Limit theorems for a random walk with memory perturbed by a dynamical system
Cristian F. Coletti, Lucas R. de Lima, Renato J. Gava, Denis A. Luiz
aa r X i v : . [ m a t h . P R ] M a y Limit theorems for a random walk with memoryperturbed by a dynamical system
Cristian F. Coletti ∗ Lucas R. de Lima ∗ Renato J. Gava † Denis A. Luiz ∗ Abstract
We introduce a new random walk with unbounded memory obtainedas a mixture of the Elephant Random Walk and the Dynamic RandomWalk which we call the Dynamic Elephant Random Walk (DERW). Asa consequence of this mixture the distribution of the increments of theresulting random process is time dependent. We prove a strong law oflarge numbers for the DERW and, in a particular case, we provide anexplicit expression for its speed. Finally, we give sufficient conditions forthe central limit theorem and the law of the iterated logarithm to hold.
In this work we introduce a new random walk with memory. The model isobtained as a mixture of two well known random walks, namely the ElephantRandom Walk and the Dynamic Random Walk and it is inspired from the theoryof Markov switching models where switching among regimes occurs randomlyaccording to a Markov process, see Hamilton [18]. The mixture of models hasalready been considered in the literature of interacting particle systems. Forinstance the mixture of spin-flip dynamics and symmetric exclusion processesare known under the name of diffusion-reaction processes, see Belitsky et al. [4]and references there in, and have been intensively studied.Recently, the Elephant Random Walk (ERW) and other random walks withmemory have received considerable attention by many authors, see [2, 3, 5–10,
Key words and phrases.
Elephant Random Walk, Dynamical System, Strong Law of LargeNumbers, Central Limit Theorem. ∗ Centro de Matem´atica, Computa¸c˜ao e Cogni¸c˜ao, Universidade Federal do ABC, Av. dosEstados, 5001, 09210-580 Santo Andr´e, S˜ao Paulo, Brazil ( [email protected] , [email protected] , [email protected] ). † Departamento de Estat´ıstica, Universidade Federal de S˜ao Carlos, Rod. Washington Luiz,km 235, S˜ao Carlos, S˜ao Paulo, 13565-905, Brazil ( [email protected] ) Funding:
C.F.C. thanks FAPESP (grant
14, 15, 21, 22, 25]. The ERW was introduced by Sch¨utz and Trimper [23] and itsdynamic is defined as follows. The elephant starts at the origin and moves onestep to the right with probability q and one step to the left with probability1 − q . At time n + 1, it chooses at random and with equal probability a number n ′ from the set { , . . . , n } . Then the walker takes, with probability p , one stepin the same direction of the step given at time n ′ and, with probability 1 − p , ittakes one step in the opposite direction. See section 2 for a formal definition ofthe ERW.The Dynamic Random Walk (DRW) is a non-homogeneous Markov chainwhich was introduced by Guillotin-Plantard [16] and whose transition probabil-ity of each step is time dependent. In this paper we consider DRW on Z whichevolves in the following manner: at each time a walker takes one step to theright or one step to the left with probability given by a function of the orbit ofa given discrete-time dynamical system. See section 2 for a formal definition ofthe DRW.This paper is organized as follows. In section 2 we introduce the DynamicElephant Random Walk. In section 3 we state our main results and we exhibitsome examples where these results hold. In section 4 we prove the strong law oflarge numbers, the central limit theorem and the law of the iterated logarithmfor the Dynamic Elephant Random Walk. We begin this section by defining the Elephant Random Walk (ERW) ( V n ) n ≥ with V = 0 and increments ( W n ) n ≥ . Let p, q ∈ [0 , P E [ W = 1] = q, P E [ W = −
