Need a Probability Theory book that also focusses on Analysis I am in search for a Probability Theory book which also contains elements and proofs from Analysis. A non-Measure Theoretic approach is most desirable. I have gone through great books like Ross but I would like some book which also discusses some Analysis, which is generally not found in more problem-oriented books like Ross, Hoel Port Stone, etc.
As an example, books having problems of the form: 

If $X$ is a non-negative continuous random variable with finite fourth order moment, show that $\lim_{x\to\infty}x^4(1-F(x))=0$ where $F$ is the c.d.f. of $X$.

I would like to solve questions of this form, and the books I have consulted do not dwell much on the Analysis-related questions.
Any such book's example will be appreciated.

If you want a probability book that uses real analysis but not measure theory, then you want an older book back before measure theory became so central to the subject.  I recommend Feller's two volume work, An Introduction to Probability and Its Applications.
