1. Example 7 from class, continued:

X_0 = 0 with probability 1.

X_1 = -1 with probability 1/2, and = +1 with probability 1/2.

For k>=2, X_k | X_(k-1) = [X_(k-1) + sgn(X_(k-1)] with probability (k-1)/k, and = 0 with probability 1/k.

(a) Draw a few sample paths. (done in lecture)

(b) Find the distribution of X_k. (done in lecture)

(c) Find the distribution of X_infinity, where X_infinity = the limit as k -> infinity in distribution of X_k. (done in lecture)

(d) Find E [ X_k | X_(k-1) ].

(e) Find the mean and standard deviation of X_k.

(f) Find the limit as k -> infinity of the mean and standard deviation of X_k.

(g) Find the mean and standard deviation of X_infinity. Why aren't these the same as your answers in (f)?

2. Suppose that {X_i, i>=0} is a sequence of i.i.d. Normal random variables with mean 0 and variance 1. Define Y_i = X_i + X_(i-1), i>= 1. Define Y_infinity = the limit as k -> infinity in distribution of Y_k. Does Y_infinity exist? What is its distribution?

3. Suppose that {X_i, i>=1} is a sequence of random variables.

Let sigma denote a constant. Define alpha = sqrt(2 pi) sigma. Define beta = -1/(2 sigma^2). Define a_n = (n-1)sigma / n.

X_n has a density function:

f(x) = [(1-1/n) / alpha] e^[beta (x-a_n)^2] + [sigma/n] e^[-sigma x] 1(x>=0)

where 1(x>=0) = 0 if x<0 and =1 if x>=0.

Define X_infinity = the limit as k -> infinity in distribution of X_k. Does X_infinity exist? What is its distribution?