1] = 1 − q, and P E [ W n = η | W , . . . , W n − ] = 12 n n − X k =1 (1 − (2 p − W k η ) . The random variables ( V n ) n ≥ and ( W n ) n ≥ are related by the formula V n = P ni =1 W i where n ≥
1. Denote by p En ( . ) the probability mass function governingthe law of the increments of the ERW.Then we define the Dynamic Random Walk (DRW). Let ( X , A , µ, T ) be adynamical system where ( X , A , µ ) is a probability space and T : X → X is a µ -invariant transformation, i.e. , µ ( A ) = µ (cid:0) T ( A ) (cid:1) for all A ∈ A . The DRW isthe stochastic process ( Y n ) n ≥ with Y = 0 and increments ( Z n ) n ≥ defined by P Dx [ Z n = 1] = f ( T n x ) and P Dx [ Z n = −
1] = 1 − f ( T n x )where f : X → [0 ,
1] is A -measurable. Here, T = id and, for n ≥ , T n := T ◦ T n − . The random variables ( Y n ) n ≥ and ( Z i ) i ≥ are related by the formula2 n = P ni =1 Z i where n ≥
1. Denote by p Dx,n ( . ) the probability mass functiongoverning the law of the increments of the DRWNow we introduce the Dynamic Elephant Random Walk (DERW) on Z .Let ( X , A , µ, T ) and f : X → [0 ,
1] be as in the definition of the DRW. Fix g : R ⊂ R × N → [0 ,
1] and p, q ∈ [0 , S n ) n ≥ with S = 0 and increments ( X n ) n ≥ defined by P x [ X n = η ] = g ( α, n ) p En ( η ) + (cid:0) − g ( α, n ) (cid:1) p Dx,n ( η ) . Here S n = P ni =1 X i . We will denote by E x the operator expectation inducedby P x . Let ( S n ) n ≥ be the DERW with increments ( X n ) n ≥ . Set F n := σ h X , . . . , X n i with F = {∅ , X } . We will exclude the case where g ( α,
2) = 1 and p = 0without further notice, which is technically inconvenient for our results. Toavoid cumbersome notation, we will write for short α n := g ( α, n ) for a given α ∈ R . Set ℓ inf ( α ) := lim inf n → + ∞ g ( α, n ) , ℓ sup ( α ) := lim sup n → + ∞ g ( α, n ) , and ℓ ( α ) := lim n → + ∞ g ( α, n ) , when it exists.It is convenient to consider the DERW satisfying the following condition: p · ℓ sup ( α ) < . (T) Theorem 3.1 (Strong Law of Large Numbers) . If the DERW satisfies condition (T) , then S n − E x [ S n ] n → P x − a.s. for every x ∈ X . It is worth remarking that if p = 1 / ℓ ( α ) exists then we maycompute the speed of the DERW. Corollary 3.2.
Let ( S n ) n ≥ be the DERW. If p = 1 / and ℓ ( α ) > then, for µ -almost every x ∈ X , lim n →∞ S n n = (1 − ℓ ( α ))(2 E µ [ f |I ]( x ) − P x − a.s.. where I stands for the σ -algebra of T -invariant sets. Corollary 3.2 provides sufficient conditions for the existence of lim n → + ∞ S n n . Inthe following example we provide a method to compute explicitly the speed ofthe DERW under some mild conditions.3 xample 1. Consider a DERW satisfying condition (T). In view of Birkhoff’sErgodic Theorem, let
E ⊂ X be a set of full measure such that for every x ∈ E ,lim n → + ∞ n n − X k =0 f ( T n x ) = E µ [ f |I ]( x ) . (3.1)Assume that, for a given x ∈ E , ℓ ( α ), lim n → + ∞ f ( T n x ) and lim n → + ∞ S n n ( P x − a.s. )do exist. From Ces`aro Mean Convergence Theorem we have thatlim n →∞ n n X k =1 f ( T n x ) = lim n →∞ f ( T n x ) . Denote by S x the P x -a.s. limit of S n /n . Then, invoking Theorem 3.1 we getlim n → + ∞ E x [ S n ] n = S x P x − a.s. It is easy to verify thatlim n →∞ E x [ X n ] = (2 p − ℓ ( α ) S x + (1 − ℓ ( α ))(2 E µ [ f |I ]( x ) − , see equation (4.1). Invoking again Ces`aro Mean Convergence Theorem we getlim n →∞ E x [ S n ] n = (2 p − ℓ ( α ) S x + (1 − ℓ ( α ))(2 E µ [ f |I ]( x ) − . Therefore S x = 1 − ℓ ( α )1 − (2 p − ℓ ( α ) (2 E µ [ f |I ]( x ) − p = 1 / a n := n − Y j =1 (cid:18) p − g ( α, j + 1) j (cid:19) , and set ( A n ) n ≥ to be the sequence of non-negative constants defined by A n := n X k =1 a k (1 − E x [ X k ] ) . We state below a version of the Central Limit Theorem for the DERWfollowed by some variations of the same result.
