤ |\frac{V_{k}^{2}}{V_{n}^{2}} − \frac{B_{k}^{2}}{B_{n}^{2}} | + |\frac{V_{k−1}^{2}}{V_{n}^{2}} − \frac{B_{k−1}^{2}}{B_{n}^{2}} | + \frac{B_{k}^{2} − B_{k−1}^{2}}{B_{n}^{2}}
≤ 2 max_{1≤i≤k} |β(i) − τ(i)| + \frac{E X_{k}^{2}}{B_{n}^{2}} ,

whence

\frac{1}{δ_{n}} max_{1≤k≤n} \frac{X_{k}^{2}}{V_{n}^{2}} ≤ \frac{2}{δ_{n}} max_{1≤k≤n} |β(k) − τ(k)| + \frac{1}{δ_{n}B_{n}^{2}} max_{1≤k≤n} E X_{k}^{2}. (31)

Next

max_{1≤k≤n} \frac{E X_{k}^{2}}{B_{n}^{2}} ≤ max_{1≤k≤n} (\frac{E |X_{k}|^{2+ε}}{B_{n}^{2+ε}} )^{2/(2+ε)}
≤ (∑_{k=1}^{n} \frac{E |X_{k}|^{2+ε}}{B_{n}^{2+ε}} )^{2/(2+ε)} = L_{n,ε}^{2/(2+ε)}.  (32)

Recalling that the sequence (δ_{n}) is such that δ_{n}^{−1}L_{n,ε}^{2/(2+ε)} → 0, the conclusion follows from (31), Lemma 4 and (32).

Lemma 6. Let X_{1}, …, X_{n}, … be independent random variables with mean zero and finite variances σ_{1}^{2}, …, σ_{n}^{2}, … respectively. Suppose that sup_{i≥1} σ_{i}^{2} =: σ^{2} < ∞ and that for some T > 0,

κ(T) := sup_{i≥1} P(X_{i}^{2} ≤ T) < 1.  (33)

Let d > 0 be such that

θ(T) := κ(T) + e^{−dT} < 1.  (34)

Then we have the estimates

a) |E X_{i}e^{−λX_{i}^{2}} | ≤ λ^{1/2}σ_{i}^{2}.

b) For p ≥ 2, E |X_{i}^{p}e^{−λX_{i}^{2}} | ≤ a(p)λ^{1−p/2}σ_{i}^{2}, with a(p) = 2(p/2 − 1)^{p/2}e^{1−p/2} for p ≥ 3 and a(2) = 1.

c) If λ ≥ d, then E e^{−λX_{i}^{2}} ≤ θ(T).

d) If 0 < λ < d, then E e^{−λX_{i}^{2}} ≤ 1 − λd^{−1}(1 − θ(T)).

Proof. As sup_{a>0} a^{−1/2}(1 − e^{−a}) ≤ 1 and E X_{i} = 0, we get a) writing

|E X_{i}e^{−λX_{i}^{2}} | = |E X_{i}(e^{−λX_{i}^{2}} − 1)| ≤ λ^{1/2}σ_{i}^{2}.

The proof of b) is obvious if p = 2. For p ≥ 3, we use the bound

sup_{x>0} x^{s} exp(−λx^{2}) = \frac{s^{s/2}}{(2λ)^{s/2}} exp(−s/2), (s ≥ 1),

which gives directly the result with s = p − 2. To check c), just note that if λ ≥ d,

E e^{−λX_{i}^{2}} = E e^{−λX_{i}^{2}}I{X_{i}^{2} ≥ T } + P(X_{i}^{2} ≤ T) ≤ κ(T) + e^{−dT} = θ_{T} .

To prove d), integration by parts gives

E e^{−λX_{i}^{2}} = 1 − λ ∫_{0}^{∞} e^{−λt}P(X_{i}^{2} ≥ t) dt
≤ 1 − λ ∫_{0}^{∞} e^{−dt}P(X_{i}^{2} ≥ t) dt
= 1 − \frac{λ}{d} ∫_{0}^{∞} de^{−dt}P(X_{i}^{2} ≥ t) dt
= 1 − \frac{λ}{d} + \frac{λ}{d}E e^{−dX_{i}^{2}}.

Then d) follows applying c) with λ = d to estimate E e^{−dX_{i}^{2}}.

