Pronouns and Case - Me Doing, Or My Doing?

Which of these sentences is correct?
I hope you don’t mind me asking? I hope you don’t mind my asking?
The same problem occurs with nouns:
I was amazed at Paul saying that. I was amazed at Paul’s saying that.
Purists point out that asking and saying in these sentences are gerunds – that is, they act like nouns – and so should take a possessive, just as in sentences like I hope you don’t mind my question and I was amazed at Paul’s insolence.
Nevertheless, idiom often favours me asking and Paul saying that, especially in informal contexts.
Certainly the possessive form –my,Paul’s – is more formal, and is clearly preferable at the beginning of a sentence:
His saying that was very strange. ? Him saying that was very strange.
And it must be used if the -ing word is so noun-like that it is followed by of plus a noun or pronoun:
We enjoyed his parodying of the minister.
On the other hand, possessives are awkward or impossible in some contexts:
You can depend on something/?? Something’s turning up. She did it without either of us/X our knowing. She did it without her brother or sister/?? Sister’s knowing. I caught him/X his rifling through the files lying on my desk.
If you are worried in particular cases, avoid the problem altogether by rewording – I hope you don’t mind that I asked; You can depend on it that something will turn up, and so on.
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