Theorem 3.3 (Central Limit Theorem) . Let ( S n ) n ≥ be the DERW with p ≤ / or ℓ sup ( α ) < p − . For all x ∈ X , one has that if p < and ℓ inf ( α ) > ,then S n − E x [ S n ] a n A n D −→ N (0 , . orollary 3.4. Let the hypothesis of p < and ℓ inf ( α ) > in Theorem 3.3 bereplaced by condition (T) jointly with(D ) If p = 1 , then lim inf n → + ∞ f ( T n ( x )) > and lim sup n → + ∞ f ( T n ( x )) < − ℓ sup ( α )1 − ℓ inf ( α ) ; and(D ) If ℓ inf ( α ) = 0 , then lim inf n → + ∞ f ( T n ( x )) > , lim sup n → + ∞ f ( T n ( x )) < − p · ℓ sup ( α ) ,and ℓ sup ( α ) < .Then the conclusion of Theorem 3.3 remains true. Corollary 3.5.
Let ( S n ) n ≥ be the DERW. If p = 1 / and ℓ ( α ) > , then, for µ -almost surely x ∈ X , S n − n (1 − ℓ ( α ))(2 E µ [ f |I ]( x ) − q n (cid:0) − (1 − ℓ ( α )) (4 E µ [ f |I ]( x ) − E µ [ f |I ]( x ) + 1) (cid:1) D −→ N (0 , . Theorem 3.6 (Law of Iterated Logarithm) . Under the conditions of Theorem3.3 or Corollary 3.4, lim sup n →∞ | S n − E x [ S n ] | a n A n p log log( A n ) = √ P x − a.s. (3.2)We also obtain an almost sure convergence result for the DERW in theregime where the central limit theorem does not hold. Theorem 3.7.
Let ( S n ) n ≥ be the DERW with p > / and ℓ inf ( α ) > p − .Then, for all x ∈ X , one has that S n − E x [ S n ] a n → M P x − a.s., where M is a non-degenerated zero mean random variable. We finish this section with an example of a DERW where the central limittheorem holds.
Example 2.