Lemma 7. Put T_{n} := V_{n}^{−1}S_{n}. Under the assumptions of Lemma 6, we have for every q ≥ 2

C(q) := sup_{n} E |T_{n}|^{q} < ∞.

Proof. Let q ≥ 2 be an integer. The basic idea, which we borrow from [3] is to use the following representation

∫_{0}^{∞} λ^{q/2−1}e^{−a^{2}λ}dλ = \frac{Γ(q/2)}{a^{q}},

which gives

E |T_{n}|^{q} = \frac{1}{Γ(q/2)}E |S_{n}|^{q} ∫_{0}^{∞} e^{−λV_{n}^{2}}λ^{q/2−1} dλ = \frac{1}{Γ(q/2)}(J_{1}(d) + J_{2}(d)),

where for d > 0

J_{1}(d) := E |S_{n}|^{q} ∫_{d}^{∞} e^{−λV_{n}^{2}}λ^{q/2−1} dλ;
J_{2}(d) := E |S_{n}|^{q} ∫_{0}^{d} e^{−λV_{n}^{2}}λ^{q/2−1} dλ.

To evaluate J_{1}(d) and J_{2}(d), we use the estimates of Lemma 6 with d > 0 choosen as in (34).

Estimate for J_{1}(d). By Cauchy-Schwarz inequality, |S_{n}| ≤ n^{1/2}V_{n}, whence

J_{1}(d) ≤ n^{q/2}E ∫_{d}^{∞} V_{n}^{q}e^{−λV_{n}^{2}}λ^{q/2−1} dλ
= n^{q/2}E ∫_{dV_{n}^{2}}^{∞} e^{−u}u^{q/2−1} du

≤ n^{q/2}E exp(\frac{−dV_{n}^{2}}{2} ) ∫_{dV_{n}^{2}}^{∞} e^{−u/2}u^{q/2−1} du

≤ 2^{q/2}Γ(q/2)n^{q/2}E exp(\frac{−dV_{n}^{2}}{2} ).  (35)

Now using Lemma 6 d) with λ = d/2, we see that

E exp(\frac{−dX_{k}^{2}}{2} ) ≤ 1 − \frac{1}{2}(1 − θ(T)) = \frac{1 + θ(T)}{2} < 1.

By independence of the X_{k}’s and (35) it follows that

J_{1}(d) ≤ 2^{q/2}Γ(q/2)n^{q/2}(\frac{1 + θ(T)}{2})^{n} = o(1),

hence for fixed q,

sup_{n≥1} J_{1}(d) < ∞.  (36)

Estimate for J_{2}(d). From the multinomial formula we get

|S_{n}|^{q} = ∑_{r=1}^{q} ∑′ \binom{q}{m_{1}, …, m_{r}} ∑_{1≤i_{1}<···<i_{r}≤n} X_{i_{1}}^{m_{1}} ⋯ X_{i_{r}}^{m_{r}},

where ∑′ extends over m_{1}, …, m_{r} ≥ 1 such that m_{1} + ⋯ + m_{r} = q. So, we have to estimate the terms

I_{i_{1},...,i_{r}}(m_{1}, …, m_{r}) = E {X_{i_{1}}^{m_{1}} ⋯ X_{i_{r}}^{m_{r}}∫_{0}^{d} e^{−λV_{n}^{2}}λ^{q/2−1} dλ} = ∫_{0}^{d}Aλ^{q/2−1} dλ,

where by independence

A := E {X_{i_{1}}^{m_{1}} ⋯ X_{i_{r}}^{m_{r}}e^{−λV_{n}^{2}}}
= ∏_{m_{j}=1} E {X_{i_{j}}e^{−λX_{i_{j}}^{2}} } ∏_{m_{j}≥2} E {X_{i_{j}}^{m_{j}}e^{−λX_{i_{j}}^{2}} } ∏_{k∉{i_{1},...,i_{r}}} E e^{−λX_{k}^{2}}.

Denoting by ν the number of factors in the first product, we have respectively r − ν and n − r factors in the second and third products. Using a), b) and d) in Lemma 6, we obtain

|A| ≤ σ^{2ν}λ^{ν/2}\{ ∏_{m_{j}≥2} a(m_{j})\}σ^{2(r−ν)}λ^{μ}(1 − \frac{λ}{d}(1 − θ(T)))^{n−r},

with

μ = ∑_{m_{j}≥2}(1 − \frac{m_{j}}{2} ) = r − ν − \frac{1}{2} ∑_{m_{j}≥2} m_{j} = r − \frac{q}{2} − \frac{ν}{2} .

