Some processes in the universe are cyclical.  To put it differently, they are self-reinforcing. That is, once this process happens, it becomes more likely that it will happen again. Examples abound, including the chain reaction of nuclear fusion that goes on in stars - the energy released by one instance of hydrogen fusing into helium heats up neighboring hydrogen nuclei to the point that they cross the threshold of potential and undergo fusion themselves - which releases even more energy, causing other nuclei to fuse, and so on. 

There are all kinds of cyclical processes in nature, not only at the nuclear level, but also at the chemical level - self-catalyzing chemical reactions, i.e., chemical reactions whose products catalyze the same chemical reaction to happen again. They are rarer, but still possible. Actually fire (rapid oxidation) is a simplistic example. 

Cyclical chemical processes need not immediately cause themselves to repeat, either - it can be a complex chain, or web of causation (event a causes b, c, and d, which each are contributing factors to events e, f, g, h, i, j, and k, which in turn lead to several other events, and so on - so long as, eventually, one or more of these processes in turn makes a once again more likely). For that matter, there are also processes that do not directly cause themselves to repeat, but which lead to something that is, in some way, similar to themselves. Think of patterns of crystallization - as crystals form, and grow, there will often be quite beautiful symmetries and patterns, as a shape replicates itself, or variations of itself, at larger or small scales, to varying degrees of precise similitude, in complex growths of fractal self-similarity. As these molecular patterns of chemicals spread and replicate, those that replicate more exactly, more numerously, and more quickly will tend to proliferate, while those that spread more slowly or are not as precisely the same pattern will become outnumbered. Nonetheless, even among the most precise self-replicators, variations will appear, and some of them will result in more efficient self-replication. 

These processes might begin among a wide variety of chemicals, but then as a pattern varies, it may pick up little bits of other types of chemicals, whatever may be available nearby, and some of them may prove to make the pattern more efficient at self-replication. Eventually the original chemical component may become unnecessary, and the pattern may switch entirely to the new chemical substrate. Who knows what chemicals started this process, but they probably became more efficient replicators, when they switched substrates to proteins, made out of amino acids, which can form quite naturally in "primordial soup" with electric strikes, as has been proven in the lab. It is quite possible to be a very efficient replicator made entirely out of protein, with no RNA, no DNA, no nucleic acids at all - witness, for example, the prions that cause mad cow disease. But eventually replicators probably evolved to become even more efficient and precise as self-replicators, as they again switched substrates, first to RNA, and then to DNA.

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