Let X = { ( x, y ) ∈ R : x + y ≤ } be endowed with the Lebesgue σ -algebra A . Denote by µ the uniform probability measure defined on A | S where S := { ( x, y ) ∈ R : x + y = 1 } . Namely, µ ( I ( t )) = t π ∧ t ≥ I ( t ) := { (cos β, sin β ) : β ∈ [0 , t ) } .Consider the following system of ordinary differential equations on X (cid:26) dxdt = − y + bx ( x + y − dydt = x + by ( x + y − . (3.3)Assume that b < x, y ) = (0 , φ t ( x, y ) := φ ( t, x, y )the solutions of (3.3) satisfying φ = Id . It is easy to verify, by passing to polarcoordinates if necessary, that φ t : X → X is µ -invariant for any t ≥ x -1-0.8-0.6-0.4-0.200.20.40.60.81 y Figure 1: Phase portrait of φ t ( x, y ) for b = − m, n ≥ φ n ◦ φ m = φ n + m (see [19, p. 175]). Set T := φ . In polar coordinates we have that T ( r, θ ) = e b q e b + r − , θ + 1 ! , if 0 < r < , θ + 1) , if r = 1 . Let f : X → [0 ,
1] be given in polar coordinates by f ( r, θ ) = cos θ . It followsfrom Birkhoff’s Ergodic Theorem that E µ [ f |I ] = lim n →∞ n n X k =1 cos ( θ + k ) ! S = 12 S µ − a.s. where the σ -algebra of all T -invariant sets I coincides with A . If g : [0 , × N → [0 ,
1] is given by g ( α, n ) = α − n +1 αn + 1 , then ℓ ( α ) = α . Assuming that p = 1 / n →∞ S n n = (cid:16) − α (cid:17) ( S − P ( x,y ) − a.s. for every ( x, y ) ∈ X . A straightforward computation yields E µ [ f |I ] = 38 S µ − a.s. S n − n (1 − α )( S − q n (cid:0) − α ) (1 − S ) (cid:1) D −→ N (0 ,
1) (3.4)for all ( x, y ) ∈ E , where E is as in (3.1). Hence, if ( x, y ) ∈ E \ S , then S n + n (1 − α ) √ n D −→ N (cid:18) , (cid:16) − α (cid:17) (cid:19) , Moreover, if ( x, y ) ∈ E ∩ S , then S n √ n D −→ N (cid:18) , (cid:16) − α (cid:17) (cid:19) . The following equation can be obtained by straightforward computation. Itgives us an explicit expression for the conditional expectation with respect tothe natural filtration of the increments of the Dynamic Elephant Random Walk. E x [ X n +1 | F n ] = α n +1 (2 p − n S n + (1 − α n +1 )(2 f ( T n +1 x ) −
1) (4.1)Before proving the Strong Law of Large Numbers, we state the lemma below.
Lemma 4.1.
Assume that the DERW satisfies (T) . Then ( n/a n ) n ≥ is non-decreasing and lim n → + ∞ a n n = 0 . Proof.
We begin by observing that a n n = n − Y k =1 k + (2 p − α k +1 ) k + 1 = n − Y k =1 (cid:18) − − (2 p − α k +1 k + 1 (cid:19) . (4.2)Let b n := − (2 p − g ( α,n +1) n +1 ∈ [0 , n → + ∞ a n n = 0 if, and only if, P n b n = ∞ . Since the DERW satisfies condition (T), we have that P n b n = ∞ .This finishes the proof of the second statement.Since (2 p − α k ≤
1, it follows from equation (4.2) that n + 1 a n +1 − na n = n !(1 − (2 p − α n +1 ) Q nk =1 ( k + (2 p − α k +1 ) ≥ , which yields the conclusion of the first statement.7et M n be the random variable defined by M n := S n − E x [ S n ] a n for n ≥ a n >
0. In subsection 4.4 we show that the sequence ( M n ) n is a martingalewith respect to the natural filtration.Let ( Y n ) n be the martingale difference sequence associated to ( M n ) n , i.e. , Y := M and Y n := M n − M n − for n ≥
2. Since Y n = M n − E x [ M n | F n − ] P x − a.s. , it follows from the linearity of the conditional expectation that Y n = X n − E x [ X n | F n − ] a n P x − a.s. (4.3)for all n ≥
1. Since we have by (4.1) that | E x [ X n | F n − ] | ≤ P x − a.s. , onehas, for all n ≥
1, that | Y n | ≤ a n P x − a.s. (4.4)for every x ∈ X .We now turn to the proof of Theorem 3.1 and Corollary 3.2. Proof of Theorem 3.1.
