Me asking is kind of sign.

Thanks, CalifJim. but what do you mean by the red part?

CalifJim wrote:

I just realized that "asking" could be regarded as modifying "Me" and "Me" here is the subject. No.  I don't think that analysis does the job.  me by itself can't be a subject.  At best, the implied clause me asking would be the subject.   But the use of that sort of subject puts the statement in a lower register. The following all have the same basic meaning. That I should ask is a kind of sign. For me to ask is a kind of sign. Me asking is a kind of sign. My asking is a kind of sign. (Subjects underlined.) CJ

The point is to understanding "Me asking ..." and "Him leaving ...".

By my understanding of what CalifJim said, "me" itself can not be a subject. But the constructure "me asking" as a whole can be a subject, where "asking" modifies "me". So if you regard "me asking" as a short phrase, it should be easier to understand why it can be a subject, and it is as natural as the following sentence:

The girl asking question is clever.

Therefore, I think your second sentence "Him leaving home ..." should be right.

But I think both of them are not as normal as "My asking ..." and "His leaving ..."

Am I right, great teachers?

Nemessos wrote:

The following all have the same basic meaning. Me asking is a kind of sign. My asking is a kind of sign. Calif Jim, does that mean both the above are grammatically correct, including 'Me asking is a kind of sign.' ?? What about : 'His leaving home plunged his widowed mother into a deep depression' and 'Him leaving home plunged his widowed mother into a deep depression' Thanks!

By my understanding of what CalifJim said, "me" itself can not be a subject. Yes.  That's right. But the constructure construction "me asking" as a whole can be a subject Yes, where "asking" modifies "me" No, it's not a matterof modifying. "me asking" as a whole can be a subject where me asking is a clause with the internal structure having me as the subject of the clause and asking the verb of the clause.  Thus, that I (should) ask becomes me asking.
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An aside:

The use of objective case for the subject of a subordinate clause is common in English.  I want him to write a letter.  What do I want?  that he (will) write a letter.  The subject of the subordinate clause (he) becomes him; the verb write becomes the infinitive to write:  I want { him to write a letter }.

In a case like I resent him opening the letter, the transformation is different, but the principle is the same.  What do I resent?  that he opened the letter.  The subject of the subordinate clause (he) becomes him; the verb opened becomes the gerund opening:  I resent { him opening the letter }.  Alternately, the subject becomes his:  I resent his opening the letter.  In this version the clause-like nature of what is resented is less obvious.
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So if you regard "me asking" as a short phrase, it should be easier to understand why it can be a subject, and it is as natural as the following sentence:

The girl asking a/the question is clever.  This example does show modification.  Most likely, in the typical interpretation,  asking the question modifies the girl.

But these examples show the clause ideas explained above.

I resent [him / his] asking the question.
Do you object to [me / my] asking the question?