Denoting by K(q) a constant dominating all possible products ∏_{m_{j}≥2} a(m_{j}), we have now

|A| ≤ K(q)σ^{2r}λ^{r−q/2}(1 − c_{0}λ)^{n−r},

where c_{0} := d^{−1}(1 − θ(T)). Therefore

I_{i_{1},...,i_{r}}(m_{1}, …, m_{r}) ≤ K(q)σ^{2r} ∫_{0}^{d} λ^{r−1}(1 − c_{0}λ)^{n−r} dλ.

This last integral may be evaluated by comparison with the Beta function B(x, y) = ∫_{0}^{1} t^{x−1}(1 − t)^{y−1} dt and using the well-known identity B(x, y) = Γ(x)Γ(y)/Γ(x + y). This leads to

∫_{0}^{d} λ^{r−1}(1 − c_{0}λ)^{n−r} dλ = \frac{1}{c_{0}^{r}} ∫_{0}^{1−θ(T)} u^{r−1}(1 − u)^{n−r} du
≤ \frac{1}{c_{0}^{r}} ∫_{0}^{1} u^{r−1}(1 − u)^{n−r} du
= \frac{1}{c_{0}^{r}rC_{n}^{r}} .

Going back to J_{2}(d) we get then

J_{2}(d) ≤ ∑_{r=1}^{q} ∑′ \binom{q}{m_{1}, …, m_{r}} ∑_{1≤i_{1}<···<i_{r}≤n} \frac{K(q)σ^{2r}}{rc_{0}^{r}} \frac{1}{C_{n}^{r}} .

As the number of terms in ∑_{1≤i_{1}<···<i_{r}≤n} is C_{n}^{r}, we finally obtain

J_{2}(d) ≤ K(q) ∑_{r=1}^{q} \frac{σ^{2r}}{rc_{0}^{r}} ∑′ \binom{q}{m_{1}, …, m_{r}}

and this bound is clearly a constant depending only on q. The proof is completed.

Proof of Theorem 2. Write p := 2+ε, M := sup_{k≥1} E |X_{k}|^{p}. Because E X_{k}^{2} ≥

T^{2}P(|X_{k}| > T) ≥ T^{2}(1−κ(T)), we have B_{n}^{2+ε} ≥ n^{1+ε/2}T^{2+ε}(1−κ(T))^{1+ε/2}. This gives for L_{n,ε} defined by (26)

L_{n,ε} ≤ {MT^{−2−ε}(1 − κ(T))^{−1−ε/2}}n^{−ε/2}.  (37)

Hence the conditions of Lemmas 4 to 7 are satisfied and we can choose δ_{n} = n^{−γ} with 0 < γ < ε/(2 + ε). First we prove that

V_{n}^{−1}ζ_{n} ⟶_{n→∞}^{fdd} W.  (38)

Consider the process (ξ_{n}(t), t ∈ [0, 1]) defined by linear interpolation between the points (β^{2}(k), S(k)), k = 0, …, n, with β defined by (25). From (37), Lindeberg’s condition is satisfied, so by Prokhorov [5] theorem, B_{n}^{−1}ξ_{n} converges in distribution to W in C[0, 1]. Hence

B_{n}^{−1}ξ_{n} ⟶_{n→∞}^{fdd} W.  (39)

Consider the following change of time θ_{n}. When V (n) > 0, θ_{n} is the map from [0, 1] onto [0, 1] which interpolates linearly between the points (β^{2}(k), τ^{2}(k)), k = 0, 1, …, n. When V_{n} = 0, we simply take θ_{n} = I, the identity on [0, 1]. With the usual convention S_{k}/V_{n} := 0 for V_{n} = 0, we always have

V_{n}^{−1}ζ_{n}(θ_{n}(t)) = V_{n}^{−1}ξ_{n}(t), 0 ≤ t ≤ 1.  (40)

Since Lindeberg’s condition is satisfied, one easily see that B_{n}^{−2}V_{n}^{2} ⟶_{n→∞}^{P} 1.

