The strong law of large numbers for homogeneous random processes

68 Chapter 4. Strong law of large numbers for random processes

4.4 The strong law of large numbers for homogeneous

4.4. The strong law of large numbers for homogeneous random

processes with independent increments 69

{X_t, t ≥ 0}, functions Yn = sup

n≤t≤n+1

|X_t−Xn|, n ∈ N, are independent identically distributed random variables. We denote by [s]the integer part ofs >0. For any t >1and s∈(0, t]we have

|X_s| ≤ |X_[s]|+Y_[s]≤

[s]

X

k=1

|X_k−Xk−1|+Y_[s]≤2

[t]

X

k=1

Y_k and consequently

E( sup

0≤s≤t

|X_s|)^α≤2^α

[t]

X

k=1

EY_k^α= 2^α[t]EY₁^α.

This implies the second statement of (4.4.1) for α∈(0,1), ifEY₁^α <∞.

Let us prove thatEY₁^α<∞. We first show that all mediansmsof random variable Xsare limited. We assume the opposite: that, e.g., lim

n→∞msn =∞for some sequence s_n∈[0,1], n∈N.We may assume that sequence{s_n}_n≥1 converges to some number s∈[0,1]. By homogeneity, random process{X_t, t≥0} is stochastically continuous.

The distribution functions of random variables Xsn, n ∈ N, thus converge to the distribution function of random variable X_s. By Theorem 1.1.1 in work [28], the following inequalities are valid:

l_s≤lim inf

n→∞ l_s,n ≤lim sup

n→∞

r_s,n ≤r_s,

where l_s, l_s,n, r_s, r_s,n – the minimum and maximum medians of random variables Xs and Xsn. We arrive at a contradiction, because both ls and rs are finite, and, consequently,d= sup

0≤s≤1

|m_s|<∞. By the symmetrization inequality in [30, p. 261]

we have

P( sup

0≤s≤1

|X_s−ms| ≥y)≤4P(|X₁| ≥y).

Using integration by parts, we obtain the inequality E( sup

0≤s≤1

|X_s−ms|)^α ≤4E|X₁|^α. Thus

Y1 = sup

0≤s≤1

|X_s| ≤ sup

0≤s≤1

|X_s−ms|+d, then

EY₁^α≤4E|X₁|^α+d^α<∞.

(ii). We continue to assume that X₀ = 0. Random variable X_n is the sum X_n =

n

P

k=1

(X_k −Xk−1) of independent identically distributed random variables. With assumption lim

n→∞(X_n−cn)/n^1/α = 0 a.s. Hence, due to the Kolmogorov theorem

70 Chapter 4. Strong law of large numbers for random processes

for α= 1 and the Zygmund–Marcinkiewicz theorem for some α ∈(0,2), α6= 1, we getE|X₂−X₁|^α<∞andc=E(X₂−X₁)forα∈[1,2). It remains to be notes that the random variables X1 andX2−X1 are identically distributed and, consequently, E|X₁|^α =E|X₂−X1|^α and c=EX1. The theorem is proved.

Appendix A

Some results from probability and linear algebra

A.1 Probability theory

Theorem A.1.1 (Central Limit Theorem). Let Z1, ..., Zn be independent random variables withEZi = 0and finite third moment, and letσ²=Pn

i=1E|Z_i|². Consider a standard normal variable g. The for every t >0:

P 1 σ

n

X

i=1

Zi ≤t

!

−P(g≤t)

≤Cσ⁻³

n

X

i=1

E|Z_i|³, where C is an absolute constant.

Lemma A.1.2. Let eventE(X, Y) depends on independent random vectorsX and Y then

P(E(X, Y))≤(P(E(X, Y), E(X, Y⁰))^1/2, where Y⁰ is an independent copy of Y.

Proof. See in [12].

Lemma A.1.3. Let Z₁, ..., Z_n be a sequence of random variables and p₁, ..., p_n be non-negative real numbers such that

n

X

i=1

pi= 1, then for every ε >0

P(

n

X

i=1

p_iZ_i≤ε)≤2

n

X

i=1

p_iP(Z_i ≤2ε).

Proof. See in [43].

We recall definition of Levy concentration function

Definition A.1.4. Levy concentration function of random variable Z with values fromR^d is a function

L(Z, ε) = sup

v∈R^d

P(||Z−v||₂< ε).

72 Appendix A. Some results from probability and linear algebra

Lemma A.1.5. Let SJ =P

i∈Jξi, where J ⊂[n], andI ⊂J then L(S_J, ε)≤ L(S_I, ε).

Proof. Let us fix arbitrary v. From independence of ξi we conclude P(|S_J−v| ≤ε)≤EP(|S_I+S_J/I −v| ≤ε|{ξ_i}_i∈I)≤sup

u∈R

P(|S_I−u| ≤ε).

