To begin with, we should cite the definition of story of initiation that Mordecai provides:
An initiation story may be said to show its young protagonist experiencing a significant change of knowledge about the world or himself, or a change of character, or of both, and this change must point or lead him towards an adult world. (...) it should give some evidence that the change is at least likely to have permanent effects. (Mordecai,1960:223)
To continue, the stages by which Daisy accomplishes her decisive initiation are depicted by Joseph Campbell in his book The Hero with a Thousand Faces. On our opinion when the story begins, Daisy has already crossed the threshold, in others words, she has accepted the call of the adventure in Europe. This is the stage of departure. Being an American girl, what she sees as an adventure is the search for sociability and for being accepted as she had been in America. Daisy is the archetypical innocent uncontaminated heroine:
There isn't any society; or, if there is, I don't know where it keeps itself. Do you? I suppose there is some society somewhere, but I haven't seen anything of it. I'm very fond of society, and I have always had a great deal of it (...) I used to go to New York every winter. In New York I had lots of society. Last winter I had seventeen dinners given me; and three of them were by gentlemen (...) I have (...) more gentleman friends; and more young lady friends too," (...) She paused again for an instant; she was looking at Winterbourne with all her prettiness in her lively eyes and in her light, slightly monotonous smile. "I have always had," she said, "a great deal of gentlemen's society. (James, 1879: 11)
Moving forward along the story we readers witness the stage of initiation proper. Daisy undergoes several experiences, that is to say, the trials or tests in Campbell’s terms. There are several crucial episodes outlining these tests. Many of them are mainly decisions taken by Daisy, which are seen as inexcusable mistakes by the American European society, though seen as natural behaviour by Daisy, quite the opposite to what she herself qualifies as ‘stiff’.
As a way of example, Daisy has to cope with Mrs. Costello’s disdainful rejection, who refuses to become personally acquainted with her. Most importantly, Daisy herself deduces this fact through Winterbourne’s hesitant words. This is not a minor detail, because it is by her capacity of deduction that Daisy’s increasing emotional maturity is made evident:
I shall be ever so glad to know your aunt." Winterbourne was embarrassed. (...) he said; "but I am afraid those headaches will interfere." (...) "But I suppose she doesn't have a headache every day," she said sympathetically. (...). "She tells me she does," he answered at last, not knowing what to say. Miss Daisy Miller stopped and stood looking at him. (...) "She doesn't want to know me!" she said suddenly. "Why don't you say so? You needn't be afraid. I'm not afraid!" (...)You needn't be afraid," she repeated. "Why should she want to know me?" (...) "Gracious! she IS exclusive!" she said. (James, 1879:18)
At Mrs. Walker’s, one of the society matrons, Daisy makes a succession of social mistakes, such as asking Mrs. Walker, who was having a party, to bring her friend Mr. Giovanelli with her....