I remember your talking about how your father died.
I remember you talking about how your father died.
Are both these sentences grammatically correct? What's the difference in their meaning?
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Both sentences are correct.
In the first example, "I remember your talking about how your father died." "talking" is a gerund or verbal noun. (The talking about the subject becomes the import portion of the sentence.)
In the second example "I remember you talking about how your father died." "talking" is a present participle. (What is remembered is the important part of the sentence.)
The word, 'your' is used for possessions. You can see the actual definition here;
"belonging to or associated with the person or people that the speaker is addressing"
1 - I remember|your talking| about how your father died.
Your talking is a pronoun + a noun gerund and the phrase functions like a quasi-direct object. You can extract the noun and use it elsewhere and it becomes a subject.
|Your talking| about how your father died| reminded me of sad times.
In English, nouns gerunds are used all the time. |My playing| [of a musical instrument] was annoying to my family. They do not remember |my playing| fondly.
2) I remember |you talking| about how your father died. |You talking| about that subject upset me.
This is just more colloquial. And traditionally talking would be considered a participle though I would say it is a direct object. However, you can also use it as a noun. [You or Your or no pronoun]Talking too much can be a problem.
The sentences are semantically equivalent. There is no difference in meaning. They both convey the same information and the noun gerund or participle (depending on how you look at it) can be used as a subject.
Verbs that take ING in this group are forget, remember, stop and try because they can be followed by ING or TO, but do change meaning.
Please also bear in mind that pronoun + gerund can be used with all the possessive pronouns (formal) and object pronouns (colloquially): your talking, his leaving, her grieving, my arriving, their disagreeing, our shouting are all further examples of a pronoun+gerund and can all be used as subjects. Also please note that the her can only be her. The other pronouns remain the same.
I remember your/you talking about x.
I remember his/him talking about x.
I remember her talking about x.[no pronoun change: possessive is the same as the object pronoun]
I remember their/them talking about x.
I remember our/us talking about x.
I remember my,me talking about x.
To show that the use of you is more colloquial, look at the first person pronoun in the sentence: there is no doubt that I remember me talking about x is more colloquial than I remember my talking about x. And sorry but the only reference I have is in my head. Me talking about that is important is used and heard but if you were getting your Phd in English, you would want to say: My talking about that is important.
Your is more precise and should be used in formal writing. It is more precise because it is the talking that is being remembered, not you.
You is more colloquial and might be more appropriate for everyday dialogue. Still, I would prefer to say your instead of you. There's nothing wrong with being precise in colloquial English. Say what you mean, not what makes you "fit in."
Briefly speaking... ("you" - objective, "your" - possessive)