in 4. Suppose that {Xk, k > 1} is a sequence of i.i.d. random variables with...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n > 0 let Sn denote the partial sumi Let Fn denote the information contained in X1, ,Xn. (1) Verify that Sn nu is a martingale. (2) Assume that μ 0, verify that Sn-nơ2 is a martingale.
3. Suppose X1,X2, are independent identically distributed random variables with mean μ and variance σ2. Let So = 0 and for n...
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
2. Let Xn,n0,1,2,... denote a biased random walk given by Xo 0 and Xn+1 Xn + YTHI, where (X } are 1.1.d. random variables with N(-1,1) distribution. Show that Mn X22n Xn (n -1) is a martingale.
Let Xn is a simple random walk (p = 1/2) on {0, 1, · · · , 100} with absorbing bound- aries. Suppose X0 = 50. Let T = min{j : Xj = 0 or 100}. Let Fn denote the information contained in X1,··· ,Xn. (1) Verify that Xn is a martingale. (2) Find P (XT = 100). (3) Let Mn = Xn2 − n. Verify that Mn is also a martingale. (4) It is known that Mn and T...
Suppose that the random variables X,..Xn are i.i.d. random variables, each uniform on the interval [0, 1]. Let Y1 = min(X1, ,X, and Yn = mar(X1,-..,X,H a. Show that Fri (y) = P(Ks y)-1-(1-Fri (y))". b. Show that and Fh(y) = P(, y) = (1-Fy(y))". c. Using the results from (a) and (b) and the fact that Fy (y)-y by property of uniform distribution on [0, 11, find EMI and EIYn]
Let us consider the biased random walk S,-X1 +X2+···+X,, with S: is a sequence of independent randon variables with P(X,--1)-g,P(X.-1) p+q=1 and Mt Show that M.-Sn-b- calculate Elr), where τ = inf{n : S" = a or-6) with a, b > 0. 0. where Xi, X2 p. where g)n is a martingale. Use this martingale to
2. Let Xn, n > 1, be a sequence of independent r.v., and Øn (t) = E (eitX»), ER be their characteristic functions. Let Yn = {k=0 Xk, n > 0, X0 = 0, and 8. () = {1*: (),ER. k = 1 a) Let t be so that I1=1 løk (t)) > 0. Show that _exp{itYn} ?, n > 0, On (t) is a martingale with respect to Fn = (Xo, ...,Xn), n > 0, and sup, E (M,|2)...
Exercise 4 Let the Xc's be i.i.d. real random variables, uniform on [0,1]. What is the limit of (X3 +..+ X3)/(X1 .+Xn) as n -> co? In which sense?
Exercise 8.41. The random variables X1,..., Xn are i.i.d. We also know that ElXl] = 0. EĮKY = a and Elx?| = b. Let Xn-Xi+n+Xn. Find the third moment of Xn
Exercise 8.41. The random variables X1,..., Xn are i.i.d. We also know that ElXl] = 0. EĮKY = a and Elx?| = b. Let Xn-Xi+n+Xn. Find the third moment of Xn