For other learners who might come across the same problem, can you tell us how it is explained in the book and what your interpretation of ゆきがふってきた is?
Toritoribe, thank you, I appreciate the help. 3 things though.
1. The strict definition from the book left me a little confused so I did some further research online and cracked open my Dictionary of Basic Japanese Grammar.
2 To make sure I've nailed down the usage correctly, I am going to be as broad as possible, so if you see mistakes please point them out. Also because I'm going to be as broad as possible this might be better in its own thread.
3 I'm the sort of person that has trouble understanding social rules in person, much less on the internet, so if I'm rude by mistake please just tell me upfront.
Here we go.
So the definition given is "The te form of the verb followed by the verb kuru or kimasu, expresses an action that moves toward the location of the speaker, or proceeds up to a current point in time, and by convention is written in hiragana.
This left me a little confused as I couldn't quite see how to it worked ゆきがふってきた. Google on the hand gave me list of resources, this one is probably the most user friendly
Requested Lesson 「〜ていく+〜てくる」(~teiku + ~tekuru) | Maggie Sensei
So to summarize there seems to be three general uses of tekuru.
1. Doing something and coming or doing something while coming - e.g. or 私は学校にはしってきた - I ran to school
2. Expressing the idea that some action takes place while moving towards the speaker - e.g.
いぬを走ってきた - the dog ran towards the speaker
3. Expressing the idea of change over some period of time, with differences depending on the verb tense as follows - tekita past tense expresses the idea that something has changed in the past tense and that the change might continue, while tekuru expresses from now something has changed is likely to keep changing though user could see it potentially stop changing - I'm a little iffy on this one, as
grammar - Difference between -ていく and -てくる - Japanese Language Stack Exchange offers information that seems to conflict with whats on the maggie-sensei site gives, and
http://www.imabi.net/l90teikutekuru.htm is just confusing.
To return to the original question, I think that the most literal translation of ゆきがふってきた might be "The snow at some point in the past started falling and has now stopped, though that might change?
If anyone can read Japanese this probably explains it more clearly than I can
初級を教える人のための日本語文法ハンドブック - 庵功雄 - Google Books