Q1. Let {Xt |t ∈ [0, 1]} be a stochastic process such that EX2 t <...

Q1. Let {Xt |t ∈ [0, 1]} be a stochastic process such that EX2 t < ∞ for all t ∈ [0, 1] which is strictly stationary. Show that it is stationary

. Q2. Let {Xt |t ∈ I} be strictly stationary. Prove or disprove that process is with stationary increments.

Q3. Let {Xt |t ∈ I} be with stationary increments. Prove or disprove that the process is stationary.

Q4. Prove or disprove that the stochastic process {Xn|n ≥ 0}, where {Xn} is i.i.d., is with independent and stationary increments.

Q5. Prove or disprove that SSRW is stationary.

Expert Solution

orchestra answered 1 year ago

show that if {X (t), t≥ 0} is a stochastic process with independent processes, then the...

show that if {X (t), t≥ 0} is a stochastic process with independent processes, then the process defined by Y (t) = X (t) - X (0), for t≥ 0, has independent increments and Y(0)=0

. Define the stochastic process Xt=2A+3Bt where PA=2=PA=-2=PB=2=PB=-2=0.5. Find PXt≥0 | t

Consider the first-order autoregressive model Xt = ρXt-1 + "єt,= 1,..,T with X0 = 0 and...

Consider the first-order autoregressive model Xt = ρXt-1 + "єt,= 1,..,T with X0 = 0 and {єt} ~ i.i.d.(0, σ 2). Explain what is the Dickey-Fuller test in the context of this model. In particular, your answer should include the null and alternative hypotheses, the definitions of both the coe¢cient test and the t test version of the Dickey-Fuller test and how the critical values are obtained. Consistency is a desirable property of statistical test. Explain the definition of a...

1. Consider the process {Xt} in which Xt = Zt + 0.5Zt-1 - 2Zt-2. Investigate the...

1. Consider the process {Xt} in which Xt = Zt + 0.5Zt-1 - 2Zt-2. Investigate the stationarity of the process under the following conditions. Calculate the ACF for the stationary models. (a) Zt ~ WN(0,(sigma)2) ; (sigma)2 < infinity (b) {Zt } is a sequence of i.i.d random variables with the following distribution: fzt(z) = 2/z3 ; z > 1

Show that the AR(2) process Xt=X(t-1)+cX(t-2)+Zt is stationary provided by-1<c<0. Find the autocorrelation function when c=-3/16....

Show that the AR(2) process Xt=X(t-1)+cX(t-2)+Zt is stationary provided by-1<c<0. Find the autocorrelation function when c=-3/16. Show that the AR(3) process Xt=X(t-1)+cX(t-2)-cX(t-2)+Zt is non-stationary for all values of c.

2. Consider the stochastic process {Xn|n ≥ 0}given by X0 = 1, Xn+1 = I{Xn =...

2. Consider the stochastic process {Xn|n ≥ 0}given by X0 = 1, Xn+1 = I{Xn = 1}Un+1 + I{Xn 6= 1}Vn+1, n ≥ 0, where {(Un, Vn)|n ≥ 1} is an i.i.d. sequence of random variables such that Un is independent of Vn for each n ≥ 1 and U1−1 is Bernoulli(p) and V1−1 is Bernoulli(q) random variables. Show that {Xn|n ≥ 1} is a Markov chain and find its transition matrix. Also find P{Xn = 2}.

Let X = [1, 0, 2, 0]tand Y = [1, −1, 0, 2]t. (a) Find a...

Let X = [1, 0, 2, 0]tand Y = [1, −1, 0, 2]t. (a) Find a system of two equations in four unknowns whose solution set is spanned by X and Y. (b) Find a system of three equations in four unknowns whose solution set is spanned by X and Y. (c) Find a system of four equations in four unknowns that has the set of vectors of the form Z + aX + bY as its general solution where...

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1,...

Q1. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3} and transition probability matrix (pij ). Let τi = min{n ≥ 1 : Xn = i}, i = 0, 1, 2, 3. Define Bij = {Xτj = i}. Is Bij ∈ σ(X0, · · · , Xτj ) ? Q2. Let {Xn|n ≥ 0} is a Markov chain with state space S = {0, 1, 2, 3}, X0 = 0, and transition...

Define the stochastic process where . Find

Give an example of a stochastic process that is: (a) Both a Markov process and a...

Give an example of a stochastic process that is: (a) Both a Markov process and a martingale. (b) A Markov process but not a martingale. (c) A martingale but not a Markov process. (d) Neither a Markov process nor a martingale.

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