I recommend doing a full assessment of the analyst firms, such as Gartner, Forrester, IDC, G2 and any others that could be important. Look at the types of quadrants and waves they publish. Once you've done a thorough review of the analyst landscape for your category, you can then look at prioritising who to influence the most at what stage.
You need to find out who the lead analysts are, get familiar with what they're talking about, arrange a call and find out what's important to them and what they're looking for. They could let you know where the gaps are and where you need to get to, guiding you through the process. This is a relationship building exercise that will inform you at which levels you need to operate.
You should also find out if any of our customers have seats with analyst firms, this way you can work with your customers to start raising awareness with the analysts from the customer side, which is very powerful.
If there is no particular magic quadrant or wave then you need to influence analysts from scratch, which is a big job. Forrester would be the easiest as they seem more "open". G2 are more accessible for building rapport and influence.
To set expectations, building relationships with analyst firms takes a long time and doesn't happen overnight. You need full 100% executive support and for everyone to understand the value in the long term. You should come up with a 5 year strategy (YES 5 YEARS!!) on how this strategy is going to evolve.
Customers are very important in this as they are always part of any review exercise the analyst firms carry out in order to add new technology providers to their reports, magic quadrants and waves. You should look at various ways to get their attention, such as through blog mentions, social media mentions and make sure that you’re constantly on their radar.