UMD Probability Qualifying Exams/Aug2009Probability

Problem 1

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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

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(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

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

 


Solution

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Problem 3

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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

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Show   is exponentially distributed

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Let   be the first time that a Poisson process   jumps.

 

  is a Poisson Process with parameter  

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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)

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Problem 4

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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

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Problem 5

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

Solution

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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

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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

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