Lemma A.1.6. Let Z be a random variable with EZ² ≥ 1 and with finite fourth moment, and put M₄⁴ := E(Z−EZ)⁴. Then for every ε ∈ (0,1) there exists p = p(M₄, ε) such that

L(Z, ε)≤p.

Proof. See in [37].

Lemma A.1.7. Let X = (X₁, ..., X_n) be a random vector in Rⁿ with independent coordinates X_k.

1. Suppose there exist numbers ε0 ≥0 and L≥0 such that L(X_k, ε)≤Lε for all ε≥ε0 and allk.

Then

L(X, ε)≤(CLε)ⁿ for all ε≥ε₀, where C is an absolute constant.

2. Suppose there exist numbers ε >0 and p∈(0,1)such that L(X_k, ε)≤Lε for all k.

Then there exist numbers ε₁ =ε₁(ε, p)>0 and p₁ =p₁(ε, p)∈(0,1)such that L(X, ε)≤(p₁)ⁿ.

Proof. See [43, Lemma 3.4].

Lemma A.1.8. There existγ >0 andδ >0 such that for alln1 and1≤i≤n, any deterministic vector v ∈ C and any subspace H of Cⁿ with 1 ≤ dim(H) ≤ n−n^1−γ, we have, denoting R := (X₁, ..., X_n) +v,

P(dist(R, H)≤ 1 2

pn−dim(H))≤exp(−n^δ).

Proof. See [41, Statement 5.1].

A.2. Linear algebra and geometry of the unit sphere 73

A.2 Linear algebra and geometry of the unit sphere

Lemma A.2.1. Let 1 ≤ m ≤ n. If A has full rank, with rows R₁, ..., R_m and H= span(Rj, j6=i), then

m

X

i=1

si(A)⁻²=

m

X

i=1

dist(Ri, Hi)⁻². Proof. See [41, Lemma A.4].

Definition A.2.2. (Compressible and incompressible vectors) Let δ, τ ∈ (0,1). A vector x ∈ Rⁿ is called sparse if |supp(x)| ≤ δn. A vector x ∈ Sⁿ⁻¹ is called compressible if x is within Euclidian distance τ from the set of all sparse vectors.

A vector x ∈ Sⁿ⁻¹ is called incompressible if it is not compressible. The sets of sparse, compressible and incompressible vectors will be denoted by Sparse = Sparse (δ), Comp = Comp (δ, τ) and Incomp = Incomp(δ, τ) respectively.

Lemma A.2.3. If x∈Incomp(δ, τ) then at least ¹₂δτ²n coordinatesx_k of x satisfy

√τ

2n ≤ |x_k| ≤ 1

√ δn. Remark A.2.4. We can fix some constant c₀ such that

1

4δτ² ≤c0 ≤ 1 4.

Then for every vector x∈Incomp(δ, τ) |spread(x)|= [2c₀n].

Proof. See in [37].

Appendix B

Methods

B.1 Moment method

In this section we present results which give the method to investigate under what conditions the convergence of moments of all fixed orders implies the weak con-vergence of the sequence of the distribution functions. Let {F_n} be a sequence of distribution functions.

Theorem B.1.1. A sequence of distribution functions {F_n} converges weakly to a limit if the following conditions are satisfied:

1. each Fn has finite moments of all orders.

2. For each fixed integer k≥0 the k-th moment of F_n converges to a finite limit β_k asn→ ∞.

3. If two right-continuous functions F and G have the same moment sequence {β_k}, then F =G+const.

Proof. See [4].

One need to verify condition 3) of the Theorem B.1.1. The following theorem gives condition that implies 3).

Theorem B.1.2 (Carleman). Let{β_k=β_k(F)}be the sequence of moments of the distribution function F. If the Carleman condition

∞

X

k=1

β_2k^−1/2k=∞.

is satisfied, thenF is uniquely determined by the moment sequence {β_k}.

Proof. See [4].

B.2 Stieltjes transform method

Definition B.2.1. The Stieltjes transform of the distribution functionG is a func-tion

S_G(α) = Z

R

dG(x)

x−α, α∈C⁺.

76 Appendix B. Methods

Theorem B.2.2 (Inversion formula). For any continuity points a < b of G, we have

G([a, b]) = lim

ε→0+

1 π

b

Z

a

ImS_G(x+iε)dx, Proof. See [4].

For the ESD of the random matrix n^−1/2Xn one has S^Xn(z) =

Z

R

1

x−zdF^Xn = 1 nTr

1

√nXn−zI −1

.

The following theorem gives the method to investigate convergence of the ESD to some limit.

Theorem B.2.3. Let F^Xn be the ESD of the random matrix n^−1/2X_n and set F^Xn =EF^Xn. Then

1. F^Xn(x) converges almost surely to F(x) in the vague topology if and only if S^Xn(z) converges almost surely toS(z) for every z in the upper half-plane;

2. F^Xn(x) converges in probability to F(x) in the vague topology if and only if S^Xn(z) converges in probability to S(z) for every z in the upper half-plane;

3. F^Xn(x) converges almost surely to F(x) in the vague topology if and only if ES^Xn(z) converges almost surely to S(z) for every z in the upper half-plane.