Set W n := a n n Y n . By (4.3), we verify that W n is a F n -measurable random variable such that E x [ W n | F n − ] = a n n E x [ Y n | F n − ] = 0 P x − a.s. Therefore, ( W n ) n is a sequence of bounded martingale differences.Since E x [ W n | F n − ] ≤ n P x − a.s. by (4.4), we have that, for all x ∈ X , ∞ X j =2 E x [ W j | F j − ] ≤ ∞ X j =2 j < ∞ P x − a.s. Thence, it follows from Theorem 2.7 of [17] that n P j =1 a j j Y j = n P j =1 W j converges P x -almost surely as n → + ∞ .We apply Lemma 4.1 and Kronecker’s lemma obtaining thatlim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) S n − E x [ S n ] n (cid:12)(cid:12)(cid:12)(cid:12) = lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12) a n M n n (cid:12)(cid:12)(cid:12)(cid:12) = lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P nj =1 Y j n/a n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =0 P x − a.s. for all x ∈ X To prove Corollary 3.2 we will use the following result on the expectation ofthe random walk.
Proposition 4.2.
Let ( S n ) n ≥ be the DERW. Then E x [ S n ] = a n α (2 q −
1) + n X k =1 (1 − α k )(2 f ( T k x ) − a k ! (4.5)8 roof. Note that S = X satisfies (4.5) for n = 1. Suppose that (4.5) holds fora given n ∈ N . Then E x [ S n +1 ] = E x [ S n ] + E x [ E x [ X n +1 | F n ]]= E x [ S n ] a n +1 a n + (1 − α n +1 )(2 f ( T n +1 x ) − a n +1 (cid:18) E x [ S n ] a n + (1 − α n +1 )(2 f ( T n +1 x ) − a n +1 (cid:19) . The conclusion follows by induction on n . Proof of Corollary 3.2.
Since p = , a n = 1 for all n ≥
1. Then it can becomputed from (4.5) that E x [ S n ] n = 1 n α (2 q −
1) + n X k =1 (1 − α k )(2 f ( T k x ) − ! . Since ℓ ( α ) = lim n →∞ α n exists, we have that1 n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 (1 − α k )(2 f ( T k x ) − − n X k =1 (1 − ℓ ( α ))(2 f ( T k x ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 (2 f ( T k x ) − ℓ ( α ) − α k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n n X k =1 | ℓ ( α ) − α k | → n → + ∞ ) . (4.6)It follows from the Birkhoff Ergodic Theorem and the Strong Law of LargeNumbers (Theorem 3.1) thatlim n →∞ E x [ S n ] n = lim n →∞ n α (2 q −
1) + n X k =1 (1 − ℓ ( α ))(2 f ( T k x ) − ! = (1 − ℓ ( α )) lim n →∞ n n X k =1 (2 f ( T k x ) − − ℓ ( α )) (cid:0) E µ [ f |I ]( x ) − (cid:1) P x − a.s. for µ -almost every x ∈ X . Define the sequence of positive terms ( B n ) n ≥ given by B n := P nk =1 1 a k . Lemma 4.3. If p ≤ / or ℓ sup ( α ) < p − , then B n ր + ∞ . Moreover, if p > / and ℓ inf ( α ) > p − , then lim n →∞ | B n | < + ∞ . roof. Let p ≤ /
4. Then a n ≤ n − Y j =1 (cid:18) / j (cid:19) ∼ n / Γ(3 / , which implies that B n ր + ∞ .Consider now p > / c n := a n . Then it follows from Raabe criterion[20, p. 285] that B n ր + ∞ when lim sup n →∞ n (cid:16) − c n +1 c n (cid:17) <
1, which is satisfiedwith (4 p − ℓ sup ( α ) < . In the same fashion, we verify that lim n →∞ B n < + ∞ if lim inf n →∞ n (cid:16) − c n +1 c n (cid:17) > p − ℓ inf ( α ) > . Proof of Theorem 3.3.
