It isn't exactly a surprising correlation: gentrification and decreases in crime. This research finds that falling crime often precedes gentrification, not the other way around.

Throughout last year's election, too often the only references to planning dealt with (often erroneous) statements about an urban crime crisis. Sad! Presidential rhetoric aside, it's a fact that, nationally, "violent crime peaked in 1991. It fell precipitously for the next decade, then more slowly through the 2000s [...] While homicides have increased recently in some cities, rates remain far below what they were 25 years ago, including in Chicago."
Emily Badger covers research suggesting an unsurprising link between gentrification and falling crime. "The new research looked at confidential geocoded data from the 1990 and 2000 censuses, and more recent American Community Surveys, to identify the neighborhoods where more than four million households moved."
Influxes of higher earners occurred after crime dropped, not before. "Because this research looked at moves that occurred after crime was already falling, the authors believe the movers were reacting to changes in crime and not simply causing it themselves."
One aspect of this could be that neighborhoods were already depopulated, limiting displacement. "Many of the urban neighborhoods studied had lost population, so they had room to grow again without pushing existing residents out." Of course, higher rents citywide still have consequences for equity.
FULL STORY: How to Predict Gentrification: Look for Falling Crime

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Safe Parking Programs Help People Access Housing
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Economic & Planning Systems, Inc.
UCLA Lewis Center for Regional Policy Studies
Mpact (formerly Rail~Volution)
Chaddick Institute at DePaul University
City of Piedmont, CA
Great Falls Development Authority, Inc.
HUDs Office of Policy Development and Research