UMD Probability Qualifying Exams/Aug2009Probability

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Problem 1[edit]

Let be i.i.d. random variables with moment generating function which is finite for all . Let .

(a) Prove that

where

and

(b) Prove that

.

(c) Assume . Use the result of (b) to establish that almost surely.


Solution[edit]

(a)

Thus far, we have not imposed any conditions on . So the above inequality will hold for all , hence for the supremum as well, which gives us the desired result.


(b) where the last equality follows from the fact that the are independent and identically distributed.

(c)

Problem 2[edit]

Let be a probability space; let be a random variable with finite second moment and let be sub -fields. Prove that


Solution[edit]

Problem 3[edit]

Let be independent homogeneous Poisson processes with rates , respectively. Let be the time of the first jump for the process and let be the random index of the component process that made the first jump. Find the joint distribution of . In particular, establish that are independent and that is exponentially distributed.



Solution[edit]

Show is exponentially distributed[edit]

Let be the first time that a Poisson process jumps.

is a Poisson Process with parameter [edit]

Proof: There are three conditions to check:

(i) almost surely

(ii) For is independent of ? This is true since both are Poisson Processes and are both independent of each other.

(iii) For is distributed Poisson with parameter ? This is true since the sum of independent Poisson processes are also poison. (see second bullet)

Joint distribution of (J,Z)[edit]

Problem 4[edit]

Let be a martingale sequence and for each let be an -measurable random variable. Define

Assuming that is integrable for each , show that is a martingale.


Solution[edit]

Problem 5[edit]

Let be an i.i.d. sequence with and . Prove that for any , the series converges almost surely.

Solution[edit]

Define . Then and . We check the three components of Kolmogorov's three-series theorem to conclude that converges almost surely.


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Problem 6[edit]

Consider the following process taking values in . Assume is an i.i.d. sequence of positive integer valued random variables and let be independent of the . Then

Solution[edit]