Now the proof of (38) due to (39) reduces in proving that

sup_{0≤t≤1} |θ_{n}(t) − t| ⟶_{n→∞}^{P} 0.  (41)

But this is a corollary of Lemma 4. To prove tightness we need to check (10) only. To this aim we have to estimate for δ > 0,

P (n, δ) := P{sup_{j≥J} 2^{αj} max_{t∈D_{j}} V_{n}^{−1}|ζ_{n}(t) − ζ_{n}(t^{−})| ≥ δ}.  (42)

Recall that for t ∈ D_{j}, t^{+} := t + 2^{−j} and t^{−} := t − 2^{−j}. Of course the full proof of tightness would require to estimate in the same way the probabilities obtained substituting t^{−} by t^{+} in (42). This follows by an easy adaptation of the arguments presented below. Define for t ∈ D

b(t) := max{k ≥ 0 : β(k) ≤ t}.  (43)

From (13) and this definition, we have for each t ∈ (0, 1),

τ(u(t)) ≤ t < τ(u(t) + 1),  β(b(t)) ≤ t < β(b(t) + 1).  (44)

Let us remark that τ(u(t)) < τ(1 + u(t)) implies that X_{1+u(t)} ≠ 0. We shall provide some estimates for |ζ_{n}(t) − ζ_{n}(t^{−})|.

Case 1: τ(u(t)) ≤ t^{−} < t < τ(1 + u(t)). Then τ(1 + u(t)) − τ(u(t)) > 2^{−j}, so |X_{1+u(t)}| > 2^{−j/2}V_{n} and writing

|ζ_{n}(t) − ζ_{n}(t^{−})| = \frac{t − t^{−}}{τ(1 + u(t)) − τ(u(t))}|X_{1+u(t)}| = 2^{−j} \frac{V_{n}}{|X_{1+u(t)}|}V_{n},

we get

|ζ_{n}(t) − ζ_{n}(t^{−})| ≤ V_{n}2^{−j/2}.  (45)

Case 2: t^{−} < τ(u(t)) ≤ t < τ(1 + u(t)). In this case τ(u(t^{−})) < τ(u(t)) hence τ(1 + u(t^{−})) ≤ τ(u(t)), giving the following configuration

t^{−} < τ(1 + u(t^{−})) ≤ τ(u(t)) ≤ t < τ(1 + u(t)).

Introduce

S(t^{−}, t) := ∑_{1+u(t^{−})<k≤u(t)} X_{k}

and note that S(t^{−}, t) = 0 when τ(1 + u(t^{−})) ≥ τ(u(t)). To compare |ζ_{n}(t) − ζ_{n}(t^{−})| with |S(t^{−}, t)|, we need some control of the remainder terms in the triangle inequality

|ζ_{n}(t) − ζ_{n}(t^{−})| ≤ |S(1 + u(t^{−})) − ζ_{n}(t^{−})| + |S(t^{−}, t)| + |ζ_{n}(t) − S(u(t))|.

This leads us to introduce and estimate for s ∈ D_{k}

I_{k}(s) := |ζ_{n}(s) − S(u(s))| = |X_{1+u(s)}|\frac{s − τ(u(s))}{τ(1 + u(s)) − τ(u(s))}, (46)

I′_{k}(s) := |S(1 + u(s)) − ζ_{n}(s)| = |X_{1+u(s)}|\frac{τ(1 + u(s)) − s}{τ(1 + u(s)) − τ(u(s))}.(47)

Estimation of I_{k}(s) and I′_{k}(s). For I_{k}(s), we are interested only in the case where s^{−} < τ(u(s)) and symmetrically for I′_{k}(s) where s^{+} > τ(1 + u(s)). The inequality s^{−} < τ(u(s)) gives s − τ(u(s)) < 2^{−k}, whence

I_{k}(s) < |X_{1+u(s)}|\frac{2^{−k}}{τ(1 + u(s)) − τ(u(s))}.  (48)

If 2^{−k} ≤ τ(1 + u(s)) − τ(u(s)) = X_{1+u(s)}^{2}V_{n}^{−2}, then V_{n} ≤ |X_{1+u(s)}|2^{k/2} and inserting this in (48) we get

I_{k}(s) ≤ |X_{1+u(s)}|\frac{V_{n}^{2}}{X_{1+u(s)}^{2}} 2^{−k} = \frac{V_{n}}{|X_{1+u(s)}|}V_{n}2^{−k} ≤ V_{n}2^{−k/2}.