We first examine the asymptotic behaviour of A n . Itfollows from Lemma 4.5 that there exists n ∈ N and c ∈ (0 ,
1) such that, if n > n , then c ( B n − B n ) ≤ A n − A n . We apply Lemma 4.3 to verify that A n ր + ∞ .Note that we are under conditions of Theorem 3.1. Therefore, S n n = E x [ S n ] n +o(1) P x − a.s. for all x ∈ X . Then it follows from (4.1) that E x [ X n | F n − ] = E x [ X n ] + o(1) P x − a.s. We apply it to (4.3) obtaining E x [ Y n | F k − ] = 1 a n (cid:0) − E x [ X n | F n − ] E x [ X n ] + E x [ X n ] + o (1) (cid:1) = 1 a n (cid:0) − E x [ X n ] + o (1) (cid:1) P x − a.s. Thus, P nk =1 E x [ Y k | F k − ] A n = A n A n + B n A n o(1) = 1 + o(1) P x − a.s., (4.7)since lim sup n → + ∞ | B n /A n | ≤ /c .We may arrive to the desired conclusion applying Corollary 3.1 of [17]. Tothis end, it suffices to verify that the conditional Lindeberg condition holds. Let S n,i be the martingale array with martingale difference sequence X n,i = Y i A n and F n,i := F i .Consider p ≥ /
2. Then a n ≥ | X n,i | ≤ A n P x − a.s. Recall that, under the given conditions, A n ր + ∞ . Hence, forevery fixed ε > P x (cid:2) | X n,i | > ε (cid:3) = 0 for sufficiently large n .Let now p < /
2. Then a − j ≤ a − n for all j ≤ n and it is immediate to seethat | X n,i | ≤ a n A n P x − a.s. We can easily verify that lim n → + ∞ a n A n = + ∞ (see10roof of Thm. 3.6 for details). Therefore, for all ε >
0, one has P x (cid:2) | X n,i | >ε (cid:3) = 0 for sufficiently large n .Now, fix ε >
0. Then n X i =1 E x [ X n,i {| X n,i | >ε } | F n,i − ] → P x − a.s., which completes the proof. Proof of the Corollary 3.5.
First, observe that since p = 1 /
2, we have that a n =1 for all n ≥
1. We also may conclude that E x [ X k | F k − ] = E x [ X k ] P x − a.s and A n = P nk =1 (1 − E x [ X k ] ) = P nk =1 (1 − (1 − α k ) (2 f ( T k x ) − ). Consequently, M n A n = S n √ n − √ n E x [ S n ] n q n P nk =1 (1 − (1 − g ( α, k )) (2 f ( T k x ) − ) . (4.8)Now, similarly to (4.6), we get1 n n X k =1 (1 − (1 − α k ) (2 f ( T k x ) − ) ∼ n n X k =1 (1 − (1 − ℓ ( α )) (2 f ( T k x ) − ) . Note that1 n n X k =1 (1 − (1 − ℓ ( α )) (2 f ( T k x ) − )= 1 − (1 − ℓ ( α )) n " n X k =1 f ( T k x )) − f ( T k x ) + 1 ! . We can apply Birkhoff Ergodic Theorem in the last equation to conclude that1 − (1 − ℓ ( α )) (4 E µ [ f |I ] − E µ [ f |I ] + 1) µ − a.s. (4.9)In particular, lim n →∞ √ n A n = 0.The desired conclusion follows applying (4.9), Corollary 3.2 and Theorem3.3 to (4.8). Proof of Theorem 3.6.
The law of iterated logarithm for the DERW follows froma application of Theorems 1 and 2 of [24]. We have already shown (4.7).Define u n = p A n and K n = u n a n A n . Let us write the inequality (4.4)in the following way | Y n | ≤ a n = K n A n u n P x − a.s.
