Chapter 2 Flashcards
Define stochatic process or random process
Collection of Random variables X(t)defined on the common probability space , indexed by elements of parameter set T for time. The set of all possible values for X is called the state space for T
X is a fucntion in 2 variables, t and omega
Explain the trageory of a random process
The realisation or sample function/ path or the random process. For a fixed value of omega the function X(t) for all t is the trajectory
Give an example of a stochastic process with a continuous state space and continuous time
Euro/ dollar exchange rate over time
Give an example of a stochastic process with a continuous state space and discrete time
Annual inflation rates
Give an example of a stochastic process with a discrete state space and discrete time
Number of earthquakes over a year
If two variables X and Y are independent what can be said about their conditional expectation
The Expectation of X|Y is just expectation of X is X and Y are independent
When do RVs have the same probability distribution
IFF for any bounded measurable function expectation of f(x) is the same as the expectation of F(y) then we write X==^d Y
When is a stochastic process said to be a white noise
X(t) is a white noise if all X(t)s are iid we use notation Xn
How do we know when Xn white noise is symmetric
The distribution of Xn equals the distribution of -Xn
What are increments of steps of a random walk
Xi terms for all n that sum up to get the random walk
If a random walk is not a symmetric random walk what can it be called?
Biased
What does Sn/n tend to as n tends to infinity when Sn is a random walk
It tends towards E(xi) = mew because of law of large numbers
Why is the central limit theorem a stronger result than the law of large numbers?
It is dealing with the converge of the distribution not just a central value
Describe the gamblers ruin game set up
Each player has a starting capital. If A RV X is 1 - player s wins 1 unit if X is -1 - player f wins one unit. Game continues until one party loses all their capital.
Sn = x1+x2+… models amount won/lost by a player.
Define p
Probability Xt = 1
Define q
Probability Xt = -1
Define a stationary process
A process X is stationary if for any t the joint distribution of the random vectors: (X(t1),X(t2)….X(tk)) and (X(t1+t),X(t2+t)….X(tk+t)) are identical. This implies Xr and Xs have the same distirbution and so therefore the expectation and variance of Xt is constant, not dependent on t.
Is a random walk stationary?
No except in the degenerate case when Xi=0
Is white noise stationary process?
Yes as all XN’s are iid
What is the difference between the weakly stationary and strong stationarity definitions
It doesn’t say anything about the distributions
Does White noise have independent increments?
No except where the Xns are not random in degenerate case - Increment rvs are dependent
Does random walk process have independent increments?
Yes S1-S0=X1, S2-S1=X2 … etc are independent
Explain the Markov property
A sequence of RVs Zn have the markov property if for any n:
P(Zn+1=z|Z0=z0,…,Zn=zn) = P(Zn+1=z|Zn=zn) . The next RV int he sequence is only dependent on the most recent value in the sequence.
Does Random walk have a markov property?
Yes
If a process has the markov property what does this mean if we want to find a probability k steps ahead of us?
We only have to condition on the most recent stage of the sequence
Is white noise a martingale?
White noise is not a martingale it is a martingale only in degenerate case where Xn is a constant
Is a random walk a martingale?
No unless it is an unbiased random walk!
Define an increasing random process
Process that for any W X(t) is an increasing function: ie. all trajectories are increasing
Define a continuous random process
Process that for any w, X(t) is a continuous function: ie. all trajectories are continuous