If 2^{−k} > τ(1 + u(s)) − τ(u(s)), then |X_{1+u(s)}| < V_{n}2^{−k/2} and bounding the fraction in (46) by 1, we obtain directly I_{k}(s) ≤ V_{n}2^{−k/2}. So we have established that

I_{k}(s) ≤ V_{n}2^{−k/2},  s ∈ D_{k}, s^{−} < τ(u(s)).  (49)

Symmetrically, it is now clear that

I′_{k}(s) ≤ V_{n}2^{−k/2},  s ∈ D_{k}, s^{+} > τ(1 + u(s)).  (50)

Going back to the estimation of |ζ_{n}(t) − ζ_{n}(t^{−})| and noting that when t ∈ D_{j}, then t^{−} ∈ D_{j−1}, we can write

|ζ_{n}(t) − ζ_{n}(t^{−})| ≤ I′_{j−1}(t^{−}) + |S(t^{−}, t)| + I_{j}(t).  (51)

Recalling that in the present case t^{−} < τ(u(t)), we have by (49) I_{j}(t) ≤ V_{n}2^{−j/2}. To apply (50) to I′_{j−1}(t^{−}), note that (t^{−})^{+} = t^{−} + 2^{−(j−1)} = t^{+} and that t^{+} > t ≥ τ(u(t)) ≥ τ(1 + u(t^{−})). So with s = t^{−} and k = j − 1 we have s^{+} > τ(1 + u(s)) and I′_{j−1}(t^{−}) ≤ 2^{−(j−1)/2}. Finally

|ζ_{n}(t) − ζ_{n}(t^{−})| ≤ |S(t^{−}, t)| + 2^{2−j/2}V_{n}.  (52)

Hence, for J such that 2^{2+J(α−1/2)} ≤ δ/2, we have P (n, δ) ≤ P_{1}(n, δ) where

P_{1}(n, δ) = P{sup_{j≥J} 2^{αj} max_{t∈D_{j}} V_{n}^{−1}|S(t^{−}, t)| ≥ δ/2}.

Set J_{n} = ν log n, where ν > 0 will be choosed latter and log is the logarithm with basis 2. Then P_{1}(n, δ) ≤ P′_{1}(n, δ) + P′′_{1}(n, δ), where

P′_{1}(n, δ) := P{ max_{J≤j≤J_{n}} 2^{αj} max_{t∈D_{j}} V_{n}^{−1}|S(t^{−}, t)| ≥ δ/4}
P′′_{1}(n, δ) := P{ sup_{j≥J_{n}} 2^{αj} max_{t∈D_{j}} V_{n}^{−1}|S(t^{−}, t)| ≥ δ/4}.

Estimation of P′′_{1}(n, δ). If τ(1 + u(t^{−})) < τ(u(t)), then 0 < τ(u(t)) − τ(1 + u(t^{−})) < t − t^{−} = 2^{−j} and

V_{n}^{−1}|S(t^{−}, t)| = \frac{|V_{u(t)}^{2} − V_{1+u(t^{−})}^{2}|^{1/2}}{V_{n}} \frac{|S(t^{−}, t)|}{|V_{u(t)}^{2} − V_{1+u(t^{−})}^{2}|^{1/2}}
≤ |τ(u(t)) − τ(1 + u(t^{−}))|^{1/2} max_{1≤k<l≤n} T_{k,l},
≤ 2^{−j/2} max_{1≤k<l≤n} T_{k,l},  (53)

where

T_{k,l} := \begin{cases}|S(l) − S(k)|(V_{k}^{2} − V_{l}^{2})^{−1/2} & if V_{k}^{2} < V_{l}^{2}, \\ 0 & if V_{k}^{2} = V_{l}^{2}. \end{cases}

We note that if τ(1 + u(t^{−})) ≥ τ(u(t)), then S(t^{−}, t) = 0 and the bound (53) remains trivially valid. By Lemma 7 we have for any q ≥ 2,

E | max_{1≤k<l≤n} T_{k,l}|^{q} = E max_{1≤k<l≤n} |T_{k,l}|^{q} ≤ ∑_{1≤k<l≤n} E |T_{k,l}|^{q} ≤ C(q)n^{2}.

Together with (53), this gives

P′′_{1}(n, δ) ≤ P{ sup_{j≥J_{n}} 2^{−j/2αj} max_{1≤k<l≤n} T_{k,l} ≥ δ/4}
= P{ max_{1≤k<l≤n} T_{k,l} ≥ δ2^{J_{n}(1/2−α)−2}}
≤ 4^{q}C(q)δ^{−q}n^{2}2^{−qJ_{n}(1/2−α)} = 4^{q}C(q)δ^{−q}n^{2−q(1/2−α)ν}.

