Can someone explain what exactly the 'possessive perfect' is? The book I read gave the following example:
Turiu atsinešęs maisto. have:PRS.1SG bring:PTCP.PST.ACT.NOM.SG.M food:GEN ‘I’ve brought some food with me.’
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It is basically pretty similar thing to English present perfect, i.e. the auxiliary verb have + some sort of past or passive participle of a content verb to express some sort of past.
But in many non-Western European languages, this has been just an emerging feature with stronger influences here and there (I believe this is something present in these languages for a few centuries but it may be reinforced by further contaminations from the many Western languages that have this construction). Normally, Lithuanian has just a synthetic past and Slavic languages form the past with auxiliary verb be + past participle. Thus, this relatively new construction is called "possessive perfect" since it uses the verb have, which has not served as an auxiliary until now. Funny thing is that this construction also appears in Russian which is known for its lack of the possessive verb, so accordingly the possessive past uses their periphrastic possession у меня (есть) + passive participle.
In my experience (mainly based on Czech), the function of this is restricted to accomplishedness (which is something slightly different than perfectivity in Slavic languages) and sometimes announcing of new recent past event. In Czech, the verb have is also combinable with other adjectives indicating accomplishedness, e.g. hotový (done, finished) even though they are not verbal participles.
EDIT: Cursory search on the internet shows a nice study of this construction developping in Baltic and North Russian area: Seržant, I. The so-called possessive perfect in North Russian and the Circum-Baltic area. A diachronic and areal account