11t is clear that K n is F n − measurable. In order to get (3.2) we need to showthat K n → n → ∞ .Recall that g ( α, <
1. Observe that a k ≥ (1 − g ( α, k − Y i =2 (cid:18) − i (cid:19) = (1 − g ( α, k − , which implies that A n ≤ n X k =1 a k ≤ (cid:0) − g ( α, (cid:1) n X k =1 ( k − ≤ (cid:0) − g ( α, (cid:1) n . (4.10)In other words, u n ≤ q n − g ( α, .Let us now take care of a n A n . Since there exists c ′ > B n /A n isbounded by 1 /c ′ , we get a n A n ≥ c ′ a n B n = 1 c ′ a n n X j =1 a j = 1 c ′ n − X j =1 a n a j = 1 c ′ n − X j =1 n − Y i = j (cid:18) p − g ( α, i + 1) i (cid:19) , which yields n − Y i = j (cid:18) p − g ( α, i + 1) i (cid:19) ≥ n − Y i = j (cid:18) − i (cid:19) = j − n − . Thence, a n A n ≥ c ′ n − X j =1 (cid:18) j − n − (cid:19) ≥ n c ′ . (4.11)Combining (4.11) and (4.10) we are able to show that K n = 2 u n a n A n ≤ vuut c ′ log log (cid:16) n (1 − g ( α, (cid:17) n → n → ∞ , and the claim (3.2) follows. Proof of Theorem 3.7.
We first observe that lim n → + ∞ B n < + ∞ by Lemma 4.3.It follows from (4.4) that P nk =1 E x [ Y k ] ≤ B n P x − a.s. . Then by Theorem12.1 of [26], for all x ∈ X , M n = n X k =1 Y k = S n − E x [ S n ] a n → M P x − a.s and in L . E x [ M n ] = 0 for all n , | E x [ M ] | = | E x [ M − M n ] | ≤ E x [ | M − M n | ]. Then | E x [ M ] | ≤ E x [ | M − M n | ] / → n → ∞ . Furthermore, since ( Y n ) n ≥ is a bounded martingale difference sequence in L and it converges almost surely,Var x [ M ] = lim n →∞ Var x [ M n ] = ∞ X k =1 E x [ Y k ] > . This leads us to conclude that M is a non-degenerated zero mean random vari-able. Proposition 4.4.
The sequence of random variables ( M n ) n ≥ defines a zeromean martingale.Proof. The zero mean property being straightforward, we only prove that E x [ M n +1 | F n ] = M n . Indeed, it follows from equation (4.1) that E x [ M n +1 | F n ] = S n − E x [ S n ] a n +1 + E x [ X n +1 | F n ] − E x [ X n +1 ] a n +1 = 1 a n +1 (cid:18) S n (cid:18) α n (2 p − n (cid:19) − (cid:18) α n (2 p − n (cid:19) E x [ S n ] (cid:19) = S n − E x [ S n ] a n P x − a.s. which finishes the proof. Lemma 4.5.
Let the DERW be defined with p < and ℓ inf ( α ) > . Then lim inf n → + ∞ Var x [ X n ] > . (4.12) Futhermore, if the DERW jointly satisfies (T) , (D ) , and (D ) , then (4.12) stillholds.Proof. Note that (4.12) is equivalent to lim sup n →∞ | E x [ X n ] | <
1. Since S n − n − ∈ [ − , n → + ∞ (cid:0) α n (1 − p ) + (1 − α n ) f ( T n ( x )) (cid:1) > n → + ∞ (cid:0) α n p + (1 − α n ) f ( T n ( x )) (cid:1) < − p ) ℓ inf ( α ) + (cid:0) − ℓ sup ( α ) (cid:1) lim inf n → + ∞ f (cid:0) T n ( x ) (cid:1) > p · ℓ sup ( α ) + (cid:0) − ℓ inf ( α ) (cid:1) lim sup n → + ∞ f (cid:0) T n ( x ) (cid:1) < p < ℓ inf ( α ) >
0. Then (4.15) is immediately satisfied. Since f ( T n ( x )) ≤
1, we verify (4.14) by noting thatlim sup n → + ∞ ( α n p + (1 − α n ) f ( T n ( x ))) ≤ − (1 − p ) ℓ inf ( α n ) < . Now, if p = 1 or ℓ inf ( α ) = 0, then (4.15) and (4.16) are straightforwardlysatisfied since conditions (T), (D ), and (D ) hold.We finish this section by proving Corollary 3.4. Proof of Corollary 3.4.
The proof follows in the same lines as those of the proofof Theorem 3.3 and it is an immediate consequence of Lemma 4.5 and Theorem3.1.
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