Since the estimate of Lemma 7 holds for any q ≥ 2, we uses the freedom to choose q > 4/(ν − 2αν) to get the convergence

lim_{n→∞} P′′_{1}(n, δ) = 0.  (54)

Estimation of P′_{1}(n, δ). Recall that δ_{n} = n^{−γ} with 0 < γ < ε/(2 + ε). Introduce the event

Ω_{n} := {b(t − δ_{n}) ≤ u(t) ≤ b(t + δ_{n}), ∀t ∈ [0, 1]} ∩ {V_{n}^{−2} max_{1≤i≤n} X_{i}^{2} ≤ δ_{n}}.

Observe that u(t) ≥ b(t + δ_{n}) + 1 yields

V_{n}^{−2} ∑_{j=1}^{b(t+δ_{n})+1} X_{j}^{2} ≤ t = t + δ_{n} − δ_{n} ≤ B_{n}^{−2} ∑_{j=1}^{b(t+δ_{n})+1} σ_{j}^{2} − δ_{n},

which means that τ(b(t + δ_{n}) + 1) ≤ β(b(t + δ_{n}) + 1) − δ_{n}, whence we have on the event {u(t) ≥ b(t + δ_{n}) + 1}

max_{1≤k≤n} |τ(k) − β(k)| ≥ δ_{n}.  (55)

Similarly (55) holds also true on the event {u(t) ≤ b(t − δ_{n}) − 1}. Hence

P(Ω_{n}^{c}) ≤ P{ max_{1≤m≤n} |τ(m) − β(m)| ≥ δ_{n}} + P{V_{n}^{−2} max_{1≤k≤n} X_{k}^{2} ≥ δ_{n}} → 0

and we have to deal only with

P_{2}(n, δ) := P\{ max_{J≤j≤J_{n}} 2^{αj} max_{t∈D_{j}} V_{n}^{−1}|S(t^{−}, t)| ≥ \frac{δ}{4}, Ω_{n}\}.

Let us introduce some abbreviations. For any set A ⊂ {1, …, n}, put

Σ(X, A) := ∑_{k∈A} X_{k} T (A) := \frac{Σ(X, A)}{Σ(X^{2}, A)^{1/2}} ,

with the convention T (A) := 0 when Σ(X^{2}, A) = 0. Define also the following index sets

I_{t} := (b(t^{−} + δ_{n}) + 1, b(t − δ_{n})]
J_{1,t} := [b(t^{−} − δ_{n}), b(t^{−} + δ_{n})]  J_{1,t}(ℓ) := (ℓ + 1, b(t^{−} + δ_{n}) + 1]
J_{2,t} := [b(t − δ_{n}), b(t + δ_{n})]  J_{2,t}(ℓ) := (b(t − δ_{n}), ℓ].

Then we have

P_{2}(n, δ) ≤ P_{3}(n, δ) + P_{4}(n, δ) + P_{5}(n, δ),

where

P_{3}(n, δ) := P\{ max_{J≤j≤J_{n}} 2^{αj} max_{t∈D_{j}} V_{n}^{−1}|Σ(X, I_{t})| ≥ \frac{δ}{12}, Ω_{n}\},
P_{4}(n, δ) := P\{ max_{J≤j≤J_{n}} 2^{αj} max_{t∈D_{j}} max_{ℓ∈J_{1,t}} V_{n}^{−1}|Σ(X, J_{1,t}(ℓ))| ≥ \frac{δ}{12}, Ω_{n}\},
P_{5}(n, δ) := P\{ max_{J≤j≤J_{n}} 2^{αj} max_{t∈D_{j}} max_{ℓ∈J_{2,t}} V_{n}^{−1}|Σ(X, J_{2,t}(ℓ))| ≥ \frac{δ}{12}, Ω_{n}\}.

We estimate first P_{3}(n, δ). On the event Ω_{n} we have b(t^{−} + δ_{n}) ≥ u(t^{−}) and b(t−δ_{n}) ≤ u(t) hence Σ(X^{2}, I_{t}) ≤ V_{u(t)}^{2}−V_{1+u(t^{−})}^{2} and V_{n}^{−2}Σ(X^{2}, I_{t}) ≤ t−t^{−}. It follows that

V_{n}^{−1}|Σ(X, I_{t})| ≤ 2^{−j/2}|T (I_{t})|, on Ω_{n}

and

P_{3}(n, δ) ≤ P\{ max_{J≤j≤J_{n}} 2^{(α−1/2)j} max_{t∈D_{j}}|T (I_{t})| ≥ \frac{δ}{12}\}.

