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I have provided a solution to the attached problem. I do not understand or like the solution - I was hoping you could provide an alternate solution or expand upon the solution I have provided in more detail.
Exercise (moment-generating function).
? Let X be a nonnegative random variable, and assume that
is finite for every (real numbers). Assume further that E [XetX] < ∞ for every . The purpose of this exercise is to show that and, in particular,
? Recall the definition of the derivative:
? The limit above is taken over a continuous variable s, but we can choose a sequence of numbers converging to t and compute:
where now we are taking a limit of the expectations of the sequence of random variables:
If this limit turns out to be the same, regardless of how we choose the sequence that converges to t, then this limit is also the same as
? The Mean Value Theorem from calculus states that if f(t) is a differentiable function, then for any two numbers s and t, there is a number θ between s and t such that:
If we fix and define , then this becomes
where θ(ω) is a number depending on ω (i.e., a random variable lying
between t and s).
1) Use the Dominated Convergence Theorem (defined in Supplement below) and equation (*) to show that:
This establishes that the desired formula .
2) Suppose that the random variable X can take both positive and negative values and and for every . Show that once again:
(Hint: Consider a random variable X that can take both + and - values. For such a random variable, we define the positive and negative parts of X by:
Use this notation to write X = X+ - X-.)
1) By (*), The last inequality is by X ≥ 0 and the fact that θ is between t and sn, and hence smaller than 2t for n sufficiently large. So, by the Dominated Convergence Theorem:
2) Since , for every ,
for every . Similarly, we have for every . So, like Solution 1), we have:
for n sufficiently large.
So, by the Dominated Convergence Theorem:
Definition 2: Let f1, f2, f3, ... be a sequence of real-valued, Borel-measurable functions defined on . Let f be another real-valued, Borel-measurable function defined on . We say that f1, f2, f3, ... converges to f almost everywhere and write:
if the set for which the sequence of numbers f1(x), f2(x), f3(x), ... does not have a limit f(x) is a set with Lebesgue measure zero.
Dominated Convergence Theorem:
Let X1, X2, ... be a sequence of random variables converging almost surely to a random variable X. If there is another random variable Y such that and almost surely for every n, then:
Let f1, f2, ... be a sequence of Borel-measurable functions on converging almost everywhere to a function f. If there is another function g such that and almost everywhere for every n, then:
Definitions from Shreve's Continuous Time Models
The explanations are written in the attached pdf file.
I followed the line ...
Measure Theory and Dominated Convergence Theorem are investigated.