Consider the random variable $X_n$ defined by $X_n = 2^n$ with probability $\frac{1}{n}$ and $0$ else, for each $n &gt;0 $. It's easy to show that $X_n$ converges to $0$ in probability. How do we show that $S_n / n $, the sample mean, fail to converge to $0$? 


Question: Can a wizard choose where to place their soul piece when creating a horcrux?
Answer: Yes, a witch or wizard creating a Horcrux can choose the object into which the Horcrux is concealed. Speaking specifically of Voldemort's Horcruxes, Harry and Dumbledore discuss vessels for Horcruxes:


  "And they could be anything?" said Harry. "They could be old tin cans or, I dunno, empty potions bottles . . . ."
  
  "You are thinking of Portkeys, Harry, which must be ordinary objects, easy to overlook. But would Lord Voldemort use tin cans or old potions bottles to guard his own precious soul?"
  
  Half-Blood Prince - Page 504 - US Hardcover


What is a little murky is whether or not an ordinary object can be used to create a Horcrux, or if an item of significance (Slytherin's locket; Ravenclaw's diadem; Gryffindor's sword; Hufflepuff's cup; etc) is required. The above passage is referring specifically to Voldemort. 

Simply committing a murder does not create a Horcrux. It is one step required. Reference Tom Riddle's conversation with Horace Slughorn while Riddle was still a student at Hogwarts: 


  "But how do you do it?"
  
  "By an act of evil -- the supreme act of evil. By committing murder. Killing rips the soul apart. The wizard intent upon creating a Horcrux would use the damage to his advantage: He would encase the torn portion--"
  
  "Encase? But how--?"
  
  "There is a spell, do not ask me, I don't know!" said Slughorn . . .
  
  Half-Blood Prince - Page 498 - US Hardcover


J.K. Rowling has not revealed the exact steps necessary in making a Horcrux; according to the HP Wiki, J.K. Rowling plans on releasing the exact spell and process in her Harry Potter Encyclopedia; the process is apparently very gruesome. 

An accidental Horcrux can be created, such as in the case of Harry.


  "[...]on the night Lord Voldemort tried to kill him, when Lily cast her own life between them as a shield, the Killing Curse rebounded upon Lord Voldemort, and a fragment of Voldemort's soul was blasted apart from the whole, and latched itself onto the only living soul left in that collapsing building. Part of Lord Voldemort lives inside Harry[...]"
  
  Deathly Hallows - Page 686 - US Hardcover
  
  "You were the seventh Horcrux, Harry, the Horcrux he never meant to make. He had rendered his soul so unstable that it broke apart when he committed those acts of unspeakable evil, the murder of your parents, the attempted killing of a child. But what escaped from that room was even less than he knew. He left more than his body behind. He left part of himself latched to you, the would-be victim who had survived."
  
  Albus Dumbledore - Deathly Hallows - Page 709 - US Hardcover


So, technically, when Voldemort cast the Killing Curse, it rebounded against him, severing yet another part of his soul. That part of Voldemort's soul sought out a living host, the only available vessel being Harry; it attached itself to Harry. How Voldemort managed to create this Horcrux without the apparently necessary accompanying spell being incanted, I am unsure.

ETA: Regarding Horcruxes, JKR has this little tidbit up on Pottermore:


  Professor Quirrell served as a temporary Horcrux when Lord Voldemort's soul occupied his body.