By Lemma 7, E (max_{t∈D_{j}} |T (I_{t})|)^{q} ≤ C(q)2^{j}, so we get

P_{3}(n, δ) ≤ C(q)(\frac{12}{δ})^{q} ∑_{j=J}^{J_{n}} 2^{−jq(1/2−α)}2^{j} ≤ C(q)(\frac{12}{δ})^{q} ∑_{j=J}^{∞} 2^{(1−q(1/2−α))j}.

Now choosing q > (1/2 − α)^{−1} gives

lim_{J→∞} lim sup_{n→∞} P_{3}(n, δ) = 0.  (56)

To estimate P_{4} and P_{5}, we need to control the number of integers in both sets J_{1,t} and J_{2,t}. From the Hypotheses (i) and (ii) of Theorem 2, we have σ_{0}^{2} := inf_{i≥1} σ_{i}^{2} > 0 and σ^{2} := sup_{i≥1} σ_{i}^{2} < ∞. In particular B_{n}^{2} = O(n). The number of integers in J_{2,t} is b(t+δ_{n})−b(t−δ_{n})+1. From the definition of b, we get B_{n}^{−2}(B_{b(t+δ_{n})}^{2}−B_{1+b(t−δ_{n})}^{2}) ≤ 2δ_{n}, so σ_{0}^{2}(b(t+δ_{n})−b(t−δ_{n})−1) ≤ 2δ_{n}B_{n}^{2}. As σ_{0}^{2}n ≤ B_{n}^{2} ≤ σ^{2}n and nδ_{n} goes to infinity, we have a constant c such that

#J_{2,t} ≤ 2 + \frac{2}{σ_{0}^{2}} δ_{n}B_{n}^{2} ≤ cnδ_{n}.  (57)

The same estimate holds true for #J_{1,t}. We have also on the event Ω_{n}

Σ(X^{2}, J_{1,t}(ℓ)) ≤ 5δ_{n}V_{n}^{2}, Σ(X^{2}, J_{2,t}(ℓ)) ≤ 5δ_{n}V_{n}^{2}.

Indeed recalling that on Ω_{n}, V_{n}^{−2} max_{i≤n} X_{i}^{2} ≤ δ_{n},

Σ(X^{2}, J_{1,t}(ℓ)) ≤ Σ(X^{2}, [u(t^{−} − 2δ_{n}), u(t^{−} + 2δ_{n})])
≤ V_{n}^{2}(δ_{n} + \frac{1}{V_{n}^{2}} Σ(X^{2}, [u(t^{−} − 2δ_{n}) + 1, u(t^{−} + 2δ_{n})]))
≤ V_{n}^{2}(δ_{n} + (t^{−} + 2δ_{n}) − (t^{−} − 2δ_{n})) = 5δ_{n}V_{n}^{2}.

Replacing t^{−} by t gives the same holds true with J_{2,t}(ℓ). Now we have on Ω_{n},

V_{n}^{−1}|Σ(X, J_{i,t}(ℓ))| ≤ \sqrt{5}δ_{n}^{1/2}|T (J_{i,t}(ℓ))|, i = 1, 2.  (58)

Let us recall here that δ_{n} = n^{−γ} and 2^{J_{n}} = n^{ν}. By (57), (58) and Lemma 7 we have with K(q) := cC(q)(12\sqrt{5})^{q},

P_{4}(n, δ) ≤ K(q)δ^{−q}nδ_{n}^{1+q/2} ∑_{j=J}^{J_{n}} 2^{j(1+αq)}
≤ K(q)δ^{−q}n^{1−γ(1+q/2)}J_{n}2^{J_{n}(1+αq)}
≤ K(q)νδ^{−q}n^{1−γ(1+q/2)}n^{ν(1+αq)} log n

Choosing 0 < ν < γ/(2α) and q large enough, we obtain

lim_{n→∞} P_{4}(n, δ) = 0.

It is worth noticing here that the convergence (54) was valid with any ν > 0. The same estimate for P_{5}(n, δ) completes the proof.

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