Proof. See [42].

B.3 Logarithmic potential

Definition B.3.1. The logarithmic potential U_m of measure m(·) is a function Um:C→(−∞,+∞] defined for all z∈C by

U_m(z) =− Z

C

log|z−w|m(dw).

Definition B.3.2. The function f :T →R, where T=C or T =R, is uniformly integrable in probability with respect to the sequence of random measures {m_n}_n≥1 on (T,B(T))if for all ε >0:

t→∞lim lim

n→∞P





 Z

|f|>t

|f(x)|m_n(dx)> ε





= 0.

B.3. Logarithmic potential 77

Lets1(n^−1/2Xn−zI)≥s2(n^−1/2Xn−zI)≥...≥sn(n^−1/2Xn−zI)be the singular values ofn^−1/2X_n−zIand define the empirical spectral measure of singular values by

νn(z, B) = 1

n#{i≥1 :si(n^−1/2Xn−zI)∈B}, B∈ B(R).

We can rewrite the logarithmic potential of µn via the logarithmic moments of measureνn by

Uµn(z) =− Z

C

log|z−w|µ_n(dw) =−1 nlog

det 1

√nXn−zI

=− 1

2nlog det 1

√nXn−zI ∗

√1

nXn−zI

=−

∞

Z

0

logxνn(dx).

This allows us to consider the Hermitian matrix (n^−1/2Xn−zI)^∗(n^−1/2Xn−zI) instead of the asymmetric matrix n^−1/2X.

Lemma B.3.3. Let(Xn)n≥1 be a sequence ofn×nrandom matrices. Suppose that for a.a. z∈C there exists a probability measure ν_z on [0,∞) such that

a) ν_n−−−→^weak ν_z as n→ ∞ in probability

b) log is uniformly integrable in probability with respect to {ν_n}_n≥1. Then there exists a probability measureµ such that

a) µn

−−−→weak µ asn→ ∞ in probability b) for a.a. z∈C

Uµ(z) =−

∞

Z

0

logxνz(dx).

Proof. See [6, Lemma 4.3] for the proof.

Appendix C

Stochastic processes

In this chapter we introduce all necessary definitions and theorems from the theory of stochastic processes. See [18] for the discussion of stochastic processes.

C.1 Some facts from stochastic processes

Definition C.1.1. Function of two variables X(t, ω) = ξ(t), defined for all t ∈ T, ω ∈Ω, taking values in a metric space X, F-measurable for all t ∈T, is called stochastic process. The set T is a domain of stochastic process and the spaceX is a codomain of stochastic process.

Definition C.1.2. Stochastic processes X1(t, ω) and X2(t,Ω) determined on the common probability space are stochastically equivalent if for all t∈T

P(X₁(t, ω)6=X₂(t, ω)) = 0.

Let us consider a sequence of random variablesX_i, then it is well known thatsup_iX_i is a random variable. It follows immediately from

{ω ∈Ω : sup

i

Xi > x}=

∞

[

i=1

{ω ∈Ω :Xi > x} ∈ F.

Now let us consider stochastic processX_t. In this case it may occur thatsup_tX_t is not a random variable. To overcome this difficulty we should introduce the definition of a separable process.

Definition C.1.3. Stochastic process is called separable it there exist the countable and dense set of points {t_j}_j≥1 ⊂T and the set N ⊂Ω, P(N) = 0, such that for all open G⊂T and all closed set F ∈X the sets

{ω:Xtj(ω)∈F, tj ∈G}, {ω:Xt(ω)∈F, t∈G}

differ only on subsets of N.

Separability is not very strict condition. Under rather general assumptions on T and X there exists a separable process which is equivalent to a given one.

Theorem C.1.4. Let X be a separable locally compact space and T – arbitrary separable space. For all Xt(ω) defined on T, taking values in X, there exists a stochastically equivalent copyX˜_t(ω)taking values inX˜ which is a compact extension of X.

80 Appendix C. Stochastic processes

Proof. See [18].

In this thesis we consider the case T = R+ and X = R. Then the statement of Theorem C.1.4 is automatically satisfied.

Definition C.1.5. Stochastic process X_t is called process with independent incre-ments if for all t0 < t1 < ... < tk from T the random variables X(t0), X(t1)− X(t0), ..., X(t_k)−X(tk−1) are independent.

Definition C.1.6. Stochastic processXt is called homogenous if Law(X_t+s−X_s) = Law(X_t−X₀), s, t∈T.

Definition C.1.7. Stochastic process X_t, t ∈ T defined on the stochastic basis (Ω,F,(F_t)t∈T,P) is called martingale, if Xt is F_t-measurable, EXt < ∞, t ∈ T, and

E(X_t|F_s) =X_s, s≤t, s, t∈T.

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