Need a Probability Theory book that also focusses on Analysis

Question

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.

Answer 1

Ash's book on Probability Theory is fantastic and covers what you want, although at a bit higher level, with some measure theory:

http://www.amazon.com/Probability-Measure-Theory-Second-Robert/dp/0120652021/ref=asap_bc?ie=UTF8

Answer 2

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.